The quadratic formula (r2)

Percentage Accurate: 52.1% → 89.2%

Time: 7.5s

Alternatives: 9

Speedup: 2.5×

Specification

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

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

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(4.0 * Float64(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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-174}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 10^{+33}:\\ \;\;\;\;\frac{b}{-2 \cdot a} - \frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+40)
   (/ c (- b))
   (if (<= b 1.8e-174)
     (/ (* 2.0 c) (+ (- b) (sqrt (fma -4.0 (* c a) (* b b)))))
     (if (<= b 1e+33)
       (- (/ b (* -2.0 a)) (/ (sqrt (fma (* a c) -4.0 (* b b))) (* 2.0 a)))
       (fma (/ (fma (/ a b) (/ c b) 1.0) b) c (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+40) {
		tmp = c / -b;
	} else if (b <= 1.8e-174) {
		tmp = (2.0 * c) / (-b + sqrt(fma(-4.0, (c * a), (b * b))));
	} else if (b <= 1e+33) {
		tmp = (b / (-2.0 * a)) - (sqrt(fma((a * c), -4.0, (b * b))) / (2.0 * a));
	} else {
		tmp = fma((fma((a / b), (c / b), 1.0) / b), c, (-b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+40)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.8e-174)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))));
	elseif (b <= 1e+33)
		tmp = Float64(Float64(b / Float64(-2.0 * a)) - Float64(sqrt(fma(Float64(a * c), -4.0, Float64(b * b))) / Float64(2.0 * a)));
	else
		tmp = fma(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / b), c, Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+40], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.8e-174], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+33], N[(N[(b / N[(-2.0 * a), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[N[(N[(a * c), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.00000000000000024e40

   1. Initial program 9.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -8.00000000000000024e40 < b < 1.79999999999999999e-174

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 66.3

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

   7. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if 1.79999999999999999e-174 < b < 9.9999999999999995e32

   1. Initial program 89.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 89.4

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

   4. Applied rewrites 89.4%

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

   6. Applied rewrites 89.5%

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

   if 9.9999999999999995e32 < b

   1. Initial program 62.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + c \cdot \left(\frac{1}{b} + \frac{a \cdot c}{{b}^{3}}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 2: 89.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-273}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 10^{+33}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{a}{b}, \frac{c}{b}, 1\right)}{b}, c, \frac{-b}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+40)
   (/ c (- b))
   (if (<= b -1.25e-273)
     (/ (* 2.0 c) (+ (- b) (sqrt (fma -4.0 (* c a) (* b b)))))
     (if (<= b 1e+33)
       (/ (- (- b) (sqrt (fma (* -4.0 c) a (* b b)))) (* 2.0 a))
       (fma (/ (fma (/ a b) (/ c b) 1.0) b) c (/ (- b) a))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+40) {
		tmp = c / -b;
	} else if (b <= -1.25e-273) {
		tmp = (2.0 * c) / (-b + sqrt(fma(-4.0, (c * a), (b * b))));
	} else if (b <= 1e+33) {
		tmp = (-b - sqrt(fma((-4.0 * c), a, (b * b)))) / (2.0 * a);
	} else {
		tmp = fma((fma((a / b), (c / b), 1.0) / b), c, (-b / a));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+40)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.25e-273)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))));
	elseif (b <= 1e+33)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) / Float64(2.0 * a));
	else
		tmp = fma(Float64(fma(Float64(a / b), Float64(c / b), 1.0) / b), c, Float64(Float64(-b) / a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+40], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.25e-273], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+33], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a / b), $MachinePrecision] * N[(c / b), $MachinePrecision] + 1.0), $MachinePrecision] / b), $MachinePrecision] * c + N[((-b) / a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.00000000000000024e40

   1. Initial program 9.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -8.00000000000000024e40 < b < -1.24999999999999991e-273

   1. Initial program 47.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 40.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 78.2%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 86.1

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

   12. Applied rewrites 86.1%

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

   if -1.24999999999999991e-273 < b < 9.9999999999999995e32

   1. Initial program 83.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 83.9

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

   4. Applied rewrites 83.9%

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

   if 9.9999999999999995e32 < b

   1. Initial program 62.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + c \cdot \left(\frac{1}{b} + \frac{a \cdot c}{{b}^{3}}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow3 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. lower-/.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-neg.f64 98.7

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

   5. Applied rewrites 98.7%

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

Alternative 3: 89.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq -1.25 \cdot 10^{-273}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{elif}\;b \leq 13500000000000:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+40)
   (/ c (- b))
   (if (<= b -1.25e-273)
     (/ (* 2.0 c) (+ (- b) (sqrt (fma -4.0 (* c a) (* b b)))))
     (if (<= b 13500000000000.0)
       (/ (- (- b) (sqrt (fma (* -4.0 c) a (* b b)))) (* 2.0 a))
       (+ (/ (- b) a) (/ c b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+40) {
		tmp = c / -b;
	} else if (b <= -1.25e-273) {
		tmp = (2.0 * c) / (-b + sqrt(fma(-4.0, (c * a), (b * b))));
	} else if (b <= 13500000000000.0) {
		tmp = (-b - sqrt(fma((-4.0 * c), a, (b * b)))) / (2.0 * a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+40)
		tmp = Float64(c / Float64(-b));
	elseif (b <= -1.25e-273)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))));
	elseif (b <= 13500000000000.0)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(Float64(-4.0 * c), a, Float64(b * b)))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+40], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, -1.25e-273], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 13500000000000.0], N[(N[((-b) - N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -8.00000000000000024e40

   1. Initial program 9.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -8.00000000000000024e40 < b < -1.24999999999999991e-273

   1. Initial program 47.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 40.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 78.2%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 86.1

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

   12. Applied rewrites 86.1%

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

   if -1.24999999999999991e-273 < b < 1.35e13

   1. Initial program 84.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 84.5

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

   4. Applied rewrites 84.5%

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

   if 1.35e13 < b

   1. Initial program 63.0%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.4

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

   5. Applied rewrites 97.4%

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

   7. Applied rewrites 97.4%

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

Alternative 4: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{+40}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-163}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8e+40)
   (/ c (- b))
   (if (<= b 2.3e-163)
     (/ (* 2.0 c) (+ (- b) (sqrt (fma -4.0 (* c a) (* b b)))))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8e+40) {
		tmp = c / -b;
	} else if (b <= 2.3e-163) {
		tmp = (2.0 * c) / (-b + sqrt(fma(-4.0, (c * a), (b * b))));
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8e+40)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.3e-163)
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + sqrt(fma(-4.0, Float64(c * a), Float64(b * b)))));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8e+40], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.3e-163], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.00000000000000024e40

   1. Initial program 9.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -8.00000000000000024e40 < b < 2.2999999999999999e-163

   1. Initial program 53.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 66.3

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

   7. Applied rewrites 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in a around 0

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

       1. lower-*.f64 82.8

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

   12. Applied rewrites 82.8%

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

   if 2.2999999999999999e-163 < b

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.1

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

   5. Applied rewrites 88.1%

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

   7. Applied rewrites 88.1%

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

Alternative 5: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e-62)
   (/ c (- b))
   (if (<= b 1.7e-174)
     (/ (+ b (sqrt (* (* c a) -4.0))) (* 2.0 (- a)))
     (+ (/ (- b) a) (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-62) {
		tmp = c / -b;
	} else if (b <= 1.7e-174) {
		tmp = (b + sqrt(((c * a) * -4.0))) / (2.0 * -a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.85d-62)) then
        tmp = c / -b
    else if (b <= 1.7d-174) then
        tmp = (b + sqrt(((c * a) * (-4.0d0)))) / (2.0d0 * -a)
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e-62) {
		tmp = c / -b;
	} else if (b <= 1.7e-174) {
		tmp = (b + Math.sqrt(((c * a) * -4.0))) / (2.0 * -a);
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.85e-62:
		tmp = c / -b
	elif b <= 1.7e-174:
		tmp = (b + math.sqrt(((c * a) * -4.0))) / (2.0 * -a)
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e-62)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 1.7e-174)
		tmp = Float64(Float64(b + sqrt(Float64(Float64(c * a) * -4.0))) / Float64(2.0 * Float64(-a)));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.85e-62)
		tmp = c / -b;
	elseif (b <= 1.7e-174)
		tmp = (b + sqrt(((c * a) * -4.0))) / (2.0 * -a);
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e-62], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 1.7e-174], N[(N[(b + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * (-a)), $MachinePrecision]), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8499999999999999e-62

   1. Initial program 11.6%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if -1.8499999999999999e-62 < b < 1.7000000000000001e-174

   1. Initial program 68.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 1.7000000000000001e-174 < b

   1. Initial program 70.4%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

   7. Applied rewrites 87.4%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\frac{b + \sqrt{\left(c \cdot a\right) \cdot -4}}{2 \cdot \left(-a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a} + \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ c (- b)) (+ (/ (- b) a) (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = c / -b;
	} else {
		tmp = (-b / a) + (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 <= (-1d-310)) then
        tmp = c / -b
    else
        tmp = (-b / a) + (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = c / -b;
	} else {
		tmp = (-b / a) + (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = c / -b
	else:
		tmp = (-b / a) + (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(Float64(-b) / a) + Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = c / -b;
	else
		tmp = (-b / a) + (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(c / (-b)), $MachinePrecision], N[(N[((-b) / a), $MachinePrecision] + N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 28.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if -9.999999999999969e-311 < b

   1. Initial program 70.5%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 78.2%

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

Alternative 7: 68.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-308}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.05e-308) (/ c (- b)) (/ (- b) a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-308) {
		tmp = c / -b;
	} else {
		tmp = -b / 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) :: tmp
    if (b <= (-1.05d-308)) then
        tmp = c / -b
    else
        tmp = -b / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.05e-308) {
		tmp = c / -b;
	} else {
		tmp = -b / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.05e-308:
		tmp = c / -b
	else:
		tmp = -b / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.05e-308)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(-b) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.05e-308)
		tmp = c / -b;
	else
		tmp = -b / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.05e-308], N[(c / (-b)), $MachinePrecision], N[((-b) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.0499999999999998e-308

   1. Initial program 28.1%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 70.5

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

   5. Applied rewrites 70.5%

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

   if -1.0499999999999998e-308 < b

   1. Initial program 70.5%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 77.9

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

   5. Applied rewrites 77.9%

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

Alternative 8: 35.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in b around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. lower-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 36.8

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

5. Applied rewrites 36.8%

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

Alternative 9: 10.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in c around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 39.9

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

5. Applied rewrites 39.9%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 10.4%

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

Developer Target 1: 70.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\\ \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + t\_0}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* 4.0 (* a c))))))
   (if (< b 0.0)
     (/ c (* a (/ (+ (- b) t_0) (* 2.0 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0d0 * a)))
    else
        tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)));
	} else {
		tmp = (-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 = c / (a * ((-b + t_0) / (2.0 * a)))
	else:
		tmp = (-b - t_0) / (2.0 * a)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))
	tmp = 0.0
	if (b < 0.0)
		tmp = Float64(c / Float64(a * Float64(Float64(Float64(-b) + t_0) / Float64(2.0 * a))));
	else
		tmp = 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 = c / (a * ((-b + t_0) / (2.0 * a)));
	else
		tmp = (-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[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Less[b, 0.0], N[(c / N[(a * N[(N[((-b) + t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025007 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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