Quadratic roots, narrow range

Percentage Accurate: 55.8% → 91.7%

Time: 4.5s

Alternatives: 8

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: 55.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -10:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\left(\left(c \cdot c\right) \cdot c\right) \cdot \left(-5 \cdot \left(a \cdot \frac{c}{b \cdot b}\right) - 2\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)} \cdot a - \frac{c \cdot c}{b \cdot b}\right) \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -10.0)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (-
     (*
      (-
       (*
        (/
         (* (* (* c c) c) (- (* -5.0 (* a (/ c (* b b)))) 2.0))
         (* (* b b) (* b b)))
        a)
       (/ (* c c) (* b b)))
      a)
     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)) <= -10.0) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = ((((((((c * c) * c) * ((-5.0 * (a * (c / (b * b)))) - 2.0)) / ((b * b) * (b * b))) * a) - ((c * c) / (b * b))) * a) - 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)) <= -10.0)
		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(Float64(Float64(Float64(Float64(Float64(c * c) * c) * Float64(Float64(-5.0 * Float64(a * Float64(c / Float64(b * b)))) - 2.0)) / Float64(Float64(b * b) * Float64(b * b))) * a) - Float64(Float64(c * c) / Float64(b * b))) * a) - c) / 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], -10.0], 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[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] * N[(N[(-5.0 * N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $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 (/.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)) < -10

   1. Initial program 55.8%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* 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 55.9

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

   3. Applied rewrites 55.9%

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

   if -10 < (/.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 55.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 90.7%

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

   11. Taylor expanded in c around 0

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

       1. lower-*.f64 N/A

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

       2. pow3 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 90.7

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

   13. Applied rewrites 90.7%

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

Alternative 2: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, a + a, \left(a + a\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}{\left(a + a\right) \cdot \left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{a \cdot \left(\left(c \cdot c\right) \cdot c\right)}{b \cdot b} - c \cdot c}{b \cdot b} \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5)
   (/
    (fma (- b) (+ a a) (* (+ a a) (sqrt (fma (* -4.0 a) c (* b b)))))
    (* (+ a a) (+ a a)))
   (/
    (-
     (* (/ (- (* -2.0 (/ (* a (* (* c c) c)) (* b b))) (* c c)) (* b b)) a)
     c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5) {
		tmp = fma(-b, (a + a), ((a + a) * sqrt(fma((-4.0 * a), c, (b * b))))) / ((a + a) * (a + a));
	} else {
		tmp = (((((-2.0 * ((a * ((c * c) * c)) / (b * b))) - (c * c)) / (b * b)) * a) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5)
		tmp = Float64(fma(Float64(-b), Float64(a + a), Float64(Float64(a + a) * sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))))) / Float64(Float64(a + a) * Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(a * Float64(Float64(c * c) * c)) / Float64(b * b))) - Float64(c * c)) / Float64(b * b)) * a) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5], N[(N[((-b) * N[(a + a), $MachinePrecision] + N[(N[(a + a), $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(a * N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] - c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 55.0%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 56.1%

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

   if 3.5 < b

   1. Initial program 55.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 90.7%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 87.5

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

   10. Applied rewrites 87.5%

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

Alternative 3: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, a + a, \left(a + a\right) \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\right)}{\left(a + a\right) \cdot \left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-2 \cdot \frac{\left(a \cdot a\right) \cdot c}{b \cdot b} - a}{b \cdot b} \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5)
   (/
    (fma (- b) (+ a a) (* (+ a a) (sqrt (fma (* -4.0 a) c (* b b)))))
    (* (+ a a) (+ a a)))
   (/
    (* (- (* (/ (- (* -2.0 (/ (* (* a a) c) (* b b))) a) (* b b)) c) 1.0) c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5) {
		tmp = fma(-b, (a + a), ((a + a) * sqrt(fma((-4.0 * a), c, (b * b))))) / ((a + a) * (a + a));
	} else {
		tmp = ((((((-2.0 * (((a * a) * c) / (b * b))) - a) / (b * b)) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5)
		tmp = Float64(fma(Float64(-b), Float64(a + a), Float64(Float64(a + a) * sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))))) / Float64(Float64(a + a) * Float64(a + a)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(-2.0 * Float64(Float64(Float64(a * a) * c) / Float64(b * b))) - a) / Float64(b * b)) * c) - 1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5], N[(N[((-b) * N[(a + a), $MachinePrecision] + N[(N[(a + a), $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a + a), $MachinePrecision] * N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(-2.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5

   1. Initial program 55.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

   3. Applied rewrites 55.0%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. frac-add N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 56.1%

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

   if 3.5 < b

   1. Initial program 55.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

Alternative 4: 89.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{-2 \cdot \frac{\left(a \cdot a\right) \cdot c}{b \cdot b} - a}{b \cdot b} \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/
    (* (- (* (/ (- (* -2.0 (/ (* (* a a) c) (* b b))) a) (* b b)) c) 1.0) c)
    b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = ((((((-2.0 * (((a * a) * c) / (b * b))) - a) / (b * b)) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5)
		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(Float64(a * a) * c) / Float64(b * b))) - a) / Float64(b * b)) * c) - 1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5], 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[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5

   1. Initial program 55.8%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* 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 55.9

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

   3. Applied rewrites 55.9%

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

   if 3.5 < b

   1. Initial program 55.8%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in c around 0

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

Alternative 5: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.2)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (- (/ (fma (* c c) (/ a (* b b)) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.2) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (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 (b <= 7.2)
		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(c * c), Float64(a / Float64(b * b)), c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.2], 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[(c * c), $MachinePrecision] * N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision] / b), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b < 7.20000000000000018

   1. Initial program 55.8%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* 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 55.9

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

   3. Applied rewrites 55.9%

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

   if 7.20000000000000018 < b

   1. Initial program 55.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 81.2

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

   4. Applied rewrites 81.2%

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

   6. Applied rewrites 81.2%

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

Alternative 6: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(c \cdot c, \frac{a}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.2)
   (/ (+ (sqrt (fma (* -4.0 a) c (* 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 <= 7.2) {
		tmp = (sqrt(fma((-4.0 * a), c, (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 (b <= 7.2)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = Float64(-Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.2], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + 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 b < 7.20000000000000018

   1. Initial program 55.8%

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

   3. Applied rewrites 55.8%

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

   if 7.20000000000000018 < b

   1. Initial program 55.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 81.2

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

   4. Applied rewrites 81.2%

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

   6. Applied rewrites 81.2%

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

Alternative 7: 81.2% 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(-Float64(fma(Float64(c * c), Float64(a / Float64(b * b)), c) / 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 55.8%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow3 N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r/ N/A

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

   16. mul-1-neg N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 81.2

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

4. Applied rewrites 81.2%

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

6. Applied rewrites 81.2%

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

Alternative 8: 64.0% 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 55.8%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. 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 64.0

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

4. Applied rewrites 64.0%

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

Reproduce

herbie shell --seed 2025127 
(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)))