quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.9% → 85.5%

Time: 3.9s

Alternatives: 12

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: 51.9% 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: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, -\frac{b\_2 + b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-7)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 4.5e+124)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (fma (/ c b_2) 0.5 (- (/ (+ b_2 b_2) a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 4.5e+124) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = fma((c / b_2), 0.5, -((b_2 + b_2) / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.2e-7)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 4.5e+124)
		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(Float64(b_2 + b_2) / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.2e-7], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 4.5e+124], 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 + b$95$2), $MachinePrecision] / a), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.1999999999999998e-7

   1. Initial program 14.5%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

   if -8.1999999999999998e-7 < b_2 < 4.5000000000000004e124

   1. Initial program 77.6%

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

   if 4.5000000000000004e124 < b_2

   1. Initial program 47.6%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-pow1 N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. lower-+.f64 96.5

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

   4. Applied rewrites 96.5%

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

Alternative 2: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, -\frac{b\_2 + b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-7)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7e-29)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (fma (/ c b_2) 0.5 (- (/ (+ b_2 b_2) a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-29) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = fma((c / b_2), 0.5, -((b_2 + b_2) / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.2e-7)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7e-29)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(-Float64(Float64(b_2 + b_2) / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.2e-7], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e-29], 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 + b$95$2), $MachinePrecision] / a), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.1999999999999998e-7

   1. Initial program 14.5%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

   if -8.1999999999999998e-7 < b_2 < 6.9999999999999995e-29

   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 inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      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 62.5

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

   4. Applied rewrites 62.5%

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

   if 6.9999999999999995e-29 < b_2

   1. Initial program 65.5%

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

   2. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sqrt-pow1 N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. lower-+.f64 89.2

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

   4. Applied rewrites 89.2%

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

Alternative 3: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-7)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7e-29)
     (/ (- (- 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 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-29) {
		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 <= (-8.2d-7)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 7d-29) 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 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7e-29) {
		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 <= -8.2e-7:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 7e-29:
		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 <= -8.2e-7)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7e-29)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.2e-7)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 7e-29)
		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, -8.2e-7], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7e-29], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.1999999999999998e-7

   1. Initial program 14.5%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

   if -8.1999999999999998e-7 < b_2 < 6.9999999999999995e-29

   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 inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      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 62.5

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

   4. Applied rewrites 62.5%

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

   if 6.9999999999999995e-29 < b_2

   1. Initial program 65.5%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 89.0

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

   4. Applied rewrites 89.0%

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

Alternative 4: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.42 \cdot 10^{-47}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-7)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.42e-47) (/ (- (sqrt (* (- a) c))) a) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.42e-47) {
		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 <= (-8.2d-7)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.42d-47) 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 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.42e-47) {
		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 <= -8.2e-7:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.42e-47:
		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 <= -8.2e-7)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.42e-47)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.2e-7)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.42e-47)
		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, -8.2e-7], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.42e-47], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.1999999999999998e-7

   1. Initial program 14.5%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

   if -8.1999999999999998e-7 < b_2 < 1.41999999999999989e-47

   1. Initial program 71.1%

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

   2. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 62.3

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

   4. Applied rewrites 62.3%

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

   if 1.41999999999999989e-47 < b_2

   1. Initial program 66.8%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 87.5

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

   4. Applied rewrites 87.5%

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

Alternative 5: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.75 \cdot 10^{-150}:\\ \;\;\;\;-\frac{\sqrt{-c}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-7)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.75e-150) (- (/ (sqrt (- c)) (sqrt a))) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.75e-150) {
		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 <= (-8.2d-7)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.75d-150) 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 <= -8.2e-7) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.75e-150) {
		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 <= -8.2e-7:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.75e-150:
		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 <= -8.2e-7)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.75e-150)
		tmp = Float64(-Float64(sqrt(Float64(-c)) / sqrt(a)));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.2e-7)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.75e-150)
		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, -8.2e-7], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.75e-150], (-N[(N[Sqrt[(-c)], $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.1999999999999998e-7

   1. Initial program 14.5%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 90.8

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

   6. Applied rewrites 90.8%

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

   if -8.1999999999999998e-7 < b_2 < 1.7499999999999999e-150

   1. Initial program 66.1%

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

   2. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-sqrt.f64 N/A

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

      5. rem-square-sqrt 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 29.7

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

   4. Applied rewrites 29.7%

      \[\leadsto \color{blue}{-\sqrt{\frac{-c}{a}}} \]
   5. 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. lift-neg.f64 N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-sqrt.f64 38.8

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

   6. Applied rewrites 38.8%

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

   if 1.7499999999999999e-150 < b_2

   1. Initial program 70.7%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 79.7

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

   4. Applied rewrites 79.7%

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

Alternative 6: 72.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-146}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.7 \cdot 10^{-157}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-146)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.7e-157) (- (sqrt (/ (- c) a))) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-146) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-157) {
		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 <= (-5d-146)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.7d-157) 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 <= -5e-146) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.7e-157) {
		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 <= -5e-146:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.7e-157:
		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 <= -5e-146)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.7e-157)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-146)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.7e-157)
		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, -5e-146], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.7e-157], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999957e-146

   1. Initial program 22.0%

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

   2. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 81.2

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

   4. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 81.2

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

   6. Applied rewrites 81.2%

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

   if -4.99999999999999957e-146 < b_2 < 1.69999999999999989e-157

   1. Initial program 75.5%

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

   2. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. rem-square-sqrt N/A

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

      4. lower-sqrt.f64 N/A

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

      5. rem-square-sqrt 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 34.1

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

   4. Applied rewrites 34.1%

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

   if 1.69999999999999989e-157 < b_2

   1. Initial program 70.8%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 79.1

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

   4. Applied rewrites 79.1%

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

Alternative 7: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.5e-143)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.9e-187) (sqrt (/ (- c) a)) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-143) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-187) {
		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 <= (-6.5d-143)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.9d-187) 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 <= -6.5e-143) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.9e-187) {
		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 <= -6.5e-143:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.9e-187:
		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 <= -6.5e-143)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.9e-187)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.5e-143)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.9e-187)
		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, -6.5e-143], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-187], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.4999999999999999e-143

   1. Initial program 21.8%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 81.4

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

   6. Applied rewrites 81.4%

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

   if -6.4999999999999999e-143 < b_2 < 2.89999999999999988e-187

   1. Initial program 75.4%

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

   2. Taylor expanded in a around -inf

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

      1. rem-square-sqrt N/A

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

      2. lower-sqrt.f64 N/A

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

      3. rem-square-sqrt 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 37.4

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

   4. Applied rewrites 37.4%

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

   if 2.89999999999999988e-187 < b_2

   1. Initial program 70.9%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

Alternative 8: 72.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.5e-143)
   (* -0.5 (/ c b_2))
   (if (<= b_2 2.9e-187) (sqrt (/ (- c) a)) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.5e-143) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.9e-187) {
		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 <= (-6.5d-143)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 2.9d-187) 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 <= -6.5e-143) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.9e-187) {
		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 <= -6.5e-143:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 2.9e-187:
		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 <= -6.5e-143)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.9e-187)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.5e-143)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 2.9e-187)
		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, -6.5e-143], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-187], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.4999999999999999e-143

   1. Initial program 21.8%

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

   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. associate-*r/ N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. associate-*r/ N/A

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

      6. lower-*.f64 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. lower-/.f64 81.4

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

   4. Applied rewrites 81.4%

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

   if -6.4999999999999999e-143 < b_2 < 2.89999999999999988e-187

   1. Initial program 75.4%

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

   2. Taylor expanded in a around -inf

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

      1. rem-square-sqrt N/A

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

      2. lower-sqrt.f64 N/A

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

      3. rem-square-sqrt 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 37.4

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

   4. Applied rewrites 37.4%

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

   if 2.89999999999999988e-187 < b_2

   1. Initial program 70.9%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

Alternative 9: 49.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.3 \cdot 10^{-81}:\\ \;\;\;\;0\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.3e-81)
   0.0
   (if (<= b_2 2.9e-187) (sqrt (/ (- c) a)) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.3e-81) {
		tmp = 0.0;
	} else if (b_2 <= 2.9e-187) {
		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 <= (-6.3d-81)) then
        tmp = 0.0d0
    else if (b_2 <= 2.9d-187) 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 <= -6.3e-81) {
		tmp = 0.0;
	} else if (b_2 <= 2.9e-187) {
		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 <= -6.3e-81:
		tmp = 0.0
	elif b_2 <= 2.9e-187:
		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 <= -6.3e-81)
		tmp = 0.0;
	elseif (b_2 <= 2.9e-187)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.3e-81)
		tmp = 0.0;
	elseif (b_2 <= 2.9e-187)
		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, -6.3e-81], 0.0, If[LessEqual[b$95$2, 2.9e-187], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.30000000000000023e-81

   1. Initial program 18.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 17.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{{\left(b\_2 \cdot b\_2 - \color{blue}{c \cdot a}\right)}^{-0.5}}}{a} \]

   3. Applied rewrites 17.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{{\left(b\_2 \cdot b\_2 - c \cdot a\right)}^{-0.5}}}}{a} \]

   4. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{0} \]
   5. Step-by-step derivation

   6. Applied rewrites 25.6%

      \[\leadsto \color{blue}{0} \]

   if -6.30000000000000023e-81 < b_2 < 2.89999999999999988e-187

   1. Initial program 71.6%

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

   2. Taylor expanded in a around -inf

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

      1. rem-square-sqrt N/A

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

      2. lower-sqrt.f64 N/A

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

      3. rem-square-sqrt 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 34.5

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

   4. Applied rewrites 34.5%

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

   if 2.89999999999999988e-187 < b_2

   1. Initial program 70.9%

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

   2. Taylor expanded in b_2 around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 76.8

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

   4. Applied rewrites 76.8%

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

Alternative 10: 31.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6.3 \cdot 10^{-81}:\\ \;\;\;\;0\\ \mathbf{elif}\;b\_2 \leq 9000000:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.3e-81)
   0.0
   (if (<= b_2 9000000.0) (sqrt (/ (- c) a)) (/ (- b_2) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.3e-81) {
		tmp = 0.0;
	} else if (b_2 <= 9000000.0) {
		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 <= (-6.3d-81)) then
        tmp = 0.0d0
    else if (b_2 <= 9000000.0d0) 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 <= -6.3e-81) {
		tmp = 0.0;
	} else if (b_2 <= 9000000.0) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.3e-81:
		tmp = 0.0
	elif b_2 <= 9000000.0:
		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 <= -6.3e-81)
		tmp = 0.0;
	elseif (b_2 <= 9000000.0)
		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 <= -6.3e-81)
		tmp = 0.0;
	elseif (b_2 <= 9000000.0)
		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, -6.3e-81], 0.0, If[LessEqual[b$95$2, 9000000.0], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.30000000000000023e-81

   1. Initial program 18.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 17.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{{\left(b\_2 \cdot b\_2 - \color{blue}{c \cdot a}\right)}^{-0.5}}}{a} \]

   3. Applied rewrites 17.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{{\left(b\_2 \cdot b\_2 - c \cdot a\right)}^{-0.5}}}}{a} \]

   4. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{0} \]
   5. Step-by-step derivation

   6. Applied rewrites 25.6%

      \[\leadsto \color{blue}{0} \]

   if -6.30000000000000023e-81 < b_2 < 9e6

   1. Initial program 78.4%

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

   2. Taylor expanded in a around -inf

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

      1. rem-square-sqrt N/A

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

      2. lower-sqrt.f64 N/A

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

      3. rem-square-sqrt 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 29.2

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

   4. Applied rewrites 29.2%

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

   if 9e6 < b_2

   1. Initial program 63.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 63.3

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{{\left(b\_2 \cdot b\_2 - \color{blue}{c \cdot a}\right)}^{-0.5}}}{a} \]

   3. Applied rewrites 63.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{{\left(b\_2 \cdot b\_2 - c \cdot a\right)}^{-0.5}}}}{a} \]

   4. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{0} \]
   5. Step-by-step derivation

   6. Applied rewrites 2.4%

      \[\leadsto \color{blue}{0} \]

   7. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 40.3

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

   9. Applied rewrites 40.3%

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

Alternative 11: 24.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.32 \cdot 10^{-191}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.32e-191) 0.0 (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.32e-191) {
		tmp = 0.0;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.32d-191)) then
        tmp = 0.0d0
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.32e-191) {
		tmp = 0.0;
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.32e-191:
		tmp = 0.0
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.32e-191)
		tmp = 0.0;
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.32e-191)
		tmp = 0.0;
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.32e-191], 0.0, N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.31999999999999996e-191

   1. Initial program 25.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 24.2

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{{\left(b\_2 \cdot b\_2 - \color{blue}{c \cdot a}\right)}^{-0.5}}}{a} \]

   3. Applied rewrites 24.2%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{{\left(b\_2 \cdot b\_2 - c \cdot a\right)}^{-0.5}}}}{a} \]

   4. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{0} \]
   5. Step-by-step derivation

   6. Applied rewrites 22.2%

      \[\leadsto \color{blue}{0} \]

   if -1.31999999999999996e-191 < b_2

   1. Initial program 72.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. pow1/2 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 71.9

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{{\left(b\_2 \cdot b\_2 - \color{blue}{c \cdot a}\right)}^{-0.5}}}{a} \]

   3. Applied rewrites 71.9%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{{\left(b\_2 \cdot b\_2 - c \cdot a\right)}^{-0.5}}}}{a} \]

   4. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{0} \]
   5. Step-by-step derivation

   6. Applied rewrites 2.8%

      \[\leadsto \color{blue}{0} \]

   7. Taylor expanded in b_2 around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-/.f64 25.3

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

   9. Applied rewrites 25.3%

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

Alternative 12: 11.1% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 51.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. pow1/2 N/A

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

   6. metadata-eval N/A

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 51.4

       \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{{\left(b\_2 \cdot b\_2 - \color{blue}{c \cdot a}\right)}^{-0.5}}}{a} \]

3. Applied rewrites 51.4%

   \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{{\left(b\_2 \cdot b\_2 - c \cdot a\right)}^{-0.5}}}}{a} \]

4. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{0} \]
5. Step-by-step derivation

6. Applied rewrites 11.1%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Developer Target 1: 99.6% 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 2025134 
(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))