quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.9% → 85.8%

Time: 3.3s

Alternatives: 11

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: 52.9% 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: 85.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.4e-54)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 9.5e+87)
     (/ (- (- 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 <= -7.4e-54) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9.5e+87) {
		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 <= -7.4e-54)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 9.5e+87)
		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, -7.4e-54], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9.5e+87], 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 < -7.4000000000000006e-54

   1. Initial program 16.5%

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

   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 87.3

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

   4. Applied rewrites 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 87.3

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

   6. Applied rewrites 87.3%

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

   if -7.4000000000000006e-54 < b_2 < 9.4999999999999992e87

   1. Initial program 79.3%

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

   if 9.4999999999999992e87 < b_2

   1. Initial program 57.3%

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

   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 96.1

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

   4. Applied rewrites 96.1%

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

Alternative 2: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-56)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.7e-17)
     (/ (- (- 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 <= -1.2e-56) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e-17) {
		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 <= -1.2e-56)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.7e-17)
		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, -1.2e-56], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-17], 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 < -1.2e-56

   1. Initial program 16.8%

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

   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 86.9

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

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 87.0

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

   6. Applied rewrites 87.0%

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

   if -1.2e-56 < b_2 < 3.6999999999999997e-17

   1. Initial program 76.3%

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

   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 64.8

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

   4. Applied rewrites 64.8%

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

   if 3.6999999999999997e-17 < b_2

   1. Initial program 66.9%

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

   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 90.4

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

   4. Applied rewrites 90.4%

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

Alternative 3: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.2e-56)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.7e-17)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.2e-56) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e-17) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = (-2.0 * 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 <= (-1.2d-56)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3.7d-17) then
        tmp = (-b_2 - sqrt((-a * c))) / a
    else
        tmp = ((-2.0d0) * 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 <= -1.2e-56) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e-17) {
		tmp = (-b_2 - Math.sqrt((-a * c))) / a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.2e-56:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3.7e-17:
		tmp = (-b_2 - math.sqrt((-a * c))) / a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.2e-56)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.7e-17)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.2e-56)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3.7e-17)
		tmp = (-b_2 - sqrt((-a * c))) / a;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.2e-56], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-17], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.2e-56

   1. Initial program 16.8%

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

   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 86.9

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

   4. Applied rewrites 86.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 87.0

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

   6. Applied rewrites 87.0%

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

   if -1.2e-56 < b_2 < 3.6999999999999997e-17

   1. Initial program 76.3%

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

   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 64.8

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

   4. Applied rewrites 64.8%

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

   if 3.6999999999999997e-17 < b_2

   1. Initial program 66.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.4e-54)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 3.7e-17) (/ (- (sqrt (* (- a) c))) a) (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.4e-54) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e-17) {
		tmp = -sqrt((-a * c)) / a;
	} else {
		tmp = (-2.0 * 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 <= (-7.4d-54)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 3.7d-17) then
        tmp = -sqrt((-a * c)) / a
    else
        tmp = ((-2.0d0) * 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 <= -7.4e-54) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 3.7e-17) {
		tmp = -Math.sqrt((-a * c)) / a;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.4e-54:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 3.7e-17:
		tmp = -math.sqrt((-a * c)) / a
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.4e-54)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 3.7e-17)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.4e-54)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 3.7e-17)
		tmp = -sqrt((-a * c)) / a;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.4e-54], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.7e-17], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.4000000000000006e-54

   1. Initial program 16.5%

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

   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 87.3

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

   4. Applied rewrites 87.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 87.3

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

   6. Applied rewrites 87.3%

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

   if -7.4000000000000006e-54 < b_2 < 3.6999999999999997e-17

   1. Initial program 76.3%

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

   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 63.2

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

   4. Applied rewrites 63.2%

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

   if 3.6999999999999997e-17 < b_2

   1. Initial program 66.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

Alternative 5: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-160}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.02 \cdot 10^{-134}:\\ \;\;\;\;-\frac{\sqrt{-c}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-160)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.02e-134) (- (/ (sqrt (- c)) (sqrt a))) (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-160) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.02e-134) {
		tmp = -(sqrt(-c) / sqrt(a));
	} else {
		tmp = (-2.0 * 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 <= (-2d-160)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.02d-134) then
        tmp = -(sqrt(-c) / sqrt(a))
    else
        tmp = ((-2.0d0) * 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 <= -2e-160) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.02e-134) {
		tmp = -(Math.sqrt(-c) / Math.sqrt(a));
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-160:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.02e-134:
		tmp = -(math.sqrt(-c) / math.sqrt(a))
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-160)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.02e-134)
		tmp = Float64(-Float64(sqrt(Float64(-c)) / sqrt(a)));
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-160)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.02e-134)
		tmp = -(sqrt(-c) / sqrt(a));
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-160], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.02e-134], (-N[(N[Sqrt[(-c)], $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2e-160

   1. Initial program 23.8%

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

   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 79.0

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

   4. Applied rewrites 79.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 79.0

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

   6. Applied rewrites 79.0%

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

   if -2e-160 < b_2 < 1.02e-134

   1. Initial program 76.0%

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

   2. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 36.7

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

   6. Applied rewrites 36.7%

      \[\leadsto -\sqrt{\frac{-c}{a}} \]
   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. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 45.6

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

   8. Applied rewrites 45.6%

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

   if 1.02e-134 < b_2

   1. Initial program 72.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 81.1

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

   4. Applied rewrites 81.1%

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

Alternative 6: 71.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-174}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-129}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-174)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1e-129) (sqrt (/ (- c) a)) (/ (* -2.0 b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-174) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-129) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (-2.0 * 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 <= (-4.6d-174)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1d-129) then
        tmp = sqrt((-c / a))
    else
        tmp = ((-2.0d0) * 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 <= -4.6e-174) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-129) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.6e-174:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1e-129:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.6e-174)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1e-129)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.6e-174)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1e-129)
		tmp = sqrt((-c / a));
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.6e-174], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1e-129], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.5999999999999998e-174

   1. Initial program 24.8%

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

   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 77.8

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

   4. Applied rewrites 77.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 77.8

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

   6. Applied rewrites 77.8%

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

   if -4.5999999999999998e-174 < b_2 < 9.9999999999999993e-130

   1. Initial program 76.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 76.5%

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

   4. Taylor expanded in a around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 37.1

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

   6. Applied rewrites 37.1%

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

   if 9.9999999999999993e-130 < b_2

   1. Initial program 71.9%

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

   2. Taylor expanded in a around 0

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

      1. lower-*.f64 81.5

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

   4. Applied rewrites 81.5%

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

Alternative 7: 53.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-174}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-174)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 5.5e+50) (sqrt (/ (- c) a)) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-174) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.5e+50) {
		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 <= (-4.6d-174)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 5.5d+50) 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 <= -4.6e-174) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 5.5e+50) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.6e-174:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 5.5e+50:
		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 <= -4.6e-174)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 5.5e+50)
		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 <= -4.6e-174)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 5.5e+50)
		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, -4.6e-174], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5.5e+50], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.5999999999999998e-174

   1. Initial program 24.8%

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

   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 77.8

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

   4. Applied rewrites 77.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   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 77.8

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

   6. Applied rewrites 77.8%

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

   if -4.5999999999999998e-174 < b_2 < 5.4999999999999998e50

   1. Initial program 82.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 82.6%

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

   4. Taylor expanded in a around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 28.6

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

   6. Applied rewrites 28.6%

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

   if 5.4999999999999998e50 < b_2

   1. Initial program 62.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 62.1%

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

   4. Taylor expanded in c around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. +-commutative N/A

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 43.9

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

   9. Applied rewrites 43.9%

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

Alternative 8: 53.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-174}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-174)
   (* -0.5 (/ c b_2))
   (if (<= b_2 5.5e+50) (sqrt (/ (- c) a)) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-174) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5.5e+50) {
		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 <= (-4.6d-174)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 5.5d+50) 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 <= -4.6e-174) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 5.5e+50) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.6e-174:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 5.5e+50:
		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 <= -4.6e-174)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 5.5e+50)
		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 <= -4.6e-174)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 5.5e+50)
		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, -4.6e-174], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 5.5e+50], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.5999999999999998e-174

   1. Initial program 24.8%

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

   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 77.8

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

   4. Applied rewrites 77.8%

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

   if -4.5999999999999998e-174 < b_2 < 5.4999999999999998e50

   1. Initial program 82.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 82.6%

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

   4. Taylor expanded in a around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 28.6

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

   6. Applied rewrites 28.6%

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

   if 5.4999999999999998e50 < b_2

   1. Initial program 62.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 62.1%

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

   4. Taylor expanded in c around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. +-commutative N/A

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 43.9

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

   9. Applied rewrites 43.9%

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

Alternative 9: 31.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \mathbf{elif}\;b\_2 \leq 5.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.8e+127)
   (* (/ c b_2) 0.5)
   (if (<= b_2 5.5e+50) (sqrt (/ (- c) a)) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.8e+127) {
		tmp = (c / b_2) * 0.5;
	} else if (b_2 <= 5.5e+50) {
		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 <= (-4.8d+127)) then
        tmp = (c / b_2) * 0.5d0
    else if (b_2 <= 5.5d+50) 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 <= -4.8e+127) {
		tmp = (c / b_2) * 0.5;
	} else if (b_2 <= 5.5e+50) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.8e+127:
		tmp = (c / b_2) * 0.5
	elif b_2 <= 5.5e+50:
		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 <= -4.8e+127)
		tmp = Float64(Float64(c / b_2) * 0.5);
	elseif (b_2 <= 5.5e+50)
		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 <= -4.8e+127)
		tmp = (c / b_2) * 0.5;
	elseif (b_2 <= 5.5e+50)
		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, -4.8e+127], N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[b$95$2, 5.5e+50], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.8000000000000004e127

   1. Initial program 5.1%

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

   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 37.6

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

   7. Applied rewrites 37.6%

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

   if -4.8000000000000004e127 < b_2 < 5.4999999999999998e50

   1. Initial program 64.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 64.1%

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

   4. Taylor expanded in a around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 24.3

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

   6. Applied rewrites 24.3%

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

   if 5.4999999999999998e50 < b_2

   1. Initial program 62.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 62.1%

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

   4. Taylor expanded in c around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. +-commutative N/A

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 43.9

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

   9. Applied rewrites 43.9%

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

Alternative 10: 27.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.5 \cdot 10^{+50}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.5e+50) (sqrt (/ (- c) a)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.5e+50) {
		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 <= 5.5d+50) 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 <= 5.5e+50) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5.5e+50:
		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 <= 5.5e+50)
		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 <= 5.5e+50)
		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, 5.5e+50], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.4999999999999998e50

   1. Initial program 49.7%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 49.1%

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

   4. Taylor expanded in a around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 21.5

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

   6. Applied rewrites 21.5%

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

   if 5.4999999999999998e50 < b_2

   1. Initial program 62.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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 62.1%

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

   4. Taylor expanded in c around -inf

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

      1. sub-div N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. +-commutative N/A

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 43.9

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

   9. Applied rewrites 43.9%

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

Alternative 11: 15.5% 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 52.9%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
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 52.5%

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

4. Taylor expanded in c around -inf

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

   1. sub-div N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

   5. negate-sub N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. +-commutative N/A

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

6. Applied rewrites 32.0%

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

7. Taylor expanded in b_2 around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lower-/.f64 15.5

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

9. Applied rewrites 15.5%

   \[\leadsto \frac{-b\_2}{\color{blue}{a}} \]
10. 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 2025110 
(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))