Quadratic roots, full range

Percentage Accurate: 52.7% → 87.3%

Time: 8.1s

Alternatives: 8

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+164}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e+164)
   (/ (- b) a)
   (if (<= b 1.05e-156)
     (/ (+ (- b) (sqrt (fma (* c -4.0) a (* b b)))) (+ a a))
     (if (<= b 4.8e+36)
       (/
        (/ (fma (* -4.0 c) a 0.0) (+ (sqrt (fma (* -4.0 c) a (* b b))) b))
        (+ a a))
       (/ (- c) b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e+164) {
		tmp = -b / a;
	} else if (b <= 1.05e-156) {
		tmp = (-b + sqrt(fma((c * -4.0), a, (b * b)))) / (a + a);
	} else if (b <= 4.8e+36) {
		tmp = (fma((-4.0 * c), a, 0.0) / (sqrt(fma((-4.0 * c), a, (b * b))) + b)) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.8e+164)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.05e-156)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) / Float64(a + a));
	elseif (b <= 4.8e+36)
		tmp = Float64(Float64(fma(Float64(-4.0 * c), a, 0.0) / Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b)) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.8e+164], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.05e-156], N[(N[((-b) + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.8e+36], N[(N[(N[(N[(-4.0 * c), $MachinePrecision] * a + 0.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.80000000000000021e164

   1. Initial program 45.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.80000000000000021e164 < b < 1.05000000000000006e-156

   1. Initial program 84.8%

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

   4. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 84.8

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

   6. Applied rewrites 84.8%

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

   if 1.05000000000000006e-156 < b < 4.79999999999999985e36

   1. Initial program 43.8%

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

   4. Applied rewrites 43.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 43.8

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

   6. Applied rewrites 43.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

   8. Applied rewrites 74.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - \left(-b\right)}}}{a + a} \]

   if 4.79999999999999985e36 < b

   1. Initial program 16.3%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 96.0

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

   5. Applied rewrites 96.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+164}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-156}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-4 \cdot c, a, 0\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{+164}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(c \cdot -4, a, b \cdot b\right)}}{a + a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.8e+164)
   (/ (- b) a)
   (if (<= b 1.25e-42)
     (/ (+ (- b) (sqrt (fma (* c -4.0) a (* b b)))) (+ a a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.8e+164) {
		tmp = -b / a;
	} else if (b <= 1.25e-42) {
		tmp = (-b + sqrt(fma((c * -4.0), a, (b * b)))) / (a + a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.8e+164)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 1.25e-42)
		tmp = Float64(Float64(Float64(-b) + sqrt(fma(Float64(c * -4.0), a, Float64(b * b)))) / Float64(a + a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.8e+164], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 1.25e-42], N[(N[((-b) + N[Sqrt[N[(N[(c * -4.0), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a + a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.80000000000000021e164

   1. Initial program 45.9%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

   if -4.80000000000000021e164 < b < 1.25000000000000001e-42

   1. Initial program 78.8%

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

   4. Applied rewrites 78.8%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 78.8

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

   6. Applied rewrites 78.8%

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

   if 1.25000000000000001e-42 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.48 \cdot 10^{-86}:\\ \;\;\;\;-\mathsf{fma}\left(\frac{-c}{b \cdot b}, b, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{\left(c \cdot a\right) \cdot -4}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.48e-86)
   (- (fma (/ (- c) (* b b)) b (/ b a)))
   (if (<= b 6.6e-43)
     (/ (+ (- b) (sqrt (* (* c a) -4.0))) (* 2.0 a))
     (/ (- c) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.48e-86) {
		tmp = -fma((-c / (b * b)), b, (b / a));
	} else if (b <= 6.6e-43) {
		tmp = (-b + sqrt(((c * a) * -4.0))) / (2.0 * a);
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.48e-86)
		tmp = Float64(-fma(Float64(Float64(-c) / Float64(b * b)), b, Float64(b / a)));
	elseif (b <= 6.6e-43)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(c * a) * -4.0))) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.48e-86], (-N[(N[((-c) / N[(b * b), $MachinePrecision]), $MachinePrecision] * b + N[(b / a), $MachinePrecision]), $MachinePrecision]), If[LessEqual[b, 6.6e-43], N[(N[((-b) + N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[((-c) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4800000000000001e-86

   1. Initial program 77.8%

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

   3. Taylor expanded in b around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 90.5

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

   5. Applied rewrites 90.5%

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

   if -1.4800000000000001e-86 < b < 6.60000000000000031e-43

   1. Initial program 63.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{2 \cdot a} \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

   if 6.60000000000000031e-43 < b

   1. Initial program 18.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 4: 67.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.7 \cdot 10^{-222}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.7e-222) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-222) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.7d-222) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.7e-222) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.7e-222:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.7e-222)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.7e-222)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.7e-222], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.7000000000000001e-222

   1. Initial program 76.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      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 68.7

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

   5. Applied rewrites 68.7%

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

   if 1.7000000000000001e-222 < b

   1. Initial program 28.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower-neg.f64 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 5: 43.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5.5e-136) (/ (- b) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.5e-136) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 5.5d-136) then
        tmp = -b / a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.5e-136) {
		tmp = -b / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.5e-136:
		tmp = -b / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.5e-136)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5.5e-136)
		tmp = -b / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.5e-136], N[((-b) / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 5.4999999999999999e-136

   1. Initial program 73.5%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r/ N/A

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

      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 63.5

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

   5. Applied rewrites 63.5%

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

   if 5.4999999999999999e-136 < b

   1. Initial program 26.4%

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

   4. Applied rewrites 26.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. frac-2neg N/A

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

      6. lift-neg.f64 N/A

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

      7. remove-double-neg N/A

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

      8. frac-add N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 8.3%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{-2 \cdot b + 2 \cdot b}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

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

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

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

      3. metadata-eval N/A

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

      4. +-inverses N/A

         \[\leadsto \frac{\frac{-1}{4} \cdot \color{blue}{0}}{a} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{a} \]

      6. +-inverses N/A

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

      7. metadata-eval N/A

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

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

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

      9. lower-/.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. +-inverses 26.6

          \[\leadsto \frac{\color{blue}{0}}{a} \]

   9. Applied rewrites 26.6%

      \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-136}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 11.4% 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 53.8%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{-2 \cdot a}{b} \cdot c}}{2 \cdot a} \]

   4. associate-*r/ N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 31.1

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

5. Applied rewrites 31.1%

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

7. Applied rewrites 31.1%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{2 \cdot a}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{a + a}} \]

   3. flip-+ N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{\frac{a \cdot a - a \cdot a}{a - a}}} \]

   4. +-inverses N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\frac{\color{blue}{0}}{a - a}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\frac{\color{blue}{\mathsf{neg}\left(0\right)}}{a - a}} \]

   6. +-inverses N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\frac{\mathsf{neg}\left(\color{blue}{\left(a \cdot a - a \cdot a\right)}\right)}{a - a}} \]

   7. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{\mathsf{neg}\left(\frac{a \cdot a - a \cdot a}{a - a}\right)}} \]

   8. flip-+ N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\mathsf{neg}\left(\color{blue}{\left(a + a\right)}\right)} \]

   9. count-2-rev N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\mathsf{neg}\left(\color{blue}{2 \cdot a}\right)} \]

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

       \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot a}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{-2} \cdot a} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{-2 \cdot a}} \]

9. Applied rewrites 12.4%

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

10. Taylor expanded in a around 0

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

    1. lower-/.f64 12.3

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

12. Applied rewrites 12.3%

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

Alternative 7: 2.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{-2 \cdot a}{b} \cdot c}}{2 \cdot a} \]

   4. associate-*r/ N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 31.1

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

5. Applied rewrites 31.1%

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

7. Applied rewrites 31.1%

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

   1. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{2 \cdot a}} \]

   2. count-2-rev N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{a + a}} \]

   3. flip-+ N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{\frac{a \cdot a - a \cdot a}{a - a}}} \]

   4. +-inverses N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\frac{\color{blue}{0}}{a - a}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\frac{\color{blue}{\mathsf{neg}\left(0\right)}}{a - a}} \]

   6. +-inverses N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\frac{\mathsf{neg}\left(\color{blue}{\left(a \cdot a - a \cdot a\right)}\right)}{a - a}} \]

   7. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{\mathsf{neg}\left(\frac{a \cdot a - a \cdot a}{a - a}\right)}} \]

   8. flip-+ N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\mathsf{neg}\left(\color{blue}{\left(a + a\right)}\right)} \]

   9. count-2-rev N/A

      \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\mathsf{neg}\left(\color{blue}{2 \cdot a}\right)} \]

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

       \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot a}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{-2} \cdot a} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{Rewrite=>}\left(lower-*.f64, \left(\frac{-2}{b} \cdot a\right)\right) \cdot c}{\color{blue}{-2 \cdot a}} \]

9. Applied rewrites 12.4%

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

10. Taylor expanded in b around -inf

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

    1. lower-/.f64 2.5

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

12. Applied rewrites 2.5%

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

Alternative 8: 11.6% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 53.8%

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

4. Applied rewrites 53.9%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. div-add N/A

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

   5. frac-2neg N/A

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

   6. lift-neg.f64 N/A

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

   7. remove-double-neg N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

6. Applied rewrites 32.4%

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

7. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{4} \cdot \frac{-2 \cdot b + 2 \cdot b}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

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

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

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

   3. metadata-eval N/A

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

   4. +-inverses N/A

      \[\leadsto \frac{\frac{-1}{4} \cdot \color{blue}{0}}{a} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{a} \]

   6. +-inverses N/A

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

   7. metadata-eval N/A

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

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

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

   9. lower-/.f64 N/A

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

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

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

   11. metadata-eval N/A

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

   12. +-inverses 12.8

       \[\leadsto \frac{\color{blue}{0}}{a} \]

9. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\frac{0}{a}} \]

10. Final simplification 12.8%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))