quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 87.2%

Time: 4.7s

Alternatives: 15

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 87.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -6.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.5e+18)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 -6.5e-130)
     (/ (/ (* a c) (fma -1.0 b_2 (sqrt (fma b_2 b_2 (* (- a) c))))) a)
     (if (<= b_2 5e+73)
       (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
       (/ (+ b_2 b_2) (- a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.5e+18) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -6.5e-130) {
		tmp = ((a * c) / fma(-1.0, b_2, sqrt(fma(b_2, b_2, (-a * c))))) / a;
	} else if (b_2 <= 5e+73) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.5e+18)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -6.5e-130)
		tmp = Float64(Float64(Float64(a * c) / fma(-1.0, b_2, sqrt(fma(b_2, b_2, Float64(Float64(-a) * c))))) / a);
	elseif (b_2 <= 5e+73)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.5e+18], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -6.5e-130], N[(N[(N[(a * c), $MachinePrecision] / N[(-1.0 * b$95$2 + N[Sqrt[N[(b$95$2 * b$95$2 + N[((-a) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5e+73], N[(N[(b$95$2 + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.5e18

   1. Initial program 10.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 94.5

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 94.5

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 94.5%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -1.5e18 < b_2 < -6.5000000000000002e-130

   1. Initial program 48.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      7. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\color{blue}{\frac{b\_2 \cdot b\_2 - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)} \cdot \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}}{a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a} \]
   6. Step-by-step derivation

      1. lower-*.f64 77.1

         \[\leadsto \frac{\frac{a \cdot \color{blue}{c}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a} \]

   7. Applied rewrites 77.1%

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, \color{blue}{b\_2 \cdot b\_2}\right)}\right)}}{a} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c + b\_2 \cdot b\_2}}\right)}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\color{blue}{\left(-a\right) \cdot c} + b\_2 \cdot b\_2}\right)}}{a} \]

      4. pow2 N/A

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\left(-a\right) \cdot c + \color{blue}{{b\_2}^{2}}}\right)}}{a} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\color{blue}{{b\_2}^{2} + \left(-a\right) \cdot c}}\right)}}{a} \]

      6. pow2 N/A

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(-a\right) \cdot c}\right)}}{a} \]

      7. lower-fma.f64 77.2

         \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}\right)}}{a} \]

   9. Applied rewrites 77.2%

      \[\leadsto \frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}\right)}}{a} \]

   if -6.5000000000000002e-130 < b_2 < 4.99999999999999976e73

   1. Initial program 80.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 4.99999999999999976e73 < b_2

   1. Initial program 53.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -6.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}\right)}}{a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -2.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{c \cdot a}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right) \cdot a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e+18)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 -2.4e-126)
     (/ (* c a) (* (fma -1.0 b_2 (sqrt (fma (- a) c (* b_2 b_2)))) a))
     (if (<= b_2 5e+73)
       (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
       (/ (+ b_2 b_2) (- a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e+18) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -2.4e-126) {
		tmp = (c * a) / (fma(-1.0, b_2, sqrt(fma(-a, c, (b_2 * b_2)))) * a);
	} else if (b_2 <= 5e+73) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e+18)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -2.4e-126)
		tmp = Float64(Float64(c * a) / Float64(fma(-1.0, b_2, sqrt(fma(Float64(-a), c, Float64(b_2 * b_2)))) * a));
	elseif (b_2 <= 5e+73)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e+18], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -2.4e-126], N[(N[(c * a), $MachinePrecision] / N[(N[(-1.0 * b$95$2 + N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5e+73], N[(N[(b$95$2 + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.8e18

   1. Initial program 10.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 94.5

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 94.5

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 94.5%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.8e18 < b_2 < -2.40000000000000007e-126

   1. Initial program 48.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      4. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      7. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c} \cdot \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}{a} \]

   4. Applied rewrites 48.5%

      \[\leadsto \frac{\color{blue}{\frac{b\_2 \cdot b\_2 - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)} \cdot \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}}{a} \]

   5. Taylor expanded in a around 0

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a} \]
   6. Step-by-step derivation

      1. lower-*.f64 77.1

         \[\leadsto \frac{\frac{a \cdot \color{blue}{c}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a} \]

   7. Applied rewrites 77.1%

      \[\leadsto \frac{\frac{\color{blue}{a \cdot c}}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}{a}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right)}}}{a} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{\color{blue}{-1 \cdot b\_2 + \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{-1 \cdot b\_2 + \color{blue}{\sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}}}{a} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{-1 \cdot b\_2 + \sqrt{\mathsf{fma}\left(-a, c, \color{blue}{b\_2 \cdot b\_2}\right)}}}{a} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{-1 \cdot b\_2 + \sqrt{\color{blue}{\left(-a\right) \cdot c + b\_2 \cdot b\_2}}}}{a} \]

      7. lift-neg.f64 N/A

         \[\leadsto \frac{\frac{a \cdot c}{-1 \cdot b\_2 + \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c + b\_2 \cdot b\_2}}}{a} \]

   9. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\frac{c \cdot a}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right) \cdot a}} \]

   if -2.40000000000000007e-126 < b_2 < 4.99999999999999976e73

   1. Initial program 80.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 4.99999999999999976e73 < b_2

   1. Initial program 53.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -2.4 \cdot 10^{-126}:\\ \;\;\;\;\frac{c \cdot a}{\mathsf{fma}\left(-1, b\_2, \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}\right) \cdot a}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{-b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e+73)
     (- (/ (- b_2) a) (/ (sqrt (fma b_2 b_2 (* c (- a)))) a))
     (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+73) {
		tmp = (-b_2 / a) - (sqrt(fma(b_2, b_2, (c * -a))) / a);
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e+73)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) / a));
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+73], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-98

   1. Initial program 17.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 85.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 85.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.9e-98 < b_2 < 4.99999999999999976e73

   1. Initial program 79.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      4. pow2 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2}} - a \cdot c}}{a} \]

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

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}}{a} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}{a} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{\left(-1 \cdot a\right)} \cdot c}}{a} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -1 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(-1 \cdot a\right) \cdot c}\right)}}{a} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c\right)}}{a} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot c}\right)}}{a} \]

      13. lower-neg.f64 79.4

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(-a\right)} \cdot c\right)}}{a} \]

   4. Applied rewrites 79.4%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}}{a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}{a}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right)} - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}{a} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}}{a} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}}{a} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a\right) \cdot c}}}{a} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{{b\_2}^{2}} + \left(-a\right) \cdot c}}{a} \]

      7. +-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c + {b\_2}^{2}}}}{a} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c} + {b\_2}^{2}}}{a} \]

      9. pow2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\left(-a\right) \cdot c + \color{blue}{b\_2 \cdot b\_2}}}{a} \]

      10. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c + b\_2 \cdot b\_2}}{a} \]

      11. sub-div N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c + b\_2 \cdot b\_2}}{a}} \]

      12. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a}} - \frac{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c + b\_2 \cdot b\_2}}{a} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{-b\_2}}{a} - \frac{\sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c + b\_2 \cdot b\_2}}{a} \]

   6. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\frac{-b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}} \]

   if 4.99999999999999976e73 < b_2

   1. Initial program 53.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{-b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e+73)
     (/ (- (- b_2) (sqrt (fma b_2 b_2 (* (- a) c)))) a)
     (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+73) {
		tmp = (-b_2 - sqrt(fma(b_2, b_2, (-a * c)))) / a;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e+73)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(Float64(-a) * c)))) / a);
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+73], N[(N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[((-a) * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-98

   1. Initial program 17.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 85.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 85.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.9e-98 < b_2 < 4.99999999999999976e73

   1. Initial program 79.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} - a \cdot c}}{a} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - \color{blue}{a \cdot c}}}{a} \]

      4. pow2 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2}} - a \cdot c}}{a} \]

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

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{{b\_2}^{2} + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}}{a} \]

      6. pow2 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2} + \left(\mathsf{neg}\left(a\right)\right) \cdot c}}{a} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{\left(-1 \cdot a\right)} \cdot c}}{a} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 + \color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -1 \cdot \left(a \cdot c\right)\right)}}}{a} \]

      10. associate-*r* N/A

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(-1 \cdot a\right) \cdot c}\right)}}{a} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(\mathsf{neg}\left(a\right)\right)} \cdot c\right)}}{a} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(\mathsf{neg}\left(a\right)\right) \cdot c}\right)}}{a} \]

      13. lower-neg.f64 79.4

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{\left(-a\right)} \cdot c\right)}}{a} \]

   4. Applied rewrites 79.4%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}}{a} \]

   if 4.99999999999999976e73 < b_2

   1. Initial program 53.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, \left(-a\right) \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5e+73)
     (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
     (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+73) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 + b_2) / -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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.9d-98)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5d+73) then
        tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a
    else
        tmp = (b_2 + b_2) / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5e+73) {
		tmp = (b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-98:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5e+73:
		tmp = (b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / -a
	else:
		tmp = (b_2 + b_2) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5e+73)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.9e-98)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5e+73)
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	else
		tmp = (b_2 + b_2) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+73], N[(N[(b$95$2 + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-98

   1. Initial program 17.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 85.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 85.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.9e-98 < b_2 < 4.99999999999999976e73

   1. Initial program 79.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 4.99999999999999976e73 < b_2

   1. Initial program 53.2%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 0.029:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 0.029)
     (/ (+ b_2 (sqrt (* (- a) c))) (- a))
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 0.029) {
		tmp = (b_2 + sqrt((-a * c))) / -a;
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 0.029)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(-a) * c))) / Float64(-a));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 0.029], N[(N[(b$95$2 + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-98

   1. Initial program 17.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 85.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 85.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.9e-98 < b_2 < 0.0290000000000000015

   1. Initial program 75.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(-1 \cdot a\right) \cdot \color{blue}{c}}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot c}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(\mathsf{neg}\left(a\right)\right) \cdot \color{blue}{c}}}{a} \]

      4. lower-neg.f64 64.5

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a} \]

   5. Applied rewrites 64.5%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\left(-a\right) \cdot c}}}{a} \]

   if 0.0290000000000000015 < b_2

   1. Initial program 61.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 97.1

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 0.029:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 9.2e-39)
     (/ (- (sqrt (* (- a) c))) a)
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9.2e-39) {
		tmp = -sqrt((-a * c)) / a;
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 9.2e-39)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9.2e-39], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-98

   1. Initial program 17.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 85.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 85.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.9e-98 < b_2 < 9.20000000000000033e-39

   1. Initial program 73.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 9.20000000000000033e-39 < b_2

   1. Initial program 64.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 94.3

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -8.2 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq 8.6 \cdot 10^{-71}:\\ \;\;\;\;-t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b_2 -2.6e-74)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -8.2e-117)
       t_0
       (if (<= b_2 8.6e-71) (- t_0) (/ (+ b_2 b_2) (- a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((-c / a));
	double tmp;
	if (b_2 <= -2.6e-74) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -8.2e-117) {
		tmp = t_0;
	} else if (b_2 <= 8.6e-71) {
		tmp = -t_0;
	} else {
		tmp = (b_2 + b_2) / -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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-c / a))
    if (b_2 <= (-2.6d-74)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-8.2d-117)) then
        tmp = t_0
    else if (b_2 <= 8.6d-71) then
        tmp = -t_0
    else
        tmp = (b_2 + b_2) / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt((-c / a));
	double tmp;
	if (b_2 <= -2.6e-74) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -8.2e-117) {
		tmp = t_0;
	} else if (b_2 <= 8.6e-71) {
		tmp = -t_0;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((-c / a))
	tmp = 0
	if b_2 <= -2.6e-74:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -8.2e-117:
		tmp = t_0
	elif b_2 <= 8.6e-71:
		tmp = -t_0
	else:
		tmp = (b_2 + b_2) / -a
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(-c) / a))
	tmp = 0.0
	if (b_2 <= -2.6e-74)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -8.2e-117)
		tmp = t_0;
	elseif (b_2 <= 8.6e-71)
		tmp = Float64(-t_0);
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((-c / a));
	tmp = 0.0;
	if (b_2 <= -2.6e-74)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -8.2e-117)
		tmp = t_0;
	elseif (b_2 <= 8.6e-71)
		tmp = -t_0;
	else
		tmp = (b_2 + b_2) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -2.6e-74], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -8.2e-117], t$95$0, If[LessEqual[b$95$2, 8.6e-71], (-t$95$0), N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -2.6000000000000001e-74

   1. Initial program 16.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 88.8

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 88.8

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.6000000000000001e-74 < b_2 < -8.20000000000000063e-117

   1. Initial program 51.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 2.5

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

   5. Applied rewrites 2.5%

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

   6. Taylor expanded in c around -inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 67.4

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

   8. Applied rewrites 67.4%

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

   if -8.20000000000000063e-117 < b_2 < 8.5999999999999994e-71

   1. Initial program 72.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.9

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

   5. Applied rewrites 40.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 40.9

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

   7. Applied rewrites 40.9%

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

   if 8.5999999999999994e-71 < b_2

   1. Initial program 66.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 90.2%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -8.2 \cdot 10^{-117}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{elif}\;b\_2 \leq 8.6 \cdot 10^{-71}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-98)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 9.2e-39) (/ (- (sqrt (* (- a) c))) a) (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9.2e-39) {
		tmp = -sqrt((-a * c)) / a;
	} else {
		tmp = (b_2 + b_2) / -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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.9d-98)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 9.2d-39) then
        tmp = -sqrt((-a * c)) / a
    else
        tmp = (b_2 + b_2) / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.9e-98) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9.2e-39) {
		tmp = -Math.sqrt((-a * c)) / a;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.9e-98:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 9.2e-39:
		tmp = -math.sqrt((-a * c)) / a
	else:
		tmp = (b_2 + b_2) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.9e-98)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 9.2e-39)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.9e-98)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 9.2e-39)
		tmp = -sqrt((-a * c)) / a;
	else
		tmp = (b_2 + b_2) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.9e-98], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9.2e-39], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.9e-98

   1. Initial program 17.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 85.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 85.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 85.2

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 85.2%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.9e-98 < b_2 < 9.20000000000000033e-39

   1. Initial program 73.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if 9.20000000000000033e-39 < b_2

   1. Initial program 64.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.6%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.9 \cdot 10^{-98}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.8 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-74)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.8e-190) (sqrt (/ (- c) a)) (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-74) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.8e-190) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (b_2 + b_2) / -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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-74)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5.8d-190) then
        tmp = sqrt((-c / a))
    else
        tmp = (b_2 + b_2) / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-74) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.8e-190) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-74:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5.8e-190:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (b_2 + b_2) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-74)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.8e-190)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-74)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5.8e-190)
		tmp = sqrt((-c / a));
	else
		tmp = (b_2 + b_2) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-74], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.8e-190], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6000000000000001e-74

   1. Initial program 16.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 88.8

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 88.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 88.8

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 88.8%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -2.6000000000000001e-74 < b_2 < 5.8000000000000004e-190

   1. Initial program 64.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 33.6

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

   5. Applied rewrites 33.6%

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

   6. Taylor expanded in c around -inf

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

      1. sqrt-prod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 40.8

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

   8. Applied rewrites 40.8%

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

   if 5.8000000000000004e-190 < b_2

   1. Initial program 68.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 79.1%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-74}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.8 \cdot 10^{-190}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-283) (/ (* -0.5 c) b_2) (/ (+ b_2 b_2) (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 + b_2) / -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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d-283)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 + b_2) / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-283:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 + b_2) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-283)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-283)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 + b_2) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-283], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -8.9999999999999994e-283

   1. Initial program 30.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 68.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 68.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 68.3

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 68.3%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -8.9999999999999994e-283 < b_2

   1. Initial program 68.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
   4. Step-by-step derivation

   5. Applied rewrites 67.8%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-283}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-283) (/ (* -0.5 c) b_2) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d-283)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-283) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-283:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-283)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-283)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-283], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -8.9999999999999994e-283

   1. Initial program 30.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 68.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 68.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{c}{\color{blue}{b\_2}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      5. lower-*.f64 68.3

         \[\leadsto \frac{-0.5 \cdot c}{b\_2} \]

   7. Applied rewrites 68.3%

      \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]

   if -8.9999999999999994e-283 < b_2

   1. Initial program 68.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(c \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-1 \cdot c\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}} \cdot \sqrt{-1}}\right)}{a} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}}} \cdot \sqrt{-1}\right)}{a} \]

      7. sqrt-unprod N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      10. lower-/.f64 29.3

          \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

   5. Applied rewrites 29.3%

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}}{a} \]

   6. Taylor expanded in b_2 around inf

      \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      2. lower-neg.f64 29.1

         \[\leadsto \frac{-b\_2}{a} \]

   8. Applied rewrites 29.1%

      \[\leadsto \frac{-b\_2}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-283}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-283) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-283) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-9d-283)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-283) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-283:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-283)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-283)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-283], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -8.9999999999999994e-283

   1. Initial program 30.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]

      2. lower-/.f64 68.2

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b\_2}} \]

   5. Applied rewrites 68.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]

   if -8.9999999999999994e-283 < b_2

   1. Initial program 68.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(c \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-1 \cdot c\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}} \cdot \sqrt{-1}}\right)}{a} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}}} \cdot \sqrt{-1}\right)}{a} \]

      7. sqrt-unprod N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      10. lower-/.f64 29.3

          \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

   5. Applied rewrites 29.3%

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}}{a} \]

   6. Taylor expanded in b_2 around inf

      \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      2. lower-neg.f64 29.1

         \[\leadsto \frac{-b\_2}{a} \]

   8. Applied rewrites 29.1%

      \[\leadsto \frac{-b\_2}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 22.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.5e+18) (* (/ c b_2) 0.5) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.5e+18) {
		tmp = (c / b_2) * 0.5;
	} else {
		tmp = -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.5d+18)) then
        tmp = (c / b_2) * 0.5d0
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.5e+18) {
		tmp = (c / b_2) * 0.5;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.5e+18:
		tmp = (c / b_2) * 0.5
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.5e+18)
		tmp = Float64(Float64(c / b_2) * 0.5);
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.5e+18)
		tmp = (c / b_2) * 0.5;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.5e+18], N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.5e18

   1. Initial program 10.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{c}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} + \color{blue}{-2} \cdot \frac{b\_2}{a} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \color{blue}{\frac{1}{2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, -2 \cdot \frac{b\_2}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, \frac{1}{2}, \frac{b\_2}{a} \cdot -2\right) \]

      7. lower-/.f64 2.3

         \[\leadsto \mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right) \]

   5. Applied rewrites 2.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{c}{b\_2} \cdot \frac{1}{2} \]

      3. lift-/.f64 32.0

         \[\leadsto \frac{c}{b\_2} \cdot 0.5 \]

   8. Applied rewrites 32.0%

      \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{0.5} \]

   if -1.5e18 < b_2

   1. Initial program 65.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around -inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(c \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)\right)}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(-1 \cdot c\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}} \cdot \sqrt{-1}}\right)}{a} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}}} \cdot \sqrt{-1}\right)}{a} \]

      7. sqrt-unprod N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

      10. lower-/.f64 28.6

          \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

   5. Applied rewrites 28.6%

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}}{a} \]

   6. Taylor expanded in b_2 around inf

      \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

      2. lower-neg.f64 22.3

         \[\leadsto \frac{-b\_2}{a} \]

   8. Applied rewrites 22.3%

      \[\leadsto \frac{-b\_2}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 15.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-b\_2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (- b_2) a))

C

double code(double a, double b_2, double c) {
	return -b_2 / 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_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = -b_2 / a
end function

Java

public static double code(double a, double b_2, double c) {
	return -b_2 / a;
}

Python

def code(a, b_2, c):
	return -b_2 / a

Julia

function code(a, b_2, c)
	return Float64(Float64(-b_2) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -b_2 / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[((-b$95$2) / a), $MachinePrecision]

Derivation

1. Initial program 51.0%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in c around -inf

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(c \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)\right)}}{a} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\left(-1 \cdot c\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\frac{b\_2}{c} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}}{a} \]

   4. lower-neg.f64 N/A

      \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{\frac{b\_2}{c}} - \sqrt{\frac{a}{c}} \cdot \sqrt{-1}\right)}{a} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}} \cdot \sqrt{-1}}\right)}{a} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \color{blue}{\sqrt{\frac{a}{c}}} \cdot \sqrt{-1}\right)}{a} \]

   7. sqrt-unprod N/A

      \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

   10. lower-/.f64 22.9

       \[\leadsto \frac{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}{a} \]

5. Applied rewrites 22.9%

   \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(\frac{b\_2}{c} - \sqrt{\frac{a}{c} \cdot -1}\right)}}{a} \]

6. Taylor expanded in b_2 around inf

   \[\leadsto \frac{-1 \cdot \color{blue}{b\_2}}{a} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(b\_2\right)}{a} \]

   2. lower-neg.f64 17.0

      \[\leadsto \frac{-b\_2}{a} \]

8. Applied rewrites 17.0%

   \[\leadsto \frac{-b\_2}{a} \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{c}{t\_1 - b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + t\_1}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2025071 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))