Quadratic roots, narrow range

Percentage Accurate: 55.6% → 92.0%

Time: 6.4s

Alternatives: 11

Speedup: 3.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.6% 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: 92.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 0.0395:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right)}{b} + \frac{\mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{6} \cdot a}, -0.25, \frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b}\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 0.0395)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 2.0 a))
     (+
      (/ (fma (* (* a a) (/ (* (* c c) c) (* (* b b) (* b b)))) -2.0 (- c)) b)
      (/
       (fma
        (/ (* (pow (* c a) 4.0) 20.0) (* (pow b 6.0) a))
        -0.25
        (/ (* a (* c c)) (* (- b) b)))
       b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (b <= 0.0395) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (2.0 * a);
	} else {
		tmp = (fma(((a * a) * (((c * c) * c) / ((b * b) * (b * b)))), -2.0, -c) / b) + (fma(((pow((c * a), 4.0) * 20.0) / (pow(b, 6.0) * a)), -0.25, ((a * (c * c)) / (-b * b))) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (b <= 0.0395)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(fma(Float64(Float64(a * a) * Float64(Float64(Float64(c * c) * c) / Float64(Float64(b * b) * Float64(b * b)))), -2.0, Float64(-c)) / b) + Float64(fma(Float64(Float64((Float64(c * a) ^ 4.0) * 20.0) / Float64((b ^ 6.0) * a)), -0.25, Float64(Float64(a * Float64(c * c)) / Float64(Float64(-b) * b))) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 0.0395], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + (-c)), $MachinePrecision] / b), $MachinePrecision] + N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] * 20.0), $MachinePrecision] / N[(N[Power[b, 6.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * -0.25 + N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[((-b) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0395000000000000004

   1. Initial program 88.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 88.9%

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

   if 0.0395000000000000004 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 94.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 94.9

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

   9. Applied rewrites 94.9%

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

       1. lift-pow.f64 N/A

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

       2. sqr-pow N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. lower-*.f64 N/A

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

       6. pow2 N/A

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

       7. lift-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 94.9

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

   11. Applied rewrites 94.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0395:\\ \;\;\;\;\frac{\frac{b \cdot b - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a \cdot a\right) \cdot \frac{\left(c \cdot c\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -2, -c\right)}{b} + \frac{\mathsf{fma}\left(\frac{{\left(c \cdot a\right)}^{4} \cdot 20}{{b}^{6} \cdot a}, -0.25, \frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b}\right)}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 0.0395:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\frac{\left(c \cdot a\right) \cdot -2}{{b}^{5}} - {b}^{-3}\right) \cdot \left(c \cdot c\right), a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 0.0395)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 2.0 a))
     (fma
      (* (- (/ (* (* c a) -2.0) (pow b 5.0)) (pow b -3.0)) (* c c))
      a
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (b <= 0.0395) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (2.0 * a);
	} else {
		tmp = fma((((((c * a) * -2.0) / pow(b, 5.0)) - pow(b, -3.0)) * (c * c)), a, (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (b <= 0.0395)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(c * a) * -2.0) / (b ^ 5.0)) - (b ^ -3.0)) * Float64(c * c)), a, Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 0.0395], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(c * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] - N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0395000000000000004

   1. Initial program 88.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 88.9%

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

   if 0.0395000000000000004 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 92.0%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-flip N/A

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

      12. lower-pow.f64 N/A

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

      13. metadata-eval N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 92.0

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

   8. Applied rewrites 92.0%

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

Alternative 3: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(c \cdot c\right) \cdot a}{{b}^{3}}, -1, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.1)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma (/ (* (* c c) a) (pow b 3.0)) -1.0 (/ (- 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)) <= -0.1) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma((((c * c) * a) / pow(b, 3.0)), -1.0, (-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)) <= -0.1)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(c * c) * a) / (b ^ 3.0)), -1.0, Float64(Float64(-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], -0.1], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c * c), $MachinePrecision] * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * -1.0 + N[((-c) / b), $MachinePrecision]), $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)) < -0.10000000000000001

   1. Initial program 82.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   if -0.10000000000000001 < (/.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 46.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   4. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 89.3

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

   5. Applied rewrites 89.3%

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

Alternative 4: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-c\right) \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.1)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (fma (/ (* (- c) c) (pow b 3.0)) 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)) <= -0.1) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = fma(((-c * c) / pow(b, 3.0)), 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)) <= -0.1)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(Float64(Float64(-c) * c) / (b ^ 3.0)), a, Float64(Float64(-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], -0.1], 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[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * a + N[((-c) / b), $MachinePrecision]), $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)) < -0.10000000000000001

   1. Initial program 82.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   if -0.10000000000000001 < (/.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 46.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 89.3

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

   8. Applied rewrites 89.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(-c\right) \cdot c}{{b}^{3}}, a, \frac{-c}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 0.0395:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\left({c}^{3} \cdot a\right) \cdot -2}{b \cdot b} - c \cdot c}{b \cdot b} \cdot a - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 0.0395)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 2.0 a))
     (/
      (-
       (* (/ (- (/ (* (* (pow c 3.0) a) -2.0) (* b b)) (* c c)) (* b b)) a)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (b <= 0.0395) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (2.0 * a);
	} else {
		tmp = (((((((pow(c, 3.0) * a) * -2.0) / (b * b)) - (c * c)) / (b * b)) * a) - c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (b <= 0.0395)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64((c ^ 3.0) * a) * -2.0) / Float64(b * b)) - Float64(c * c)) / Float64(b * b)) * a) - c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 0.0395], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[Power[c, 3.0], $MachinePrecision] * a), $MachinePrecision] * -2.0), $MachinePrecision] / N[(b * b), $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 < 0.0395000000000000004

   1. Initial program 88.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 88.9%

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

   if 0.0395000000000000004 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

   8. Applied rewrites 92.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{-2 \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} - {c}^{2}}{{b}^{2}} \cdot a - c}{b} \]
   10. 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. associate-*r/ N/A

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

       4. lower-/.f64 N/A

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. lift-pow.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 N/A

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

       12. pow2 N/A

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

       13. lift-*.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 92.0

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

   11. Applied rewrites 92.0%

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

Alternative 6: 89.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}\\ \mathbf{if}\;b \leq 0.0395:\\ \;\;\;\;\frac{\frac{b \cdot b - t\_0 \cdot t\_0}{\left(-b\right) - t\_0}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{\left(\left(a \cdot a\right) \cdot c\right) \cdot -2}{{b}^{4}} - \frac{a}{b \cdot b}\right) \cdot c - 1\right) \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* -4.0 a) c (* b b)))))
   (if (<= b 0.0395)
     (/ (/ (- (* b b) (* t_0 t_0)) (- (- b) t_0)) (* 2.0 a))
     (/
      (*
       (- (* (- (/ (* (* (* a a) c) -2.0) (pow b 4.0)) (/ a (* b b))) c) 1.0)
       c)
      b))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((-4.0 * a), c, (b * b)));
	double tmp;
	if (b <= 0.0395) {
		tmp = (((b * b) - (t_0 * t_0)) / (-b - t_0)) / (2.0 * a);
	} else {
		tmp = ((((((((a * a) * c) * -2.0) / pow(b, 4.0)) - (a / (b * b))) * c) - 1.0) * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(-4.0 * a), c, Float64(b * b)))
	tmp = 0.0
	if (b <= 0.0395)
		tmp = Float64(Float64(Float64(Float64(b * b) - Float64(t_0 * t_0)) / Float64(Float64(-b) - t_0)) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * a) * c) * -2.0) / (b ^ 4.0)) - Float64(a / Float64(b * b))) * c) - 1.0) * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, 0.0395], N[(N[(N[(N[(b * b), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[((-b) - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] * -2.0), $MachinePrecision] / N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - 1.0), $MachinePrecision] * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.0395000000000000004

   1. Initial program 88.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. flip-+ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 88.9%

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

   if 0.0395000000000000004 < b

   1. Initial program 50.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 94.8%

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

   6. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 91.9%

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

Alternative 7: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.1)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 a))
   (/ (- (/ (* a (* c c)) (* (- 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)) <= -0.1) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * a);
	} else {
		tmp = (((a * (c * c)) / (-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)) <= -0.1)
		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(a * Float64(c * c)) / Float64(Float64(-b) * b)) - 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], -0.1], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / 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)) < -0.10000000000000001

   1. Initial program 82.1%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-*r* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.2

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

   4. Applied rewrites 82.2%

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

   if -0.10000000000000001 < (/.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 46.5%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 89.3

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

   8. Applied rewrites 89.3%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b} - c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.1)
   (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) (+ a a))
   (/ (- (/ (* a (* c c)) (* (- 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)) <= -0.1) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) - b) / (a + a);
	} else {
		tmp = (((a * (c * c)) / (-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)) <= -0.1)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(Float64(Float64(a * Float64(c * c)) / Float64(Float64(-b) * b)) - 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], -0.1], 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[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / 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)) < -0.10000000000000001

   1. Initial program 82.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 82.1

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

   6. Applied rewrites 82.1%

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

   if -0.10000000000000001 < (/.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 46.5%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 89.3

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

   8. Applied rewrites 89.3%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot \left(c \cdot c\right)}{\left(-b\right) \cdot b} - c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 92.4%

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

6. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. lower-/.f64 N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

   6. lower-*.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 83.3

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

8. Applied rewrites 83.3%

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

9. Final simplification 83.3%

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

Alternative 10: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a \cdot \frac{c}{b \cdot b} + 1\right) \cdot \left(-c\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (+ (* a (/ c (* b b))) 1.0) (- c)) b))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double a, double b, double c) {
	return (((a * (c / (b * b))) + 1.0) * -c) / b;
}

Python

def code(a, b, c):
	return (((a * (c / (b * b))) + 1.0) * -c) / b

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(a * Float64(c / Float64(b * b))) + 1.0) * Float64(-c)) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 92.4%

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

6. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 83.2

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

8. Applied rewrites 83.2%

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

9. Final simplification 83.2%

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

Alternative 11: 64.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 66.3

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

5. Applied rewrites 66.3%

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

Reproduce

herbie shell --seed 2025057 
(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)))