Quadratic roots, narrow range

Percentage Accurate: 55.3% → 92.3%

Time: 8.4s

Alternatives: 9

Speedup: 4.6×

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: 92.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0572:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) - \mathsf{fma}\left({\left(c \cdot a\right)}^{4}, \frac{5}{{b}^{6} \cdot a}, \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0572)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/
    (-
     (- c)
     (fma
      (pow (* c a) 4.0)
      (/ 5.0 (* (pow b 6.0) a))
      (fma
       (* (/ a (* b b)) c)
       c
       (* (* (pow b -4.0) (* (* (* (* a a) c) c) c)) 2.0))))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0572) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (-c - fma(pow((c * a), 4.0), (5.0 / (pow(b, 6.0) * a)), fma(((a / (b * b)) * c), c, ((pow(b, -4.0) * ((((a * a) * c) * c) * c)) * 2.0)))) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0572)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-c) - fma((Float64(c * a) ^ 4.0), Float64(5.0 / Float64((b ^ 6.0) * a)), fma(Float64(Float64(a / Float64(b * b)) * c), c, Float64(Float64((b ^ -4.0) * Float64(Float64(Float64(Float64(a * a) * c) * c) * c)) * 2.0)))) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0572], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-c) - N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * N[(5.0 / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * c + N[(N[(N[Power[b, -4.0], $MachinePrecision] * N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.057200000000000001

   1. Initial program 55.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 55.4

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

   3. Applied rewrites 55.4%

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

   if 0.057200000000000001 < b

   1. Initial program 55.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 91.3%

      \[\leadsto \frac{\left(-c\right) - \left(\mathsf{fma}\left(0.25, {\left(c \cdot a\right)}^{4} \cdot \frac{20}{{b}^{6} \cdot a}, \left(c \cdot c\right) \cdot \frac{a}{b \cdot b}\right) - \left(\left(\left(a \cdot a\right) \cdot c\right) \cdot \left(c \cdot c\right)\right) \cdot \left({b}^{-4} \cdot -2\right)\right)}{b} \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 91.3%

      \[\leadsto \frac{\left(-c\right) - \mathsf{fma}\left({\left(c \cdot a\right)}^{4}, \frac{20}{{b}^{6} \cdot a} \cdot 0.25, \mathsf{fma}\left(\frac{a}{b \cdot b} \cdot c, c, \left({b}^{-4} \cdot \left(\left(\left(\left(a \cdot a\right) \cdot c\right) \cdot c\right) \cdot c\right)\right) \cdot 2\right)\right)}{b} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. metadata-eval 91.3

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

   10. Applied rewrites 91.3%

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

Alternative 2: 90.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{4}} - \frac{{c}^{2}}{{b}^{2}}\right) - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.3)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/
    (-
     (*
      a
      (-
       (* -2.0 (/ (* a (pow c 3.0)) (pow b 4.0)))
       (/ (pow c 2.0) (pow b 2.0))))
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.3) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = ((a * ((-2.0 * ((a * pow(c, 3.0)) / pow(b, 4.0))) - (pow(c, 2.0) / pow(b, 2.0)))) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(Float64(-2.0 * Float64(Float64(a * (c ^ 3.0)) / (b ^ 4.0))) - Float64((c ^ 2.0) / (b ^ 2.0)))) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.3], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(N[(-2.0 * N[(N[(a * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 55.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 55.4

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

   3. Applied rewrites 55.4%

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

   if 0.299999999999999989 < b

   1. Initial program 55.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 91.3%

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

   7. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 88.2%

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

Alternative 3: 89.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(c \cdot \mathsf{fma}\left(-2, \frac{{a}^{2} \cdot c}{{b}^{5}}, -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.3)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (*
    c
    (-
     (*
      c
      (fma -2.0 (/ (* (pow a 2.0) c) (pow b 5.0)) (* -1.0 (/ a (pow b 3.0)))))
     (/ 1.0 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.3) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = c * ((c * fma(-2.0, ((pow(a, 2.0) * c) / pow(b, 5.0)), (-1.0 * (a / pow(b, 3.0))))) - (1.0 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(c * Float64(Float64(c * fma(-2.0, Float64(Float64((a ^ 2.0) * c) / (b ^ 5.0)), Float64(-1.0 * Float64(a / (b ^ 3.0))))) - Float64(1.0 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.3], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(c * N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 55.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 55.4

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

   3. Applied rewrites 55.4%

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

   if 0.299999999999999989 < b

   1. Initial program 55.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   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 88.1%

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

Alternative 4: 89.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.3:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{4}} - \frac{a}{{b}^{2}}\right) - 1\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.3)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/
    (*
     c
     (-
      (* c (- (* -2.0 (/ (* (pow a 2.0) c) (pow b 4.0))) (/ a (pow b 2.0))))
      1.0))
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.3) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = (c * ((c * ((-2.0 * ((pow(a, 2.0) * c) / pow(b, 4.0))) - (a / pow(b, 2.0)))) - 1.0)) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.3)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(c * Float64(Float64(c * Float64(Float64(-2.0 * Float64(Float64((a ^ 2.0) * c) / (b ^ 4.0))) - Float64(a / (b ^ 2.0)))) - 1.0)) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.3], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(c * N[(N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.299999999999999989

   1. Initial program 55.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 55.4

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

   3. Applied rewrites 55.4%

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

   if 0.299999999999999989 < b

   1. Initial program 55.3%

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

   2. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-2, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}}, \mathsf{fma}\left(-1, c, \mathsf{fma}\left(-1, \frac{a \cdot {c}^{2}}{{b}^{2}}, -0.25 \cdot \frac{\mathsf{fma}\left(4, {a}^{4} \cdot {c}^{4}, 16 \cdot \left({a}^{4} \cdot {c}^{4}\right)\right)}{a \cdot {b}^{6}}\right)\right)\right)}{b}} \]

   6. Applied rewrites 91.3%

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

   7. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   9. Applied rewrites 88.1%

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

Alternative 5: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00205:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot a\right) \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00205)
   (/ (+ (- b) (sqrt (fma b b (* (* -4.0 a) c)))) (* 2.0 a))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00205) {
		tmp = (-b + sqrt(fma(b, b, ((-4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00205)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(Float64(-4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.00205], N[(N[((-b) + N[Sqrt[N[(b * b + N[(N[(-4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00205000000000000017

   1. Initial program 55.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. sqr-neg-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. remove-double-neg N/A

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

      12. lift-neg.f64 N/A

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

      13. remove-double-neg N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

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

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

      17. lower-*.f64 N/A

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

      18. metadata-eval 55.4

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

   3. Applied rewrites 55.4%

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

   if -0.00205000000000000017 < (/.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 55.3%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   3. 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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. sub-negate N/A

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

      8. sub-negate-rev N/A

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

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

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

      10. distribute-lft-out N/A

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

      11. mul-1-neg N/A

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

      12. frac-2neg-rev N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 82.1%

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

Alternative 6: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00205:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00205)
   (* (- (sqrt (fma (* c -4.0) a (* b b))) b) (/ 0.5 a))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00205) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) * (0.5 / a);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00205)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.00205], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00205000000000000017

   1. Initial program 55.3%

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

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

      3. lower-*.f64 N/A

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

   3. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b\right) \cdot \frac{0.5}{a}} \]

   if -0.00205000000000000017 < (/.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 55.3%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   3. 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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. sub-negate N/A

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

      8. sub-negate-rev N/A

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

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

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

      10. distribute-lft-out N/A

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

      11. mul-1-neg N/A

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

      12. frac-2neg-rev N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 82.1%

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

Alternative 7: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.00205:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.00205)
   (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
   (/ (fma (* c c) (/ a (* b b)) c) (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.00205) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = fma((c * c), (a / (b * b)), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.00205)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], -0.00205], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -0.00205000000000000017

   1. Initial program 55.3%

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

   3. Applied rewrites 55.3%

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

   if -0.00205000000000000017 < (/.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 55.3%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   3. 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. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 82.1

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

   4. Applied rewrites 82.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. sub-negate N/A

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

      8. sub-negate-rev N/A

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

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

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

      10. distribute-lft-out N/A

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

      11. mul-1-neg N/A

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

      12. frac-2neg-rev N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 82.1%

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

Alternative 8: 82.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * c), $MachinePrecision] * N[(a / 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. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
3. 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. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 82.1

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

4. Applied rewrites 82.1%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. frac-2neg N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. sub-negate N/A

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

   8. sub-negate-rev N/A

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

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

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

   10. distribute-lft-out N/A

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

   11. mul-1-neg N/A

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

   12. frac-2neg-rev N/A

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

   13. lower-/.f64 N/A

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

6. Applied rewrites 82.1%

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

Alternative 9: 64.7% accurate, 4.6× 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. 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 64.7

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

4. Applied rewrites 64.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
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. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.7

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

6. Applied rewrites 64.7%

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

Reproduce

herbie shell --seed 2025156 
(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)))