quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 86.1%

Time: 3.8s

Alternatives: 14

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: 51.7% 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: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-b\_2}{a}\\ \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;t\_0 - \frac{\sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(-0.5, \frac{c}{b\_2}, \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- b_2) a)))
   (if (<= b_2 -3.1e-95)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 5.2e+120)
       (- t_0 (/ (sqrt (fma (- a) c (* b_2 b_2))) a))
       (- t_0 (fma -0.5 (/ c b_2) (/ b_2 a)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = -b_2 / a;
	double tmp;
	if (b_2 <= -3.1e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e+120) {
		tmp = t_0 - (sqrt(fma(-a, c, (b_2 * b_2))) / a);
	} else {
		tmp = t_0 - fma(-0.5, (c / b_2), (b_2 / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(-b_2) / a)
	tmp = 0.0
	if (b_2 <= -3.1e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.2e+120)
		tmp = Float64(t_0 - Float64(sqrt(fma(Float64(-a), c, Float64(b_2 * b_2))) / a));
	else
		tmp = Float64(t_0 - fma(-0.5, Float64(c / b_2), Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[((-b$95$2) / a), $MachinePrecision]}, If[LessEqual[b$95$2, -3.1e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.2e+120], N[(t$95$0 - N[(N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(-0.5 * N[(c / b$95$2), $MachinePrecision] + N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.09999999999999992e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.09999999999999992e-95 < b_2 < 5.1999999999999998e120

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      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

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

      7. lift-*.f64 N/A

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

      8. div-sub N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 91.4%

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

   if 5.1999999999999998e120 < b_2

   1. Initial program 49.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      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

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

      7. lift-*.f64 N/A

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

      8. div-sub N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{-b\_2}{a} - \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{b\_2} + \frac{b\_2}{a}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 2: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a} - \mathsf{fma}\left(-0.5, \frac{c}{b\_2}, \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-95)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.2e+120)
     (/ (- (- b_2) (sqrt (fma (- a) c (* b_2 b_2)))) a)
     (- (/ (- b_2) a) (fma -0.5 (/ c b_2) (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.1e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e+120) {
		tmp = (-b_2 - sqrt(fma(-a, c, (b_2 * b_2)))) / a;
	} else {
		tmp = (-b_2 / a) - fma(-0.5, (c / b_2), (b_2 / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.1e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.2e+120)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(Float64(-a), c, Float64(b_2 * b_2)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) / a) - fma(-0.5, Float64(c / b_2), Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.1e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.2e+120], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(-0.5 * N[(c / b$95$2), $MachinePrecision] + N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.09999999999999992e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.09999999999999992e-95 < b_2 < 5.1999999999999998e120

   1. Initial program 91.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.4

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

   4. Applied rewrites 91.4%

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

   if 5.1999999999999998e120 < b_2

   1. Initial program 49.0%

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

      1. lift-/.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      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

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

      7. lift-*.f64 N/A

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

      8. div-sub N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower--.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. lower-/.f64 N/A

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

      15. lift-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{-b\_2}{a} - \color{blue}{\left(\frac{-1}{2} \cdot \frac{c}{b\_2} + \frac{b\_2}{a}\right)} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 100.0

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

   7. Applied rewrites 100.0%

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

Alternative 3: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-95)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.2e+120)
     (/ (- (- b_2) (sqrt (fma (- a) c (* b_2 b_2)))) 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 <= -3.1e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e+120) {
		tmp = (-b_2 - sqrt(fma(-a, c, (b_2 * b_2)))) / 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 <= -3.1e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.2e+120)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(Float64(-a), c, Float64(b_2 * b_2)))) / 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, -3.1e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.2e+120], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $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 < -3.09999999999999992e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.09999999999999992e-95 < b_2 < 5.1999999999999998e120

   1. Initial program 91.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 91.4

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

   4. Applied rewrites 91.4%

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

   if 5.1999999999999998e120 < b_2

   1. Initial program 49.0%

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

   3. Taylor expanded in c around 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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-95)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.2e+120)
     (/ (+ b_2 (sqrt (- (* b_2 b_2) (* 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 <= -3.1e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e+120) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (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 <= -3.1e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.2e+120)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - 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, -3.1e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.2e+120], 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[(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 < -3.09999999999999992e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.09999999999999992e-95 < b_2 < 5.1999999999999998e120

   1. Initial program 91.4%

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

   if 5.1999999999999998e120 < b_2

   1. Initial program 49.0%

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

   3. Taylor expanded in c around 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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-95)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6e-14)
     (/ (+ 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 <= -3.1e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e-14) {
		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 <= -3.1e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6e-14)
		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, -3.1e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e-14], 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 < -3.09999999999999992e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.09999999999999992e-95 < b_2 < 5.9999999999999997e-14

   1. Initial program 90.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

   if 5.9999999999999997e-14 < b_2

   1. Initial program 65.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{b\_2 + \sqrt{\left(-a\right) \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-95)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6e-14)
     (/ (- (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 <= -3.8e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e-14) {
		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 <= -3.8e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6e-14)
		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, -3.8e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e-14], 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 < -3.7999999999999997e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.7999999999999997e-95 < b_2 < 5.9999999999999997e-14

   1. Initial program 90.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 80.7

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

   5. Applied rewrites 80.7%

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

   if 5.9999999999999997e-14 < b_2

   1. Initial program 65.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

Alternative 7: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-95)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6e-14) (/ (- (sqrt (* (- a) c))) a) (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e-14) {
		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 <= (-3.8d-95)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6d-14) 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 <= -3.8e-95) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e-14) {
		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 <= -3.8e-95:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6e-14:
		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 <= -3.8e-95)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6e-14)
		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 <= -3.8e-95)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6e-14)
		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, -3.8e-95], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e-14], 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 < -3.7999999999999997e-95

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -3.7999999999999997e-95 < b_2 < 5.9999999999999997e-14

   1. Initial program 90.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 80.7

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

   5. Applied rewrites 80.7%

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

   if 5.9999999999999997e-14 < b_2

   1. Initial program 65.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 94.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e-97)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.2e-113) (sqrt (/ (- c) a)) (/ (+ b_2 b_2) (- a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-97) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e-113) {
		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 <= (-2.2d-97)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5.2d-113) 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 <= -2.2e-97) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.2e-113) {
		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 <= -2.2e-97:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5.2e-113:
		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 <= -2.2e-97)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.2e-113)
		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 <= -2.2e-97)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5.2e-113)
		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, -2.2e-97], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.2e-113], 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 < -2.1999999999999999e-97

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 85.1

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

   5. Applied rewrites 85.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if -2.1999999999999999e-97 < b_2 < 5.1999999999999998e-113

   1. Initial program 88.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 88.5

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

   4. Applied rewrites 88.5%

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

   5. Taylor expanded in a around -inf

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

      1. div-sub N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. pow2 N/A

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

      10. div-sub N/A

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

      11. sqrt-prod N/A

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

      12. lower-sqrt.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 40.8%

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

   if 5.1999999999999998e-113 < b_2

   1. Initial program 69.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 88.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.2 \cdot 10^{-113}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* -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 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 34.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 65.8

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

   5. Applied rewrites 65.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 73.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (/ (* -0.5 c) b_2) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 34.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 65.8

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

   5. Applied rewrites 65.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot c}{b\_2} \]

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in a around inf

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

      1. div-sub N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. pow2 N/A

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

      10. div-sub N/A

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 28.4

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

   10. Applied rewrites 28.4%

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

Alternative 11: 48.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		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 <= (-5d-310)) 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 <= -5e-310) {
		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 <= -5e-310:
		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 <= -5e-310)
		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 <= -5e-310)
		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, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 34.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 65.8

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

   5. Applied rewrites 65.8%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 74.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. mul-1-neg N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.2

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in a around inf

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

      1. div-sub N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. pow2 N/A

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

      10. div-sub N/A

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

   7. Applied rewrites 24.7%

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

   8. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 28.4

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

   10. Applied rewrites 28.4%

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

Alternative 12: 15.7% 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 52.9%

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

   1. lift--.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. mul-1-neg N/A

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

   7. associate-*r* N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. mul-1-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. pow2 N/A

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

   14. lift-*.f64 53.0

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

4. Applied rewrites 53.0%

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

5. Taylor expanded in a around inf

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

   1. div-sub N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-neg.f64 N/A

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

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

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

   9. pow2 N/A

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

   10. div-sub N/A

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

7. Applied rewrites 20.5%

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

8. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-/.f64 14.7

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

10. Applied rewrites 14.7%

    \[\leadsto \frac{-b\_2}{\color{blue}{a}} \]
11. Add Preprocessing

Alternative 13: 6.9% accurate, 3.3× speedup

Math

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

FPCore

(FPCore (a b_2 c) :precision binary64 (/ c 0.0))

C

double code(double a, double b_2, double c) {
	return c / 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = c / 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return c / 0.0;
}

Python

def code(a, b_2, c):
	return c / 0.0

Julia

function code(a, b_2, c)
	return Float64(c / 0.0)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = c / 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := N[(c / 0.0), $MachinePrecision]

Derivation

1. Initial program 52.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 29.6%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\frac{c}{0 \cdot b\_2}} \]

8. Taylor expanded in b_2 around 0

   \[\leadsto \frac{c}{0} \]
9. Step-by-step derivation

10. Applied rewrites 6.5%

    \[\leadsto \frac{c}{0} \]
11. Add Preprocessing

Alternative 14: 0.0% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{NAN}\left(\right) \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (NAN))

Derivation

1. Initial program 52.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 29.6%

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

5. Taylor expanded in a around 0

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

7. Applied rewrites 5.6%

   \[\leadsto \color{blue}{\frac{c}{0 \cdot b\_2}} \]

8. Taylor expanded in b_2 around 0

   \[\leadsto \mathsf{NAN}\left(\right) \]
9. Step-by-step derivation

   1. lower-NAN.f64 0.0

      \[\leadsto \mathsf{NAN}\left(\right) \]

10. Applied rewrites 0.0%

    \[\leadsto \mathsf{NAN}\left(\right) \]
11. Add Preprocessing

Developer Target 1: 99.7% 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 2025073 
(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))