quad2p (problem 3.2.1, positive)

Percentage Accurate: 53.0% → 85.5%

Time: 3.6s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.0% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-a, c, b\_2 \cdot b\_2\right)} + \left(-b\_2\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e+62)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.05e-79)
     (/ (+ (sqrt (fma (- a) c (* b_2 b_2))) (- b_2)) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+62) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = (sqrt(fma(-a, c, (b_2 * b_2))) + -b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e+62)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.05e-79)
		tmp = Float64(Float64(sqrt(fma(Float64(-a), c, Float64(b_2 * b_2))) + Float64(-b_2)) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e+62], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-79], N[(N[(N[Sqrt[N[((-a) * c + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-b$95$2)), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.20000000000000015e62

   1. Initial program 61.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

   if -2.20000000000000015e62 < b_2 < 1.05e-79

   1. Initial program 79.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-neg.f64 79.8

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

   3. Applied rewrites 79.8%

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

   if 1.05e-79 < b_2

   1. Initial program 19.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e+62)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.05e-79)
     (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+62) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.2d+62)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.05d-79) then
        tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e+62) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.2e+62:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.05e-79:
		tmp = (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e+62)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.05e-79)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.2e+62)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.05e-79)
		tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e+62], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-79], 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), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.20000000000000015e62

   1. Initial program 61.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

   if -2.20000000000000015e62 < b_2 < 1.05e-79

   1. Initial program 79.8%

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

   if 1.05e-79 < b_2

   1. Initial program 19.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.55 \cdot 10^{-38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b\_2} \cdot a, -2 \cdot b\_2\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.55e-38)
   (/ (fma 0.5 (* (/ c b_2) a) (* -2.0 b_2)) a)
   (if (<= b_2 1.05e-79)
     (/ (+ (- b_2) (sqrt (* (- a) c))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.55e-38) {
		tmp = fma(0.5, ((c / b_2) * a), (-2.0 * b_2)) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = (-b_2 + sqrt((-a * c))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.55e-38)
		tmp = Float64(fma(0.5, Float64(Float64(c / b_2) * a), Float64(-2.0 * b_2)) / a);
	elseif (b_2 <= 1.05e-79)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.55e-38], N[(N[(0.5 * N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] + N[(-2.0 * b$95$2), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-79], N[(N[((-b$95$2) + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.55000000000000006e-38

   1. Initial program 69.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-neg.f64 69.8

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

   3. Applied rewrites 69.8%

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

   4. Taylor expanded in b_2 around -inf

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

      1. +-commutative N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. pow2 N/A

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

   6. Applied rewrites 89.7%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 89.7

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

   9. Applied rewrites 89.7%

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

   if -4.55000000000000006e-38 < b_2 < 1.05e-79

   1. Initial program 75.9%

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

   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 65.7

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

   4. Applied rewrites 65.7%

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

   if 1.05e-79 < b_2

   1. Initial program 19.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.55 \cdot 10^{-38}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\left(-b\_2\right) + \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.55e-38)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.05e-79)
     (/ (+ (- b_2) (sqrt (* (- a) c))) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.55e-38) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = (-b_2 + sqrt((-a * c))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-4.55d-38)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.05d-79) then
        tmp = (-b_2 + sqrt((-a * c))) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.55e-38) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = (-b_2 + Math.sqrt((-a * c))) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.55e-38:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.05e-79:
		tmp = (-b_2 + math.sqrt((-a * c))) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.55e-38)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.05e-79)
		tmp = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.55e-38)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.05e-79)
		tmp = (-b_2 + sqrt((-a * c))) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.55e-38], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-79], N[(N[((-b$95$2) + N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.55000000000000006e-38

   1. Initial program 69.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 89.5

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

   4. Applied rewrites 89.5%

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

   if -4.55000000000000006e-38 < b_2 < 1.05e-79

   1. Initial program 75.9%

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

   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 65.7

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

   4. Applied rewrites 65.7%

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

   if 1.05e-79 < b_2

   1. Initial program 19.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.05 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.5e-70)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.05e-79) (/ (sqrt (* (- a) c)) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-70) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = sqrt((-a * c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-7.5d-70)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.05d-79) then
        tmp = sqrt((-a * c)) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-70) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.05e-79) {
		tmp = Math.sqrt((-a * c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.5e-70:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.05e-79:
		tmp = math.sqrt((-a * c)) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.5e-70)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.05e-79)
		tmp = Float64(sqrt(Float64(Float64(-a) * c)) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.5e-70)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.05e-79)
		tmp = sqrt((-a * c)) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.5e-70], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-79], N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.49999999999999973e-70

   1. Initial program 71.1%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if -7.49999999999999973e-70 < b_2 < 1.05e-79

   1. Initial program 74.6%

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

   2. Taylor expanded in a around inf

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. lower-sqrt.f64 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 65.9

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

   4. Applied rewrites 65.9%

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

   if 1.05e-79 < b_2

   1. Initial program 19.5%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 84.9

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

   4. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-133}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.5e-70)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 8e-133) (- (sqrt (/ (- c) a))) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-70) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 8e-133) {
		tmp = -sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-7.5d-70)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 8d-133) then
        tmp = -sqrt((-c / a))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.5e-70) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 8e-133) {
		tmp = -Math.sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.5e-70:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 8e-133:
		tmp = -math.sqrt((-c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.5e-70)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 8e-133)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.5e-70)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 8e-133)
		tmp = -sqrt((-c / a));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.5e-70], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8e-133], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.49999999999999973e-70

   1. Initial program 71.1%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 86.6

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

   4. Applied rewrites 86.6%

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

   if -7.49999999999999973e-70 < b_2 < 8.0000000000000005e-133

   1. Initial program 76.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-neg.f64 76.7

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

   3. Applied rewrites 76.7%

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

   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

   6. Applied rewrites 31.9%

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

   if 8.0000000000000005e-133 < b_2

   1. Initial program 22.4%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.48 \cdot 10^{-245}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.7 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.48e-245)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.7e-206) (sqrt (/ (- c) a)) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.48e-245) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.7e-206) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-1.48d-245)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.7d-206) then
        tmp = sqrt((-c / a))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.48e-245) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.7e-206) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.48e-245:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.7e-206:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.48e-245)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.7e-206)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.48e-245)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.7e-206)
		tmp = sqrt((-c / a));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.48e-245], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.7e-206], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.48e-245

   1. Initial program 73.7%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 72.7

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

   4. Applied rewrites 72.7%

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

   if -1.48e-245 < b_2 < 1.69999999999999992e-206

   1. Initial program 76.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. mul-1-neg N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. pow2 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-neg.f64 76.1

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

   3. Applied rewrites 76.1%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 37.7%

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

   if 1.69999999999999992e-206 < b_2

   1. Initial program 26.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 75.5

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

   4. Applied rewrites 75.5%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5e-279) (/ (* -2.0 b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5e-279) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 5d-279) then
        tmp = ((-2.0d0) * b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5e-279) {
		tmp = (-2.0 * b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5e-279:
		tmp = (-2.0 * b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5e-279)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 5e-279)
		tmp = (-2.0 * b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5e-279], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.99999999999999969e-279

   1. Initial program 74.0%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 66.7

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

   4. Applied rewrites 66.7%

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

   if 4.99999999999999969e-279 < b_2

   1. Initial program 30.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.9

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

   4. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 48.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5 \cdot 10^{-279}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5e-279) (/ (- b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5e-279) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 5d-279) then
        tmp = -b_2 / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5e-279) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5e-279:
		tmp = -b_2 / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5e-279)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 5e-279)
		tmp = -b_2 / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5e-279], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 4.99999999999999969e-279

   1. Initial program 74.0%

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

   2. Taylor expanded in a around -inf

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. lower-/.f64 N/A

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

      10. lift-neg.f64 25.9

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

   4. Applied rewrites 25.9%

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

   5. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 28.3

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

   7. Applied rewrites 28.3%

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

   if 4.99999999999999969e-279 < b_2

   1. Initial program 30.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.9

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

   4. Applied rewrites 69.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 15.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

2. Taylor expanded in a around -inf

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. sqrt-unprod N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. mul-1-neg N/A

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

   9. lower-/.f64 N/A

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

   10. lift-neg.f64 19.2

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

4. Applied rewrites 19.2%

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

5. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

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

   2. distribute-frac-neg N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 15.8

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

7. Applied rewrites 15.8%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2025099 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

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

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