quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 85.5%

Time: 3.7s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{-b\_2}{a} - \frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-111)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.2e+59)
     (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.2e+59) {
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-111)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.2e+59)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a));
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-111], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.2e+59], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.59999999999999982e-111

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -2.59999999999999982e-111 < b_2 < 1.2000000000000001e59

   1. Initial program 80.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. lift-neg.f64 N/A

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

      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. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 80.8%

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

   if 1.2000000000000001e59 < b_2

   1. Initial program 60.6%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 95.3

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

   4. Applied rewrites 95.3%

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

Alternative 2: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-111)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.2e+59)
     (/ (- (- 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 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.2e+59) {
		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 <= -2.6e-111)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.2e+59)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-111], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.2e+59], 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 < -2.59999999999999982e-111

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -2.59999999999999982e-111 < b_2 < 1.2000000000000001e59

   1. Initial program 80.8%

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

   if 1.2000000000000001e59 < b_2

   1. Initial program 60.6%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 95.3

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

   4. Applied rewrites 95.3%

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

Alternative 3: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-111)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.7e-20)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (fma (/ c b_2) 0.5 (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-20) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = fma((c / b_2), 0.5, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-111)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.7e-20)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-111], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.7e-20], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.59999999999999982e-111

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -2.59999999999999982e-111 < b_2 < 1.6999999999999999e-20

   1. Initial program 78.2%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. 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 67.2

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

   4. Applied rewrites 67.2%

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

   if 1.6999999999999999e-20 < b_2

   1. Initial program 67.3%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   3. 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 90.8

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

   4. Applied rewrites 90.8%

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

Alternative 4: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.7 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-111)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.7e-20)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-20) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-111)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.7d-20) then
        tmp = (-b_2 - sqrt((-a * c))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-20) {
		tmp = (-b_2 - Math.sqrt((-a * c))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-111:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.7e-20:
		tmp = (-b_2 - math.sqrt((-a * c))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-111)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.7e-20)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-111)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.7e-20)
		tmp = (-b_2 - sqrt((-a * c))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-111], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.7e-20], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.59999999999999982e-111

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -2.59999999999999982e-111 < b_2 < 1.6999999999999999e-20

   1. Initial program 78.2%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. 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 67.2

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

   4. Applied rewrites 67.2%

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

   if 1.6999999999999999e-20 < b_2

   1. Initial program 67.3%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 90.4

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

   4. Applied rewrites 90.4%

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

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{-111}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e-111)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.3e-77) (/ (- (sqrt (* (- a) c))) a) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.3e-77) {
		tmp = -sqrt((-a * c)) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.6d-111)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3.3d-77) then
        tmp = -sqrt((-a * c)) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e-111) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.3e-77) {
		tmp = -Math.sqrt((-a * c)) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e-111:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3.3e-77:
		tmp = -math.sqrt((-a * c)) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e-111)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.3e-77)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e-111)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3.3e-77)
		tmp = -sqrt((-a * c)) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e-111], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.3e-77], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.59999999999999982e-111

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -2.59999999999999982e-111 < b_2 < 3.29999999999999991e-77

   1. Initial program 76.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{-1}\right)}}{a} \]
   3. 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 69.2

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

   4. Applied rewrites 69.2%

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

   if 3.29999999999999991e-77 < b_2

   1. Initial program 69.8%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 86.9

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

   4. Applied rewrites 86.9%

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

Alternative 6: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -2.8 \cdot 10^{-304}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{elif}\;b\_2 \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{-\left(-c\right)}}{\sqrt{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8e-112)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 -2.8e-304)
     (- (sqrt (/ (- c) a)))
     (if (<= b_2 6.5e-35)
       (/ (sqrt (- (- c))) (sqrt (- a)))
       (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-112) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -2.8e-304) {
		tmp = -sqrt((-c / a));
	} else if (b_2 <= 6.5e-35) {
		tmp = sqrt(-(-c)) / sqrt(-a);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_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 <= (-8d-112)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-2.8d-304)) then
        tmp = -sqrt((-c / a))
    else if (b_2 <= 6.5d-35) then
        tmp = sqrt(-(-c)) / sqrt(-a)
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8e-112) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -2.8e-304) {
		tmp = -Math.sqrt((-c / a));
	} else if (b_2 <= 6.5e-35) {
		tmp = Math.sqrt(-(-c)) / Math.sqrt(-a);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8e-112:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -2.8e-304:
		tmp = -math.sqrt((-c / a))
	elif b_2 <= 6.5e-35:
		tmp = math.sqrt(-(-c)) / math.sqrt(-a)
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8e-112)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -2.8e-304)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	elseif (b_2 <= 6.5e-35)
		tmp = Float64(sqrt(Float64(-Float64(-c))) / sqrt(Float64(-a)));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8e-112)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -2.8e-304)
		tmp = -sqrt((-c / a));
	elseif (b_2 <= 6.5e-35)
		tmp = sqrt(-(-c)) / sqrt(-a);
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8e-112], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -2.8e-304], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), If[LessEqual[b$95$2, 6.5e-35], N[(N[Sqrt[(-(-c))], $MachinePrecision] / N[Sqrt[(-a)], $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -7.9999999999999996e-112

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -7.9999999999999996e-112 < b_2 < -2.7999999999999998e-304

   1. Initial program 70.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. lift-neg.f64 N/A

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

      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. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 70.1%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. sub-div N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 31.7

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

   6. Applied rewrites 31.7%

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

   if -2.7999999999999998e-304 < b_2 < 6.4999999999999999e-35

   1. Initial program 82.7%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 30.0

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

   4. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 30.0

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

   6. Applied rewrites 30.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. sqrt-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sqrt.f64 40.4

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

   8. Applied rewrites 40.4%

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

   if 6.4999999999999999e-35 < b_2

   1. Initial program 67.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

Alternative 7: 71.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b\_2 \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -2.8 \cdot 10^{-304}:\\ \;\;\;\;-t\_0\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b_2 -8e-112)
     (/ (* -0.5 c) b_2)
     (if (<= b_2 -2.8e-304)
       (- t_0)
       (if (<= b_2 6e-35) t_0 (* (/ b_2 a) -2.0))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((-c / a));
	double tmp;
	if (b_2 <= -8e-112) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -2.8e-304) {
		tmp = -t_0;
	} else if (b_2 <= 6e-35) {
		tmp = t_0;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-c / a))
    if (b_2 <= (-8d-112)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= (-2.8d-304)) then
        tmp = -t_0
    else if (b_2 <= 6d-35) then
        tmp = t_0
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt((-c / a));
	double tmp;
	if (b_2 <= -8e-112) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= -2.8e-304) {
		tmp = -t_0;
	} else if (b_2 <= 6e-35) {
		tmp = t_0;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((-c / a))
	tmp = 0
	if b_2 <= -8e-112:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= -2.8e-304:
		tmp = -t_0
	elif b_2 <= 6e-35:
		tmp = t_0
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(-c) / a))
	tmp = 0.0
	if (b_2 <= -8e-112)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= -2.8e-304)
		tmp = Float64(-t_0);
	elseif (b_2 <= 6e-35)
		tmp = t_0;
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((-c / a));
	tmp = 0.0;
	if (b_2 <= -8e-112)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= -2.8e-304)
		tmp = -t_0;
	elseif (b_2 <= 6e-35)
		tmp = t_0;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -8e-112], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -2.8e-304], (-t$95$0), If[LessEqual[b$95$2, 6e-35], t$95$0, N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -7.9999999999999996e-112

   1. Initial program 20.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 83.2

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

   6. Applied rewrites 83.2%

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

   if -7.9999999999999996e-112 < b_2 < -2.7999999999999998e-304

   1. Initial program 70.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. 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--.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-sub N/A

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

      8. lift-neg.f64 N/A

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

      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. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 70.1%

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

   4. Taylor expanded in a around inf

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

      1. lift-neg.f64 N/A

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

      2. sub-div N/A

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

      3. lift-neg.f64 N/A

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

      4. *-commutative N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. sqrt-unprod N/A

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

      8. *-commutative N/A

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

      9. lower-sqrt.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 31.7

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

   6. Applied rewrites 31.7%

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

   if -2.7999999999999998e-304 < b_2 < 5.99999999999999978e-35

   1. Initial program 82.7%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 30.0

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

   4. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 30.0

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

   6. Applied rewrites 30.0%

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

   if 5.99999999999999978e-35 < b_2

   1. Initial program 67.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

Alternative 8: 71.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-123}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-123)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6e-35) (sqrt (/ (- c) a)) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e-35) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.15d-123)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6d-35) then
        tmp = sqrt((-c / a))
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-123) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e-35) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-123:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6e-35:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-123)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6e-35)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15e-123)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6e-35)
		tmp = sqrt((-c / a));
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-123], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e-35], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.14999999999999993e-123

   1. Initial program 21.4%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 82.4

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

   4. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 82.4

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

   6. Applied rewrites 82.4%

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

   if -1.14999999999999993e-123 < b_2 < 5.99999999999999978e-35

   1. Initial program 78.5%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.7

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

   4. Applied rewrites 31.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 31.7

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

   6. Applied rewrites 31.7%

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

   if 5.99999999999999978e-35 < b_2

   1. Initial program 67.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

Alternative 9: 70.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6e-35) (sqrt (/ (- c) a)) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6e-35) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.15d-123)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 6d-35) then
        tmp = sqrt((-c / a))
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6e-35) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-123:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 6e-35:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6e-35)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15e-123)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 6e-35)
		tmp = sqrt((-c / a));
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e-35], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.14999999999999993e-123

   1. Initial program 21.4%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 82.4

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

   4. Applied rewrites 82.4%

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

   if -1.14999999999999993e-123 < b_2 < 5.99999999999999978e-35

   1. Initial program 78.5%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.7

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

   4. Applied rewrites 31.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 31.7

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

   6. Applied rewrites 31.7%

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

   if 5.99999999999999978e-35 < b_2

   1. Initial program 67.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 89.6

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

   4. Applied rewrites 89.6%

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

Alternative 10: 53.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-123}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-123)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.05e-28) (sqrt (/ (- c) a)) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.05e-28) {
		tmp = sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.15d-123)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.05d-28) then
        tmp = sqrt((-c / a))
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-123) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.05e-28) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-123:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.05e-28:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-123)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.05e-28)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15e-123)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.05e-28)
		tmp = sqrt((-c / a));
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-123], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-28], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.14999999999999993e-123

   1. Initial program 21.4%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 82.4

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

   4. Applied rewrites 82.4%

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

   if -1.14999999999999993e-123 < b_2 < 1.05000000000000003e-28

   1. Initial program 78.7%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.4

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

   4. Applied rewrites 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 31.4

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

   6. Applied rewrites 31.4%

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

   if 1.05000000000000003e-28 < b_2

   1. Initial program 67.6%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 25.3

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

   4. Applied rewrites 25.3%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 37.9

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

   7. Applied rewrites 37.9%

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

Alternative 11: 31.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.4e+146)
   (* (/ c b_2) 0.5)
   (if (<= b_2 1.05e-28) (sqrt (/ (- c) a)) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e+146) {
		tmp = (c / b_2) * 0.5;
	} else if (b_2 <= 1.05e-28) {
		tmp = sqrt((-c / a));
	} 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 <= (-2.4d+146)) then
        tmp = (c / b_2) * 0.5d0
    else if (b_2 <= 1.05d-28) then
        tmp = sqrt((-c / a))
    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 <= -2.4e+146) {
		tmp = (c / b_2) * 0.5;
	} else if (b_2 <= 1.05e-28) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.4e+146:
		tmp = (c / b_2) * 0.5
	elif b_2 <= 1.05e-28:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.4e+146)
		tmp = Float64(Float64(c / b_2) * 0.5);
	elseif (b_2 <= 1.05e-28)
		tmp = sqrt(Float64(Float64(-c) / a));
	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 <= -2.4e+146)
		tmp = (c / b_2) * 0.5;
	elseif (b_2 <= 1.05e-28)
		tmp = sqrt((-c / a));
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.4e+146], N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-28], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.4000000000000002e146

   1. Initial program 3.1%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 2.3

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

   4. Applied rewrites 2.3%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-/.f64 37.7

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

   7. Applied rewrites 37.7%

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

   if -2.4000000000000002e146 < b_2 < 1.05000000000000003e-28

   1. Initial program 59.2%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 24.8

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

   4. Applied rewrites 24.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 24.8

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

   6. Applied rewrites 24.8%

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

   if 1.05000000000000003e-28 < b_2

   1. Initial program 67.6%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 25.3

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

   4. Applied rewrites 25.3%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 37.9

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

   7. Applied rewrites 37.9%

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

Alternative 12: 27.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.05e-28) (sqrt (/ (- c) a)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.05e-28) {
		tmp = sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 1.05d-28) then
        tmp = sqrt((-c / a))
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.05e-28) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.05e-28:
		tmp = math.sqrt((-c / a))
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.05e-28)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.05e-28)
		tmp = sqrt((-c / a));
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.05e-28], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.05000000000000003e-28

   1. Initial program 45.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 22.1

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

   4. Applied rewrites 22.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 22.1

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

   6. Applied rewrites 22.1%

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

   if 1.05000000000000003e-28 < b_2

   1. Initial program 67.6%

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

   2. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. sqrt-unprod N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 25.3

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

   4. Applied rewrites 25.3%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 37.9

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

   7. Applied rewrites 37.9%

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

Alternative 13: 15.3% accurate, 3.4× 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 52.5%

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

2. Taylor expanded in c around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. associate-*r/ N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. sqrt-unprod N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 20.1

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

4. Applied rewrites 20.1%

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

5. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 15.3

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

7. Applied rewrites 15.3%

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

Developer Target 1: 99.7% accurate, 0.4× 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 2025116 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform c (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))