The quadratic formula (r1)

Percentage Accurate: 52.6% → 86.2%

Time: 4.2s

Alternatives: 11

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: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e+74)
   (fma (/ b a) -1.0 (/ c b))
   (if (<= b 5.2e-97)
     (/ (+ (- b) (pow (fma (* -4.0 a) c (* b b)) 0.5)) (* 2.0 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e+74) {
		tmp = fma((b / a), -1.0, (c / b));
	} else if (b <= 5.2e-97) {
		tmp = (-b + pow(fma((-4.0 * a), c, (b * b)), 0.5)) / (2.0 * a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.12e+74)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	elseif (b <= 5.2e-97)
		tmp = Float64(Float64(Float64(-b) + (fma(Float64(-4.0 * a), c, Float64(b * b)) ^ 0.5)) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e+74], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-97], N[(N[((-b) + N[Power[N[(N[(-4.0 * a), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.12000000000000003e74

   1. Initial program 65.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 65.7

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 65.7%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if -1.12000000000000003e74 < b < 5.20000000000000014e-97

   1. Initial program 75.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 75.8

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 75.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   if 5.20000000000000014e-97 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in 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 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e+74)
   (fma (/ b a) -1.0 (/ c b))
   (if (<= b 5.2e-97)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e+74) {
		tmp = fma((b / a), -1.0, (c / b));
	} else if (b <= 5.2e-97) {
		tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.12e+74)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	elseif (b <= 5.2e-97)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e+74], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-97], 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], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.12000000000000003e74

   1. Initial program 65.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 65.7

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 65.7%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if -1.12000000000000003e74 < b < 5.20000000000000014e-97

   1. Initial program 75.8%

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

   if 5.20000000000000014e-97 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in 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 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e+74)
   (fma (/ b a) -1.0 (/ c b))
   (if (<= b 5.2e-97)
     (/ (- (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 <= -1.12e+74) {
		tmp = fma((b / a), -1.0, (c / b));
	} else if (b <= 5.2e-97) {
		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 <= -1.12e+74)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	elseif (b <= 5.2e-97)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * a), c, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e+74], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-97], 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 < -1.12000000000000003e74

   1. Initial program 65.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 65.7

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 65.7%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.3

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

   10. Applied rewrites 98.3%

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

   if -1.12000000000000003e74 < b < 5.20000000000000014e-97

   1. Initial program 75.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 75.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 75.8

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

   6. Applied rewrites 75.8%

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

   if 5.20000000000000014e-97 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in 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 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.5e-43)
   (fma (/ b a) -1.0 (/ c b))
   (if (<= b 5.2e-97) (/ (- (sqrt (* (* a -4.0) c)) b) (+ a a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.5e-43) {
		tmp = fma((b / a), -1.0, (c / b));
	} else if (b <= 5.2e-97) {
		tmp = (sqrt(((a * -4.0) * c)) - b) / (a + a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.5e-43)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	elseif (b <= 5.2e-97)
		tmp = Float64(Float64(sqrt(Float64(Float64(a * -4.0) * c)) - b) / Float64(a + a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.5e-43], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-97], N[(N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.50000000000000068e-43

   1. Initial program 74.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 74.5

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 74.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 93.8

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 94.6

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

   10. Applied rewrites 94.6%

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

   if -7.50000000000000068e-43 < b < 5.20000000000000014e-97

   1. Initial program 69.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 69.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 69.6

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

   6. Applied rewrites 69.6%

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

   7. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 62.9

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

   9. Applied rewrites 62.9%

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

   if 5.20000000000000014e-97 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in 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 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -9e-99)
   (fma (/ b a) -1.0 (/ c b))
   (if (<= b 5.2e-97) (/ (sqrt (* (* a -4.0) c)) (+ a a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -9e-99) {
		tmp = fma((b / a), -1.0, (c / b));
	} else if (b <= 5.2e-97) {
		tmp = sqrt(((a * -4.0) * c)) / (a + a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -9e-99)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	elseif (b <= 5.2e-97)
		tmp = Float64(sqrt(Float64(Float64(a * -4.0) * c)) / Float64(a + a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -9e-99], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-97], N[(N[Sqrt[N[(N[(a * -4.0), $MachinePrecision] * c), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.0000000000000006e-99

   1. Initial program 76.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 76.6

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 76.6%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 90.5

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

   7. Applied rewrites 90.5%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if -9.0000000000000006e-99 < b < 5.20000000000000014e-97

   1. Initial program 66.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 66.5

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 66.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot c} \cdot \sqrt{-4}}}{2 \cdot a} \]
   6. Step-by-step derivation

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 62.8

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

   9. Applied rewrites 62.8%

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

   if 5.20000000000000014e-97 < b

   1. Initial program 15.7%

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

   3. Taylor expanded in 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 88.0

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

   5. Applied rewrites 88.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\sqrt{\left(a \cdot -4\right) \cdot c}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-213}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-204)
   (fma (/ b a) -1.0 (/ c b))
   (if (<= b 2e-213) (- (sqrt (/ (- c) a))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-204) {
		tmp = fma((b / a), -1.0, (c / b));
	} else if (b <= 2e-213) {
		tmp = -sqrt((-c / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-204)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	elseif (b <= 2e-213)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-204], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e-213], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999997e-204

   1. Initial program 77.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 77.6

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 77.6%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 80.3

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

   7. Applied rewrites 80.3%

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

   8. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 81.9

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

   10. Applied rewrites 81.9%

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

   if -5.9999999999999997e-204 < b < 1.9999999999999999e-213

   1. Initial program 65.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 65.1%

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

   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. +-commutative N/A

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

      2. pow1/2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. *-commutative N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 46.9

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

   7. Applied rewrites 46.9%

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

   if 1.9999999999999999e-213 < b

   1. Initial program 24.3%

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

   3. Taylor expanded in a around 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 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-213}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-204}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-213}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6e-204)
   (/ (- b) a)
   (if (<= b 2e-213) (- (sqrt (/ (- c) a))) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6e-204) {
		tmp = -b / a;
	} else if (b <= 2e-213) {
		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 <= (-6d-204)) then
        tmp = -b / a
    else if (b <= 2d-213) 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 <= -6e-204) {
		tmp = -b / a;
	} else if (b <= 2e-213) {
		tmp = -Math.sqrt((-c / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6e-204:
		tmp = -b / a
	elif b <= 2e-213:
		tmp = -math.sqrt((-c / a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6e-204)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 2e-213)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6e-204)
		tmp = -b / a;
	elseif (b <= 2e-213)
		tmp = -sqrt((-c / a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6e-204], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 2e-213], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.9999999999999997e-204

   1. Initial program 77.6%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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. lift-neg.f64 81.1

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

   5. Applied rewrites 81.1%

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

   if -5.9999999999999997e-204 < b < 1.9999999999999999e-213

   1. Initial program 65.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 65.1%

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

   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. +-commutative N/A

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

      2. pow1/2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. *-commutative N/A

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

      7. lower-sqrt.f64 N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lift-neg.f64 46.9

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

   7. Applied rewrites 46.9%

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

   if 1.9999999999999999e-213 < b

   1. Initial program 24.3%

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

   3. Taylor expanded in a around 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 77.6

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

   5. Applied rewrites 77.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{-204}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{-213}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-220}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.7e-115)
   (/ (- b) a)
   (if (<= b 1e-220) (sqrt (/ (- c) a)) (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.7e-115) {
		tmp = -b / a;
	} else if (b <= 1e-220) {
		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 <= (-3.7d-115)) then
        tmp = -b / a
    else if (b <= 1d-220) 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 <= -3.7e-115) {
		tmp = -b / a;
	} else if (b <= 1e-220) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.7e-115:
		tmp = -b / a
	elif b <= 1e-220:
		tmp = math.sqrt((-c / a))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.7e-115)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1e-220)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.7e-115)
		tmp = -b / a;
	elseif (b <= 1e-220)
		tmp = sqrt((-c / a));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.7e-115], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1e-220], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.7e-115

   1. Initial program 77.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   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. lift-neg.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -3.7e-115 < b < 9.99999999999999992e-221

   1. Initial program 69.1%

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

   3. Taylor expanded in a around 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 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 41.7

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

   8. Applied rewrites 41.7%

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

      1. count-2-rev 41.7

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. mul-1-neg N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 41.7

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

   10. Applied rewrites 41.7%

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

   if 9.99999999999999992e-221 < b

   1. Initial program 24.8%

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

   3. Taylor expanded in 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 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.7 \cdot 10^{-115}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 10^{-220}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-275}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.1e-275) (/ (- b) a) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1e-275) {
		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 <= 1.1d-275) 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 <= 1.1e-275) {
		tmp = -b / a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.1e-275:
		tmp = -b / a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1e-275)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.1e-275)
		tmp = -b / a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1e-275], N[((-b) / a), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.09999999999999994e-275

   1. Initial program 75.5%

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

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   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. lift-neg.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 1.09999999999999994e-275 < b

   1. Initial program 28.5%

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

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   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 69.8

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

   5. Applied rewrites 69.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-275}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.3 \cdot 10^{+116}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.3e+116) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.3e+116) {
		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 <= 4.3d+116) 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 <= 4.3e+116) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.3e+116:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.3e+116)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.3e+116)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.3e+116], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3e116

   1. Initial program 66.3%

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

   3. Taylor expanded in 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. lift-neg.f64 46.7

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

   5. Applied rewrites 46.7%

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

   if 4.3e116 < b

   1. Initial program 4.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. lower-pow.f64 N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 4.3

          \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

   4. Applied rewrites 4.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

   5. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 2.5

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

   7. Applied rewrites 2.5%

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

   8. Taylor expanded in a around inf

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

      1. lower-/.f64 28.7

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

   10. Applied rewrites 28.7%

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

Alternative 11: 10.6% accurate, 4.2× 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(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 54.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. lower-pow.f64 N/A

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

   8. pow2 N/A

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

   9. associate-*r* N/A

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

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

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

   11. metadata-eval N/A

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

   12. +-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. pow2 N/A

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

   17. lift-*.f64 54.5

       \[\leadsto \frac{\left(-b\right) + {\left(\mathsf{fma}\left(-4 \cdot a, c, \color{blue}{b \cdot b}\right)\right)}^{0.5}}{2 \cdot a} \]

4. Applied rewrites 54.5%

   \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left(\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}^{0.5}}}{2 \cdot a} \]

5. Taylor expanded in b around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. pow2 N/A

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

   9. lift-*.f64 N/A

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

   10. inv-pow N/A

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

   11. lower-pow.f64 37.4

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

7. Applied rewrites 37.4%

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

8. Taylor expanded in a around inf

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

   1. lower-/.f64 7.9

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

10. Applied rewrites 7.9%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform default (let ((d (- (* b b) (* (* 4 a) c)))) (let ((r1 (/ (+ (- b) (sqrt d)) (* 2 a)))) (let ((r2 (/ (- (- b) (sqrt d)) (* 2 a)))) (if (< b 0) r1 (/ c (* a r2)))))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))