Quadratic roots, medium range

Percentage Accurate: 31.9% → 95.5%

Time: 5.3s

Alternatives: 8

Speedup: 4.6×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.9% 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: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \left(\frac{-2 \cdot \left(a \cdot c\right)}{b \cdot b} - 1\right)}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-7)
   (+ (/ (- b) (+ a a)) (/ (sqrt (fma b b (* -4.0 (* c a)))) (+ a a)))
   (fma
    a
    (/ (* (* c c) (- (/ (* -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 <= 2.8e-7) {
		tmp = (-b / (a + a)) + (sqrt(fma(b, b, (-4.0 * (c * a)))) / (a + a));
	} else {
		tmp = fma(a, (((c * c) * (((-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 (b <= 2.8e-7)
		tmp = Float64(Float64(Float64(-b) / Float64(a + a)) + Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) / Float64(a + a)));
	else
		tmp = fma(a, Float64(Float64(Float64(c * c) * Float64(Float64(Float64(-2.0 * 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[b, 2.8e-7], N[(N[((-b) / N[(a + a), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(c * c), $MachinePrecision] * N[(N[(N[(-2.0 * N[(a * c), $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 < 2.80000000000000019e-7

   1. Initial program 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.0

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

   3. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 76.7%

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

   if 2.80000000000000019e-7 < b

   1. Initial program 30.5%

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

   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 96.0%

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

   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 96.0%

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

   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. lower--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 94.6

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

   10. Applied rewrites 94.6%

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

Alternative 2: 95.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot b\\ \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\left(\frac{\left(c \cdot c\right) \cdot \left(c \cdot c\right)}{t\_0 \cdot t\_0} \cdot 20\right) \cdot a}{b}, -0.25, \frac{\left(c \cdot c\right) \cdot c}{{b}^{5}} \cdot -2\right), -\frac{c \cdot c}{t\_0}\right), \frac{-c}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (* b b) b)))
   (fma
    a
    (fma
     a
     (fma
      (/ (* (* (/ (* (* c c) (* c c)) (* t_0 t_0)) 20.0) a) b)
      -0.25
      (* (/ (* (* c c) c) (pow b 5.0)) -2.0))
     (- (/ (* c c) t_0)))
    (/ (- c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (b * b) * b;
	return fma(a, fma(a, fma(((((((c * c) * (c * c)) / (t_0 * t_0)) * 20.0) * a) / b), -0.25, ((((c * c) * c) / pow(b, 5.0)) * -2.0)), -((c * c) / t_0)), (-c / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(b * b) * b)
	return fma(a, fma(a, fma(Float64(Float64(Float64(Float64(Float64(Float64(c * c) * Float64(c * c)) / Float64(t_0 * t_0)) * 20.0) * a) / b), -0.25, Float64(Float64(Float64(Float64(c * c) * c) / (b ^ 5.0)) * -2.0)), Float64(-Float64(Float64(c * c) / t_0))), Float64(Float64(-c) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]}, N[(a * N[(a * N[(N[(N[(N[(N[(N[(N[(c * c), $MachinePrecision] * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * 20.0), $MachinePrecision] * a), $MachinePrecision] / b), $MachinePrecision] * -0.25 + N[(N[(N[(N[(c * c), $MachinePrecision] * c), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] + (-N[(N[(c * c), $MachinePrecision] / t$95$0), $MachinePrecision])), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 31.9%

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

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 95.5%

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

Alternative 3: 94.0% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  a
  (/
   (*
    (* c c)
    (- (* 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) {
	return fma(a, (((c * c) * ((c * (fma(-5.0, ((a * a) * (c / (b * b))), (-2.0 * a)) / (b * b))) - 1.0)) / ((b * b) * b)), (-c / b));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

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 95.5%

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

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 95.5%

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

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. lower--.f64 N/A

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

10. Applied rewrites 95.5%

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

11. Taylor expanded in b around inf

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

    1. lower-/.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. associate-/l* N/A

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

    4. lower-*.f64 N/A

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

    5. pow2 N/A

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

    6. lift-*.f64 N/A

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

    7. pow2 N/A

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

    8. lift-/.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. pow2 N/A

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

    12. lift-*.f64 95.5

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

13. Applied rewrites 95.5%

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

Alternative 4: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{-b}{a + a} + \frac{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-7)
   (+ (/ (- b) (+ a a)) (/ (sqrt (fma b b (* -4.0 (* c a)))) (+ a a)))
   (fma a (/ (- (* c c)) (* (* b b) b)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-7) {
		tmp = (-b / (a + a)) + (sqrt(fma(b, b, (-4.0 * (c * a)))) / (a + a));
	} else {
		tmp = fma(a, (-(c * c) / ((b * b) * b)), (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-7)
		tmp = Float64(Float64(Float64(-b) / Float64(a + a)) + Float64(sqrt(fma(b, b, Float64(-4.0 * Float64(c * a)))) / Float64(a + a)));
	else
		tmp = fma(a, Float64(Float64(-Float64(c * c)) / Float64(Float64(b * b) * b)), Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-7], N[(N[((-b) / N[(a + a), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[((-N[(c * c), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.80000000000000019e-7

   1. Initial program 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.0

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

   3. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. div-add N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 76.7%

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

   if 2.80000000000000019e-7 < b

   1. Initial program 30.5%

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

   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 96.0%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 5: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{-c \cdot c}{\left(b \cdot b\right) \cdot b}, \frac{-c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-7)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (fma a (/ (- (* c c)) (* (* b b) b)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-7) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = fma(a, (-(c * c) / ((b * b) * b)), (-c / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-7)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / Float64(a + a));
	else
		tmp = fma(a, Float64(Float64(-Float64(c * c)) / Float64(Float64(b * b) * b)), Float64(Float64(-c) / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-7], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(a * N[((-N[(c * c), $MachinePrecision]) / N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + N[((-c) / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.80000000000000019e-7

   1. Initial program 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.0

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

   3. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if 2.80000000000000019e-7 < b

   1. Initial program 30.5%

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

   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 96.0%

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

   5. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 91.6

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

   7. Applied rewrites 91.6%

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

Alternative 6: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a, \frac{c \cdot c}{b \cdot b}, c\right)}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-7)
   (/ (+ (- b) (sqrt (fma b b (* -4.0 (* c a))))) (+ a a))
   (- (/ (fma a (/ (* c c) (* b b)) c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.8e-7) {
		tmp = (-b + sqrt(fma(b, b, (-4.0 * (c * a))))) / (a + a);
	} else {
		tmp = -(fma(a, ((c * c) / (b * b)), c) / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.8e-7)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(b, b, Float64(-4.0 * Float64(c * a))))) / 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[b, 2.8e-7], N[(N[((-b) + N[Sqrt[N[(b * b + N[(-4.0 * N[(c * a), $MachinePrecision]), $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 b < 2.80000000000000019e-7

   1. Initial program 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

      8. pow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 78.0

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

   3. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if 2.80000000000000019e-7 < b

   1. Initial program 30.5%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   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 91.6

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

   4. Applied rewrites 91.6%

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

   6. Applied rewrites 91.6%

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

Alternative 7: 90.7% 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 31.9%

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

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 90.7

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

4. Applied rewrites 90.7%

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

6. Applied rewrites 90.7%

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

Alternative 8: 81.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

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

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 81.0

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

4. Applied rewrites 81.0%

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

Reproduce

herbie shell --seed 2025122 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))