Quadratic roots, narrow range

Percentage Accurate: 55.6% → 91.7%

Time: 5.0s

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: 55.6% 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: 91.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot c\right) \cdot c\\ \mathbf{if}\;b \leq 0.0295:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{\left(a \cdot a\right) \cdot \left(t\_0 \cdot c\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, -5, \frac{t\_0 \cdot a}{b \cdot b} \cdot -2 - c \cdot c\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* c c) c)))
   (if (<= b 0.0295)
     (/ (+ (- b) (sqrt (* (fma -4.0 a (/ (* b b) c)) c))) (+ a a))
     (fma
      a
      (/
       (fma
        (/ (* (* a a) (* t_0 c)) (* (* b b) (* b b)))
        -5.0
        (- (* (/ (* t_0 a) (* b b)) -2.0) (* c c)))
       (* (* b b) b))
      (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double t_0 = (c * c) * c;
	double tmp;
	if (b <= 0.0295) {
		tmp = (-b + sqrt((fma(-4.0, a, ((b * b) / c)) * c))) / (a + a);
	} else {
		tmp = fma(a, (fma((((a * a) * (t_0 * c)) / ((b * b) * (b * b))), -5.0, ((((t_0 * a) / (b * b)) * -2.0) - (c * c))) / ((b * b) * b)), (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(c * c) * c)
	tmp = 0.0
	if (b <= 0.0295)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c))) / Float64(a + a));
	else
		tmp = fma(a, Float64(fma(Float64(Float64(Float64(a * a) * Float64(t_0 * c)) / Float64(Float64(b * b) * Float64(b * b))), -5.0, Float64(Float64(Float64(Float64(t_0 * a) / Float64(b * b)) * -2.0) - Float64(c * c))) / Float64(Float64(b * b) * b)), Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[b, 0.0295], N[(N[((-b) + N[Sqrt[N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(N[(a * a), $MachinePrecision] * N[(t$95$0 * c), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -5.0 + N[(N[(N[(N[(t$95$0 * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] - N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.029499999999999998

   1. Initial program 85.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. pow2 N/A

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

      3. lower--.f64 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-*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. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 85.5

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 85.3

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

   6. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 85.3

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

   8. Applied rewrites 85.3%

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

   if 0.029499999999999998 < b

   1. Initial program 53.1%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 92.2%

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

Alternative 2: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.0295:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, a, \frac{b \cdot b}{c}\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c, \frac{\mathsf{fma}\left(-5, \frac{\left(a \cdot a\right) \cdot c}{b \cdot b}, -2 \cdot a\right)}{b \cdot b}, -1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.0295)
   (/ (+ (- b) (sqrt (* (fma -4.0 a (/ (* b b) c)) c))) (+ a a))
   (fma
    a
    (/
     (*
      (* c c)
      (fma c (/ (fma -5.0 (/ (* (* a a) c) (* b b)) (* -2.0 a)) (* b b)) -1.0))
     (* (* b b) b))
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.0295) {
		tmp = (-b + sqrt((fma(-4.0, a, ((b * b) / c)) * c))) / (a + a);
	} else {
		tmp = fma(a, (((c * c) * fma(c, (fma(-5.0, (((a * a) * c) / (b * b)), (-2.0 * a)) / (b * b)), -1.0)) / ((b * b) * b)), (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.0295)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(fma(-4.0, a, Float64(Float64(b * b) / c)) * c))) / Float64(a + a));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * fma(c, Float64(fma(-5.0, Float64(Float64(Float64(a * a) * c) / Float64(b * b)), Float64(-2.0 * a)) / Float64(b * b)), -1.0)) / Float64(Float64(b * b) * b)), Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.0295], N[(N[((-b) + N[Sqrt[N[(N[(-4.0 * a + N[(N[(b * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(c * N[(N[(-5.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.029499999999999998

   1. Initial program 85.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. pow2 N/A

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

      3. lower--.f64 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-*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. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 85.5

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 85.3

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

   6. Applied rewrites 85.3%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f64 85.3

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

   8. Applied rewrites 85.3%

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

   if 0.029499999999999998 < b

   1. Initial program 53.1%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 92.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

   10. Applied rewrites 92.2%

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

   11. Taylor expanded in b around inf

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. lift-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 92.2

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

   13. Applied rewrites 92.2%

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

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.15:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)\right) \cdot \frac{1}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(2, \frac{\left(a \cdot a\right) \cdot c}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{a}{b \cdot b}\right) + c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.15)
   (* (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (/ 1.0 (+ a a)))
   (-
    (/
     (+
      (* (* c c) (fma 2.0 (/ (* (* a a) c) (* (* b b) (* b b))) (/ a (* b b))))
      c)
     b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)) <= -0.15) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) * (1.0 / (a + a));
	} else {
		tmp = -((((c * c) * fma(2.0, (((a * a) * c) / ((b * b) * (b * b))), (a / (b * b)))) + c) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a)) <= -0.15)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) * Float64(1.0 / Float64(a + a)));
	else
		tmp = Float64(-Float64(Float64(Float64(Float64(c * c) * fma(2.0, Float64(Float64(Float64(a * a) * c) / Float64(Float64(b * b) * Float64(b * b))), Float64(a / 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], -0.15], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(1.0 / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[(N[(c * c), $MachinePrecision] * N[(2.0 * N[(N[(N[(a * a), $MachinePrecision] * c), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(b * b), $MachinePrecision]), $MachinePrecision]), $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)) < -0.149999999999999994

   1. Initial program 79.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 79.8%

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

   if -0.149999999999999994 < (/.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 49.4%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. sqr-pow N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. pow2 N/A

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

   10. Applied rewrites 91.4%

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

Alternative 4: 89.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.15:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)\right) \cdot \frac{1}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(-2, \frac{a \cdot c}{b \cdot b}, -1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.15)
   (* (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (/ 1.0 (+ a a)))
   (fma
    a
    (/ (* (* c c) (fma -2.0 (/ (* a c) (* b b)) -1.0)) (* (* b 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)) <= -0.15) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) * (1.0 / (a + a));
	} else {
		tmp = fma(a, (((c * c) * fma(-2.0, ((a * c) / (b * b)), -1.0)) / ((b * 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)) <= -0.15)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) * Float64(1.0 / Float64(a + a)));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * fma(-2.0, Float64(Float64(a * c) / Float64(b * b)), -1.0)) / Float64(Float64(b * b) * b)), 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], -0.15], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(1.0 / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(-2.0 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + 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)) < -0.149999999999999994

   1. Initial program 79.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 79.8%

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

   if -0.149999999999999994 < (/.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 49.4%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around inf

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in c around 0

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

      1. lower-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. negate-sub N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 91.4

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

   10. Applied rewrites 91.4%

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

Alternative 5: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.038:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} + \left(-b\right)\right) \cdot \frac{1}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.038)
   (* (+ (sqrt (fma (* -4.0 a) c (* b b))) (- b)) (/ 1.0 (+ a a)))
   (- (/ (fma 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)) <= -0.038) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) + -b) * (1.0 / (a + a));
	} else {
		tmp = -(fma(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)) <= -0.038)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) + Float64(-b)) * Float64(1.0 / Float64(a + a)));
	else
		tmp = Float64(-Float64(fma(a, Float64(Float64(c * c) / 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], -0.038], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] * N[(1.0 / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(a * N[(N[(c * c), $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)) < -0.0379999999999999991

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      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-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 78.9%

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

   if -0.0379999999999999991 < (/.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 48.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 87.2

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

   7. Applied rewrites 87.2%

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

Alternative 6: 85.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -0.038:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -0.038)
   (/ (+ (- b) (sqrt (fma (* c a) -4.0 (* b b)))) (+ a a))
   (- (/ (fma 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)) <= -0.038) {
		tmp = (-b + sqrt(fma((c * a), -4.0, (b * b)))) / (a + a);
	} else {
		tmp = -(fma(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)) <= -0.038)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(-Float64(fma(a, Float64(Float64(c * c) / 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], -0.038], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], (-N[(N[(a * N[(N[(c * c), $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)) < -0.0379999999999999991

   1. Initial program 78.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower--.f64 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-*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. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 78.9

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

   3. Applied rewrites 78.9%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -0.0379999999999999991 < (/.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 48.0%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

   4. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

   5. Taylor expanded in b around inf

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

      1. distribute-lft-out N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 87.2

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

   7. Applied rewrites 87.2%

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

Alternative 7: 81.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{4} \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

4. Applied rewrites 90.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{\left(b \cdot b\right) \cdot b}\right), \frac{-c}{b}\right)} \]

5. Taylor expanded in b around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r/ N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. associate-/l* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 81.4

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

7. Applied rewrites 81.4%

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

Alternative 8: 64.1% 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 55.6%

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

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 64.1

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

4. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2025120 
(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)))