Quadratic roots, full range

Percentage Accurate: 50.7% → 85.8%

Time: 5.8s

Alternatives: 7

Speedup: 2.5×

Specification

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: 50.7% 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: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9.5e+143)
   (/ (- b) a)
   (if (<= b 3.5e-53)
     (/ (- (sqrt (fma (* -4.0 a) c (* b b))) b) (+ a a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9.5e+143) {
		tmp = -b / a;
	} else if (b <= 3.5e-53) {
		tmp = (sqrt(fma((-4.0 * a), c, (b * b))) - b) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9.5e+143)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.5e-53)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9.5e+143], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.5e-53], N[(N[(N[Sqrt[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.50000000000000066e143

   1. Initial program 45.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 96.4

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

   5. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -9.50000000000000066e143 < b < 3.49999999999999993e-53

   1. Initial program 82.0%

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 82.0

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

   4. Applied rewrites 82.0%

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

      1. count-2-rev N/A

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

      2. lower-+.f64 82.0

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

   6. Applied rewrites 82.0%

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

   if 3.49999999999999993e-53 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -0.5, \sqrt{a \cdot \left(-c\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e-12)
   (/ (- b) a)
   (if (<= b 3.4e-53) (/ (fma b -0.5 (sqrt (* a (- c)))) a) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e-12) {
		tmp = -b / a;
	} else if (b <= 3.4e-53) {
		tmp = fma(b, -0.5, sqrt((a * -c))) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e-12)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.4e-53)
		tmp = Float64(fma(b, -0.5, sqrt(Float64(a * Float64(-c)))) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e-12], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.4e-53], N[(N[(b * -0.5 + N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.99999999999999984e-12

   1. Initial program 70.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 91.0

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

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -7.99999999999999984e-12 < b < 3.4e-53

   1. Initial program 74.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 39.7

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right) \]

   5. Applied rewrites 39.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)} \]

   6. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 39.7

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   8. Applied rewrites 39.7%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b}{a}}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot b + \sqrt{a \cdot c} \cdot \sqrt{-1}}{a} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 67.2

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{a} \]

   11. Applied rewrites 67.2%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{a} \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. sqrt-prod N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. sqrt-unprod N/A

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

       6. *-commutative N/A

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

       7. mul-1-neg N/A

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

       8. lower-sqrt.f64 N/A

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

       9. lower-neg.f64 N/A

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

       10. lower-*.f64 67.2

           \[\leadsto \frac{\mathsf{fma}\left(b, -0.5, \sqrt{-a \cdot c}\right)}{a} \]

   13. Applied rewrites 67.2%

       \[\leadsto \frac{\mathsf{fma}\left(b, -0.5, \sqrt{-a \cdot c}\right)}{a} \]

   if 3.4e-53 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-12}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -0.5, \sqrt{a \cdot \left(-c\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b \leq -2.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-270}:\\ \;\;\;\;-t\_0\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b -2.5e-161)
     (/ (- b) a)
     (if (<= b 4.5e-270) (- t_0) (if (<= b 2.2e-93) t_0 (/ (- c) b))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((-c / a));
	double tmp;
	if (b <= -2.5e-161) {
		tmp = -b / a;
	} else if (b <= 4.5e-270) {
		tmp = -t_0;
	} else if (b <= 2.2e-93) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-c / a))
    if (b <= (-2.5d-161)) then
        tmp = -b / a
    else if (b <= 4.5d-270) then
        tmp = -t_0
    else if (b <= 2.2d-93) then
        tmp = t_0
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((-c / a));
	double tmp;
	if (b <= -2.5e-161) {
		tmp = -b / a;
	} else if (b <= 4.5e-270) {
		tmp = -t_0;
	} else if (b <= 2.2e-93) {
		tmp = t_0;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((-c / a))
	tmp = 0
	if b <= -2.5e-161:
		tmp = -b / a
	elif b <= 4.5e-270:
		tmp = -t_0
	elif b <= 2.2e-93:
		tmp = t_0
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(-c) / a))
	tmp = 0.0
	if (b <= -2.5e-161)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 4.5e-270)
		tmp = Float64(-t_0);
	elseif (b <= 2.2e-93)
		tmp = t_0;
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((-c / a));
	tmp = 0.0;
	if (b <= -2.5e-161)
		tmp = -b / a;
	elseif (b <= 4.5e-270)
		tmp = -t_0;
	elseif (b <= 2.2e-93)
		tmp = t_0;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -2.5e-161], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 4.5e-270], (-t$95$0), If[LessEqual[b, 2.2e-93], t$95$0, N[((-c) / b), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.5e-161

   1. Initial program 74.4%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 82.6

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

   5. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -2.5e-161 < b < 4.49999999999999998e-270

   1. Initial program 78.5%

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a}} \cdot \sqrt{-1} \]

      3. sqrt-unprod N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      5. lower-*.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      6. lower-/.f64 47.4

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

   7. Applied rewrites 47.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{c}{a} \cdot -1}} \]
   8. Step-by-step derivation

      1. sqrt-prod N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a}} \cdot \sqrt{-1} \]

      3. sqrt-prod N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      5. *-commutative N/A

         \[\leadsto -\sqrt{-1 \cdot \frac{c}{a}} \]

      6. associate-*r/ N/A

         \[\leadsto -\sqrt{\frac{-1 \cdot c}{a}} \]

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 47.4

         \[\leadsto -\sqrt{\frac{-c}{a}} \]

   9. Applied rewrites 47.4%

      \[\leadsto -\sqrt{\frac{-c}{a}} \]

   if 4.49999999999999998e-270 < b < 2.19999999999999996e-93

   1. Initial program 62.0%

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 62.0

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

   4. Applied rewrites 62.0%

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

   5. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a}} \cdot \sqrt{-1} \]

      3. sqrt-unprod N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      5. lower-*.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      6. lower-/.f64 31.7

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

   7. Applied rewrites 31.7%

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

   8. Taylor expanded in c around -inf

      \[\leadsto \sqrt{\frac{c}{a}} \cdot \color{blue}{\sqrt{-1}} \]
   9. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{-1 \cdot \frac{c}{a}} \]

      4. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{-1 \cdot c}{a}} \]

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 52.1

         \[\leadsto \sqrt{\frac{-c}{a}} \]

   10. Applied rewrites 52.1%

       \[\leadsto \sqrt{\frac{-c}{a}} \]

   if 2.19999999999999996e-93 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 87.7

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

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-145}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.6e-145)
   (/ (- b) a)
   (if (<= b 3.4e-53) (/ (sqrt (* a (- c))) a) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.6e-145) {
		tmp = -b / a;
	} else if (b <= 3.4e-53) {
		tmp = sqrt((a * -c)) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b <= (-5.6d-145)) then
        tmp = -b / a
    else if (b <= 3.4d-53) then
        tmp = sqrt((a * -c)) / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.6e-145) {
		tmp = -b / a;
	} else if (b <= 3.4e-53) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.6e-145:
		tmp = -b / a
	elif b <= 3.4e-53:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.6e-145)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 3.4e-53)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.6e-145)
		tmp = -b / a;
	elseif (b <= 3.4e-53)
		tmp = sqrt((a * -c)) / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.6e-145], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 3.4e-53], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.6000000000000002e-145

   1. Initial program 73.7%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.8

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

   5. Applied rewrites 84.8%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -5.6000000000000002e-145 < b < 3.4e-53

   1. Initial program 70.3%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-unprod N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 40.2

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right) \]

   5. Applied rewrites 40.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{c}{a} \cdot -4}, 0.5, \frac{b}{a} \cdot -0.5\right)} \]

   6. Taylor expanded in a around -inf

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.2

         \[\leadsto \mathsf{fma}\left(-0.5, \frac{b}{a}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   8. Applied rewrites 40.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\frac{b}{a}}, \sqrt{\frac{c}{a} \cdot -1}\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{-1}{2} \cdot b + \sqrt{a \cdot c} \cdot \sqrt{-1}}{a} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. sqrt-unprod N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 70.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{a} \]

   11. Applied rewrites 70.4%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, b, \sqrt{\left(c \cdot a\right) \cdot -1}\right)}{a} \]

   12. Taylor expanded in a around inf

       \[\leadsto \frac{\sqrt{a \cdot c} \cdot \sqrt{-1}}{a} \]
   13. Step-by-step derivation

       1. sqrt-unprod N/A

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

       2. *-commutative N/A

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

       3. mul-1-neg N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-neg.f64 N/A

          \[\leadsto \frac{\sqrt{-a \cdot c}}{a} \]

       6. lower-*.f64 69.8

          \[\leadsto \frac{\sqrt{-a \cdot c}}{a} \]

   14. Applied rewrites 69.8%

       \[\leadsto \frac{\sqrt{-a \cdot c}}{a} \]

   if 3.4e-53 < b

   1. Initial program 11.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 90.7

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

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.6 \cdot 10^{-145}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.72 \cdot 10^{-152}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.72e-152)
   (/ (- b) a)
   (if (<= b 2.2e-93) (sqrt (/ (- c) a)) (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.72e-152) {
		tmp = -b / a;
	} else if (b <= 2.2e-93) {
		tmp = sqrt((-c / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b <= (-1.72d-152)) then
        tmp = -b / a
    else if (b <= 2.2d-93) then
        tmp = sqrt((-c / a))
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.72e-152) {
		tmp = -b / a;
	} else if (b <= 2.2e-93) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.72e-152:
		tmp = -b / a
	elif b <= 2.2e-93:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.72e-152)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2.2e-93)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.72e-152)
		tmp = -b / a;
	elseif (b <= 2.2e-93)
		tmp = sqrt((-c / a));
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.72e-152], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2.2e-93], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.72e-152

   1. Initial program 73.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 84.1

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

   5. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -1.72e-152 < b < 2.19999999999999996e-93

   1. Initial program 69.5%

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

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

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in a around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a}} \cdot \sqrt{-1} \]

      3. sqrt-unprod N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      5. lower-*.f64 N/A

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

      6. lower-/.f64 37.0

         \[\leadsto -\sqrt{\frac{c}{a} \cdot -1} \]

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in c around -inf

      \[\leadsto \sqrt{\frac{c}{a}} \cdot \color{blue}{\sqrt{-1}} \]
   9. Step-by-step derivation

      1. sqrt-prod N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{c}{a} \cdot -1} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{-1 \cdot \frac{c}{a}} \]

      4. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{-1 \cdot c}{a}} \]

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 41.6

         \[\leadsto \sqrt{\frac{-c}{a}} \]

   10. Applied rewrites 41.6%

       \[\leadsto \sqrt{\frac{-c}{a}} \]

   if 2.19999999999999996e-93 < b

   1. Initial program 15.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 87.7

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

   5. Applied rewrites 87.7%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{-257}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.4e-257) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-257) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (b <= 1.4d-257) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.4e-257) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.4e-257:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.4e-257)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.4e-257)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.4e-257], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.40000000000000001e-257

   1. Initial program 73.8%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 67.0

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

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if 1.40000000000000001e-257 < b

   1. Initial program 30.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 66.3

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

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 34.7% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around -inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lower-neg.f64 37.3

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

5. Applied rewrites 37.3%

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

Reproduce

herbie shell --seed 2025044 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))