The quadratic formula (r1)

Percentage Accurate: 52.1% → 85.4%

Time: 5.1s

Alternatives: 9

Speedup: 2.2×

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: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}, a, b \cdot -0.5\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+120)
   (* -1.0 (/ b a))
   (if (<= b 3.9e-64)
     (/ (fma (/ (sqrt (fma (* c -4.0) a (* b b))) (+ a a)) a (* b -0.5)) a)
     (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+120) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.9e-64) {
		tmp = fma((sqrt(fma((c * -4.0), a, (b * b))) / (a + a)), a, (b * -0.5)) / a;
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+120)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 3.9e-64)
		tmp = Float64(fma(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) / Float64(a + a)), a, Float64(b * -0.5)) / a);
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e+120], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-64], N[(N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision] * a + N[(b * -0.5), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.50000000000000007e120

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -3.50000000000000007e120 < b < 3.8999999999999997e-64

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. add-to-fraction N/A

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

      8. lower-/.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 50.0

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

   5. Applied rewrites 50.0%

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

   if 3.8999999999999997e-64 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.5}{a}, \frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+120)
   (* -1.0 (/ b a))
   (if (<= b 3.9e-64)
     (fma b (/ -0.5 a) (/ (sqrt (fma (* c -4.0) a (* b b))) (+ a a)))
     (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+120) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.9e-64) {
		tmp = fma(b, (-0.5 / a), (sqrt(fma((c * -4.0), a, (b * b))) / (a + a)));
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+120)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 3.9e-64)
		tmp = fma(b, Float64(-0.5 / a), Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) / Float64(a + a)));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e+120], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-64], N[(b * N[(-0.5 / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.50000000000000007e120

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -3.50000000000000007e120 < b < 3.8999999999999997e-64

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. frac-2neg N/A

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

      5. mult-flip N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 50.0

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

   3. Applied rewrites 50.0%

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

   if 3.8999999999999997e-64 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+120}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}{a + a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+120)
   (* -1.0 (/ b a))
   (if (<= b 3.9e-64)
     (/ (- (sqrt (fma (* c -4.0) a (* b b))) b) (+ a a))
     (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+120) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.9e-64) {
		tmp = (sqrt(fma((c * -4.0), a, (b * b))) - b) / (a + a);
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+120)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 3.9e-64)
		tmp = Float64(Float64(sqrt(fma(Float64(c * -4.0), a, Float64(b * b))) - b) / Float64(a + a));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e+120], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-64], N[(N[(N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.50000000000000007e120

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -3.50000000000000007e120 < b < 3.8999999999999997e-64

   1. Initial program 52.1%

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

   3. Applied rewrites 52.1%

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

   if 3.8999999999999997e-64 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-115}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, b, 0.5 \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-115)
   (* -1.0 (/ b a))
   (if (<= b 3.9e-64)
     (/ (fma -0.5 b (* 0.5 (sqrt (* -4.0 (* a c))))) a)
     (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-115) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.9e-64) {
		tmp = fma(-0.5, b, (0.5 * sqrt((-4.0 * (a * c))))) / a;
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-115)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 3.9e-64)
		tmp = Float64(fma(-0.5, b, Float64(0.5 * sqrt(Float64(-4.0 * Float64(a * c))))) / a);
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-115], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-64], N[(N[(-0.5 * b + N[(0.5 * N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.0000000000000001e-115

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -2.0000000000000001e-115 < b < 3.8999999999999997e-64

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. add-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{b}{-2 \cdot a}, a, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot 0.5\right)}{a}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 47.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 47.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 33.0

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

   8. Applied rewrites 33.0%

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

   if 3.8999999999999997e-64 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{-116}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-64}:\\ \;\;\;\;\frac{0.5 \cdot \sqrt{-4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.7e-116)
   (* -1.0 (/ b a))
   (if (<= b 3.9e-64) (/ (* 0.5 (sqrt (* -4.0 (* a c)))) a) (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.7e-116) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.9e-64) {
		tmp = (0.5 * sqrt((-4.0 * (a * c)))) / a;
	} else {
		tmp = -1.0 * (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 <= (-4.7d-116)) then
        tmp = (-1.0d0) * (b / a)
    else if (b <= 3.9d-64) then
        tmp = (0.5d0 * sqrt(((-4.0d0) * (a * c)))) / a
    else
        tmp = (-1.0d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.7e-116) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.9e-64) {
		tmp = (0.5 * Math.sqrt((-4.0 * (a * c)))) / a;
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.7e-116:
		tmp = -1.0 * (b / a)
	elif b <= 3.9e-64:
		tmp = (0.5 * math.sqrt((-4.0 * (a * c)))) / a
	else:
		tmp = -1.0 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.7e-116)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 3.9e-64)
		tmp = Float64(Float64(0.5 * sqrt(Float64(-4.0 * Float64(a * c)))) / a);
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4.7e-116)
		tmp = -1.0 * (b / a);
	elseif (b <= 3.9e-64)
		tmp = (0.5 * sqrt((-4.0 * (a * c)))) / a;
	else
		tmp = -1.0 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.7e-116], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-64], N[(N[(0.5 * N[Sqrt[N[(-4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.69999999999999994e-116

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -4.69999999999999994e-116 < b < 3.8999999999999997e-64

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. add-to-fraction N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 49.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{b}{-2 \cdot a}, a, \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} \cdot 0.5\right)}{a}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. mult-flip N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 47.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 47.4

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 28.6

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

   8. Applied rewrites 28.6%

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

   if 3.8999999999999997e-64 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3 \cdot 10^{-138}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-116}:\\ \;\;\;\;0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3e-138)
   (* -1.0 (/ b a))
   (if (<= b 3.1e-116) (* 0.5 (sqrt (* -4.0 (/ c a)))) (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-138) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.1e-116) {
		tmp = 0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (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 <= (-3d-138)) then
        tmp = (-1.0d0) * (b / a)
    else if (b <= 3.1d-116) then
        tmp = 0.5d0 * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = (-1.0d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3e-138) {
		tmp = -1.0 * (b / a);
	} else if (b <= 3.1e-116) {
		tmp = 0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3e-138:
		tmp = -1.0 * (b / a)
	elif b <= 3.1e-116:
		tmp = 0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -1.0 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3e-138)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 3.1e-116)
		tmp = Float64(0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3e-138)
		tmp = -1.0 * (b / a);
	elseif (b <= 3.1e-116)
		tmp = 0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -1.0 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3e-138], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e-116], N[(0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.0000000000000001e-138

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -3.0000000000000001e-138 < b < 3.10000000000000018e-116

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 17.3

         \[\leadsto 0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   8. Applied rewrites 17.3%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 3.10000000000000018e-116 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.9 \cdot 10^{-118}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-110}:\\ \;\;\;\;-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.9e-118)
   (* -1.0 (/ b a))
   (if (<= b 2.7e-110) (* -0.5 (sqrt (* -4.0 (/ c a)))) (* -1.0 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.9e-118) {
		tmp = -1.0 * (b / a);
	} else if (b <= 2.7e-110) {
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (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 <= (-6.9d-118)) then
        tmp = (-1.0d0) * (b / a)
    else if (b <= 2.7d-110) then
        tmp = (-0.5d0) * sqrt(((-4.0d0) * (c / a)))
    else
        tmp = (-1.0d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.9e-118) {
		tmp = -1.0 * (b / a);
	} else if (b <= 2.7e-110) {
		tmp = -0.5 * Math.sqrt((-4.0 * (c / a)));
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.9e-118:
		tmp = -1.0 * (b / a)
	elif b <= 2.7e-110:
		tmp = -0.5 * math.sqrt((-4.0 * (c / a)))
	else:
		tmp = -1.0 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.9e-118)
		tmp = Float64(-1.0 * Float64(b / a));
	elseif (b <= 2.7e-110)
		tmp = Float64(-0.5 * sqrt(Float64(-4.0 * Float64(c / a))));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.9e-118)
		tmp = -1.0 * (b / a);
	elseif (b <= 2.7e-110)
		tmp = -0.5 * sqrt((-4.0 * (c / a)));
	else
		tmp = -1.0 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.9e-118], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e-110], N[(-0.5 * N[Sqrt[N[(-4.0 * N[(c / a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.9000000000000002e-118

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -6.9000000000000002e-118 < b < 2.6999999999999998e-110

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in a around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\sqrt{-4 \cdot \frac{c}{a}}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

      4. lower-/.f64 17.0

         \[\leadsto -0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}} \]

   8. Applied rewrites 17.0%

      \[\leadsto \color{blue}{-0.5 \cdot \sqrt{-4 \cdot \frac{c}{a}}} \]

   if 2.6999999999999998e-110 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* -1.0 (/ b a)) (* -1.0 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = -1.0 * (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 <= (-5d-310)) then
        tmp = (-1.0d0) * (b / a)
    else
        tmp = (-1.0d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -1.0 * (b / a);
	} else {
		tmp = -1.0 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -1.0 * (b / a)
	else:
		tmp = -1.0 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-1.0 * Float64(b / a));
	else
		tmp = Float64(-1.0 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = -1.0 * (b / a);
	else
		tmp = -1.0 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-1.0 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   8. Applied rewrites 36.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 52.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 52.0

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

      5. lift-*.f64 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f64 52.0

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. add-flip N/A

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

      11. lower--.f64 N/A

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

   3. Applied rewrites 52.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. count-2 N/A

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

      8. associate-/r* N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 52.0

          \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

   6. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 34.6

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

   8. Applied rewrites 34.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 36.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{b}{a} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-1.0d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -1.0 * (b / a);
}

Python

def code(a, b, c):
	return -1.0 * (b / a)

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = -1.0 * (b / a);
end

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f64 N/A

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

   4. lower-unsound-/.f64 52.0

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

   5. lift-*.f64 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f64 52.0

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. add-flip N/A

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

   11. lower--.f64 N/A

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

3. Applied rewrites 52.1%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. mult-flip N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. count-2 N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 52.0

       \[\leadsto \frac{\frac{0.5}{a}}{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)} - b}}} \]

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 52.0

       \[\leadsto \frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{-4 \cdot c}, a, b \cdot b\right)} - b}} \]

5. Applied rewrites 52.0%

   \[\leadsto \color{blue}{\frac{\frac{0.5}{a}}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}} \]

6. Taylor expanded in b around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 36.1

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

8. Applied rewrites 36.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
9. Add Preprocessing

Developer Target 1: 70.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - t\_0}{2 \cdot a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (< b 0.0)
     (/ (+ (- b) t_0) (* 2.0 a))
     (/ c (* a (/ (- (- b) t_0) (* 2.0 a)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b < 0.0d0) then
        tmp = (-b + t_0) / (2.0d0 * a)
    else
        tmp = c / (a * ((-b - t_0) / (2.0d0 * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b < 0.0) {
		tmp = (-b + t_0) / (2.0 * a);
	} else {
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b < 0.0:
		tmp = (-b + t_0) / (2.0 * a)
	else:
		tmp = c / (a * ((-b - t_0) / (2.0 * a)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b < 0.0)
		tmp = (-b + t_0) / (2.0 * a);
	else
		tmp = c / (a * ((-b - t_0) / (2.0 * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(c / N[(a * N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025164 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :alt
  (! :herbie-platform c (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)))