Quadratic roots, narrow range

Percentage Accurate: 55.3% → 90.9%

Time: 5.1s

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: 55.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

2. 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 90.9%

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

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 90.9%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 90.9%

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

11. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-/.f64 N/A

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

    4. pow2 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-*.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. pow2 N/A

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

    11. lift-*.f64 90.9

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

13. Applied rewrites 90.9%

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

Alternative 2: 89.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -4.0)
   (/ (+ (sqrt (fma -4.0 (* c a) (* b b))) (- b)) (+ a a))
   (fma
    a
    (/ (* (fma (* a (/ c (* b b))) -2.0 -1.0) (* c c)) (* (* 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)) <= -4.0) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * b))) + -b) / (a + a);
	} else {
		tmp = fma(a, ((fma((a * (c / (b * b))), -2.0, -1.0) * (c * c)) / ((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)) <= -4.0)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + Float64(-b)) / Float64(a + a));
	else
		tmp = fma(a, Float64(Float64(fma(Float64(a * Float64(c / Float64(b * b))), -2.0, -1.0) * Float64(c * c)) / 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], -4.0], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(N[(a * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision] * N[(c * c), $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)) < -4

   1. Initial program 83.6%

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

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 83.4

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

   3. Applied rewrites 83.4%

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

   5. Applied rewrites 83.6%

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

   if -4 < (/.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 51.5%

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

   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 93.1%

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

   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 93.1%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. negate-sub N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 90.5

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

   10. Applied rewrites 90.5%

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

Alternative 3: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -3e-5)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (* 2.0 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)) <= -3e-5) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (2.0 * 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)) <= -3e-5)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(2.0 * 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], -3e-5], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], (-N[(N[(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)) < -3.00000000000000008e-5

   1. Initial program 73.9%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*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 74.0

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

   3. Applied rewrites 74.0%

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

   if -3.00000000000000008e-5 < (/.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 38.4%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 92.7

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

   4. Applied rewrites 92.7%

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

   6. Applied rewrites 92.7%

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

Alternative 4: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \leq -3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + \left(-b\right)}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) -3e-5)
   (/ (+ (sqrt (fma -4.0 (* c a) (* b 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)) <= -3e-5) {
		tmp = (sqrt(fma(-4.0, (c * a), (b * 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)) <= -3e-5)
		tmp = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + Float64(-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], -3e-5], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b)), $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)) < -3.00000000000000008e-5

   1. Initial program 73.9%

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

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 73.6

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

   3. Applied rewrites 73.6%

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

   5. Applied rewrites 73.9%

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

   if -3.00000000000000008e-5 < (/.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 38.4%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow3 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 92.7

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

   4. Applied rewrites 92.7%

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

   6. Applied rewrites 92.7%

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

Alternative 5: 81.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

2. 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 90.9%

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

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 90.9%

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

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
9. 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. mul-1-neg N/A

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

   3. negate-sub N/A

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

   4. lower--.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. pow2 N/A

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

   12. lift-*.f64 N/A

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

   13. pow3 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lift-/.f64 81.7

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

10. Applied rewrites 81.7%

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

Alternative 6: 81.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lower-+.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 81.7

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

4. Applied rewrites 81.7%

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

Alternative 7: 81.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow3 N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*r/ N/A

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

   16. mul-1-neg N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 81.7

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

4. Applied rewrites 81.7%

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

6. Applied rewrites 81.7%

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

Alternative 8: 64.5% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.3%

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

2. Taylor expanded in 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 64.5

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

4. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025116 
(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)))