quad2p (problem 3.2.1, positive)

Percentage Accurate: 53.3% → 90.4%

Time: 4.0s

Alternatives: 11

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.3% 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: 90.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\left|b\_2\right|}, -0.5, \frac{\left|b\_2\right| - b\_2}{a}\right)\\ t_1 := \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ t_2 := \sqrt{b\_2 \cdot b\_2 - c \cdot a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-259}:\\ \;\;\;\;\frac{-b\_2}{a} + \frac{t\_2}{a}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 10^{+276}:\\ \;\;\;\;\frac{t\_2 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (fma (/ c (fabs b_2)) -0.5 (/ (- (fabs b_2) b_2) a)))
        (t_1 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
        (t_2 (sqrt (- (* b_2 b_2) (* c a)))))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -5e-259)
       (+ (/ (- b_2) a) (/ t_2 a))
       (if (<= t_1 0.0)
         (* (/ c b_2) -0.5)
         (if (<= t_1 1e+276) (/ (- t_2 b_2) a) t_0))))))

C

double code(double a, double b_2, double c) {
	double t_0 = fma((c / fabs(b_2)), -0.5, ((fabs(b_2) - b_2) / a));
	double t_1 = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	double t_2 = sqrt(((b_2 * b_2) - (c * a)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -5e-259) {
		tmp = (-b_2 / a) + (t_2 / a);
	} else if (t_1 <= 0.0) {
		tmp = (c / b_2) * -0.5;
	} else if (t_1 <= 1e+276) {
		tmp = (t_2 - b_2) / a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = fma(Float64(c / abs(b_2)), -0.5, Float64(Float64(abs(b_2) - b_2) / a))
	t_1 = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
	t_2 = sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -5e-259)
		tmp = Float64(Float64(Float64(-b_2) / a) + Float64(t_2 / a));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (t_1 <= 1e+276)
		tmp = Float64(Float64(t_2 - b_2) / a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(c / N[Abs[b$95$2], $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(N[Abs[b$95$2], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -5e-259], N[(N[((-b$95$2) / a), $MachinePrecision] + N[(t$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e+276], N[(N[(t$95$2 - b$95$2), $MachinePrecision] / a), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < -inf.0 or 1.0000000000000001e276 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a)

   1. Initial program 53.3%

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

   2. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 67.5

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

   4. Applied rewrites 67.5%

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

   if -inf.0 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < -4.99999999999999977e-259

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. div-add N/A

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

      8. lift-neg.f64 N/A

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

      9. mul-1-neg N/A

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

      10. associate-*r/ N/A

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

      11. lower-+.f64 N/A

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

      12. associate-*r/ N/A

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

      13. mul-1-neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-/.f64 N/A

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

   3. Applied rewrites 52.5%

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

   if -4.99999999999999977e-259 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < 0.0

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

   if 0.0 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < 1.0000000000000001e276

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 53.3%

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

Alternative 2: 90.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\left|b\_2\right|}, -0.5, \frac{\left|b\_2\right| - b\_2}{a}\right)\\ t_1 := \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ t_2 := \frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-259}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 10^{+276}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (fma (/ c (fabs b_2)) -0.5 (/ (- (fabs b_2) b_2) a)))
        (t_1 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
        (t_2 (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)))
   (if (<= t_1 (- INFINITY))
     t_0
     (if (<= t_1 -5e-259)
       t_2
       (if (<= t_1 0.0) (* (/ c b_2) -0.5) (if (<= t_1 1e+276) t_2 t_0))))))

C

double code(double a, double b_2, double c) {
	double t_0 = fma((c / fabs(b_2)), -0.5, ((fabs(b_2) - b_2) / a));
	double t_1 = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
	double t_2 = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_1 <= -5e-259) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (c / b_2) * -0.5;
	} else if (t_1 <= 1e+276) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	t_0 = fma(Float64(c / abs(b_2)), -0.5, Float64(Float64(abs(b_2) - b_2) / a))
	t_1 = Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
	t_2 = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_1 <= -5e-259)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (t_1 <= 1e+276)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(c / N[Abs[b$95$2], $MachinePrecision]), $MachinePrecision] * -0.5 + N[(N[(N[Abs[b$95$2], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$0, If[LessEqual[t$95$1, -5e-259], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$1, 1e+276], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < -inf.0 or 1.0000000000000001e276 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a)

   1. Initial program 53.3%

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

   2. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. div-sub N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. lower-fabs.f64 67.5

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

   4. Applied rewrites 67.5%

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

   if -inf.0 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < -4.99999999999999977e-259 or 0.0 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < 1.0000000000000001e276

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 53.3%

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

   if -4.99999999999999977e-259 < (/.f64 (+.f64 (neg.f64 b_2) (sqrt.f64 (-.f64 (*.f64 b_2 b_2) (*.f64 a c)))) a) < 0.0

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 3: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.9 \cdot 10^{+65}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.9e+65)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 2e+48)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e+65) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2e+48) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.9d+65)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 2d+48) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.9e+65) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2e+48) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.9e+65:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 2e+48:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.9e+65)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 2e+48)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.9e+65)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 2e+48)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.9e+65], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2e+48], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.8999999999999998e65

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 35.5

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

   4. Applied rewrites 35.5%

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

   if -3.8999999999999998e65 < b_2 < 2.00000000000000009e48

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-+.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 53.3%

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

   if 2.00000000000000009e48 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 4: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 0.0136:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.4e-96)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 0.0136) (/ (- (sqrt (* (- a) c)) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-96) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 0.0136) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.4d-96)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 0.0136d0) then
        tmp = (sqrt((-a * c)) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.4e-96) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 0.0136) {
		tmp = (Math.sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.4e-96:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 0.0136:
		tmp = (math.sqrt((-a * c)) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.4e-96)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 0.0136)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.4e-96)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 0.0136)
		tmp = (sqrt((-a * c)) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.4e-96], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 0.0136], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.40000000000000019e-96

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 35.5

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

   4. Applied rewrites 35.5%

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

   if -2.40000000000000019e-96 < b_2 < 0.0135999999999999992

   1. Initial program 53.3%

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

   2. Taylor expanded in a around inf

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

      1. 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 34.6

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

   4. Applied rewrites 34.6%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lower-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 34.6

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

   6. Applied rewrites 34.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lower-/.f64 34.6

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

      5. lift-+.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. sub-flip-reverse N/A

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

      8. lower--.f64 34.6

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

   8. Applied rewrites 34.6%

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

   if 0.0135999999999999992 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-96}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 0.0136:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-96)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 0.0136) (/ (sqrt (* (- a) c)) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-96) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 0.0136) {
		tmp = sqrt((-a * c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.15d-96)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 0.0136d0) then
        tmp = sqrt((-a * c)) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-96) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 0.0136) {
		tmp = Math.sqrt((-a * c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-96:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 0.0136:
		tmp = math.sqrt((-a * c)) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-96)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 0.0136)
		tmp = Float64(sqrt(Float64(Float64(-a) * c)) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15e-96)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 0.0136)
		tmp = sqrt((-a * c)) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-96], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 0.0136], N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.15e-96

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 35.5

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

   4. Applied rewrites 35.5%

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

   if -1.15e-96 < b_2 < 0.0135999999999999992

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around 0

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

      1. lower-sqrt.f64 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 30.0

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

   4. Applied rewrites 30.0%

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

   if 0.0135999999999999992 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 6: 73.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-121}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{\sqrt{-c}}{\sqrt{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-121)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 5.5e-108) (/ (sqrt (- c)) (sqrt a)) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-121) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 5.5e-108) {
		tmp = sqrt(-c) / sqrt(a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-121)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 5.5d-108) then
        tmp = sqrt(-c) / sqrt(a)
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-121) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 5.5e-108) {
		tmp = Math.sqrt(-c) / Math.sqrt(a);
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-121:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 5.5e-108:
		tmp = math.sqrt(-c) / math.sqrt(a)
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-121)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 5.5e-108)
		tmp = Float64(sqrt(Float64(-c)) / sqrt(a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-121)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 5.5e-108)
		tmp = sqrt(-c) / sqrt(a);
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-121], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 5.5e-108], N[(N[Sqrt[(-c)], $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.9999999999999998e-122

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 35.5

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

   4. Applied rewrites 35.5%

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

   if -9.9999999999999998e-122 < b_2 < 5.50000000000000031e-108

   1. Initial program 53.3%

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

   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 17.1

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

   4. Applied rewrites 17.1%

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

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 16.9

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

   6. Applied rewrites 16.9%

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

   if 5.50000000000000031e-108 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 7: 71.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.3 \cdot 10^{-158}:\\ \;\;\;\;-\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-198)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 2.3e-158) (- (sqrt (/ (- c) a))) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-198) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.3e-158) {
		tmp = -sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.2d-198)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 2.3d-158) then
        tmp = -sqrt((-c / a))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-198) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 2.3e-158) {
		tmp = -Math.sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-198:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 2.3e-158:
		tmp = -math.sqrt((-c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-198)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 2.3e-158)
		tmp = Float64(-sqrt(Float64(Float64(-c) / a)));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.2e-198)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 2.3e-158)
		tmp = -sqrt((-c / a));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-198], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.3e-158], (-N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]), N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.19999999999999994e-198

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 35.5

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

   4. Applied rewrites 35.5%

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

   if -3.19999999999999994e-198 < b_2 < 2.2999999999999999e-158

   1. Initial program 53.3%

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

   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 17.6

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

   4. Applied rewrites 17.6%

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

   if 2.2999999999999999e-158 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 8: 70.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.7 \cdot 10^{-193}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.7e-193)
   (/ (* -2.0 b_2) a)
   (if (<= b_2 1.5e-200) (sqrt (/ (- c) a)) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-193) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.5e-200) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.7d-193)) then
        tmp = ((-2.0d0) * b_2) / a
    else if (b_2 <= 1.5d-200) then
        tmp = sqrt((-c / a))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.7e-193) {
		tmp = (-2.0 * b_2) / a;
	} else if (b_2 <= 1.5e-200) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.7e-193:
		tmp = (-2.0 * b_2) / a
	elif b_2 <= 1.5e-200:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.7e-193)
		tmp = Float64(Float64(-2.0 * b_2) / a);
	elseif (b_2 <= 1.5e-200)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.7e-193)
		tmp = (-2.0 * b_2) / a;
	elseif (b_2 <= 1.5e-200)
		tmp = sqrt((-c / a));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.7e-193], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-200], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.6999999999999999e-193

   1. Initial program 53.3%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 35.5

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

   4. Applied rewrites 35.5%

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

   if -2.6999999999999999e-193 < b_2 < 1.49999999999999997e-200

   1. Initial program 53.3%

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

   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 17.1

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

   4. Applied rewrites 17.1%

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

   if 1.49999999999999997e-200 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 9: 53.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{-200}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.4e-43)
   (/ (- b_2) a)
   (if (<= b_2 1.5e-200) (sqrt (/ (- c) a)) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.4e-43) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.5e-200) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.4d-43)) then
        tmp = -b_2 / a
    else if (b_2 <= 1.5d-200) then
        tmp = sqrt((-c / a))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.4e-43) {
		tmp = -b_2 / a;
	} else if (b_2 <= 1.5e-200) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.4e-43:
		tmp = -b_2 / a
	elif b_2 <= 1.5e-200:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.4e-43)
		tmp = Float64(Float64(-b_2) / a);
	elseif (b_2 <= 1.5e-200)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.4e-43)
		tmp = -b_2 / a;
	elseif (b_2 <= 1.5e-200)
		tmp = sqrt((-c / a));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.4e-43], N[((-b$95$2) / a), $MachinePrecision], If[LessEqual[b$95$2, 1.5e-200], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.39999999999999982e-43

   1. Initial program 53.3%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) + \frac{\sqrt{{b\_2}^{2}}}{a}\right) - \frac{b\_2}{a}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) + \color{blue}{\left(\frac{\sqrt{{b\_2}^{2}}}{a} - \frac{b\_2}{a}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) \cdot c + \left(\color{blue}{\frac{\sqrt{{b\_2}^{2}}}{a}} - \frac{b\_2}{a}\right) \]

      3. div-sub N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) \cdot c + \frac{\sqrt{{b\_2}^{2}} - b\_2}{\color{blue}{a}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}, \color{blue}{c}, \frac{\sqrt{{b\_2}^{2}} - b\_2}{a}\right) \]

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c \cdot a}{\left|b\_2\right| \cdot \left(b\_2 \cdot b\_2\right)} \cdot -0.125 - \frac{0.5}{\left|b\_2\right|}, c, \frac{\left|b\_2\right| - b\_2}{a}\right)} \]

   5. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-neg.f64 15.8

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

   7. Applied rewrites 15.8%

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

   if -5.39999999999999982e-43 < b_2 < 1.49999999999999997e-200

   1. Initial program 53.3%

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

   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 17.1

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

   4. Applied rewrites 17.1%

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

   if 1.49999999999999997e-200 < b_2

   1. Initial program 53.3%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 34.3

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

   4. Applied rewrites 34.3%

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

Alternative 10: 27.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.4e-43) (/ (- b_2) a) (sqrt (/ (- c) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.4e-43) {
		tmp = -b_2 / a;
	} else {
		tmp = sqrt((-c / a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.4d-43)) then
        tmp = -b_2 / a
    else
        tmp = sqrt((-c / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.4e-43) {
		tmp = -b_2 / a;
	} else {
		tmp = Math.sqrt((-c / a));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.4e-43:
		tmp = -b_2 / a
	else:
		tmp = math.sqrt((-c / a))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.4e-43)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = sqrt(Float64(Float64(-c) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.4e-43)
		tmp = -b_2 / a;
	else
		tmp = sqrt((-c / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.4e-43], N[((-b$95$2) / a), $MachinePrecision], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -5.39999999999999982e-43

   1. Initial program 53.3%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) + \frac{\sqrt{{b\_2}^{2}}}{a}\right) - \frac{b\_2}{a}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) + \color{blue}{\left(\frac{\sqrt{{b\_2}^{2}}}{a} - \frac{b\_2}{a}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) \cdot c + \left(\color{blue}{\frac{\sqrt{{b\_2}^{2}}}{a}} - \frac{b\_2}{a}\right) \]

      3. div-sub N/A

         \[\leadsto \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) \cdot c + \frac{\sqrt{{b\_2}^{2}} - b\_2}{\color{blue}{a}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}, \color{blue}{c}, \frac{\sqrt{{b\_2}^{2}} - b\_2}{a}\right) \]

   4. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c \cdot a}{\left|b\_2\right| \cdot \left(b\_2 \cdot b\_2\right)} \cdot -0.125 - \frac{0.5}{\left|b\_2\right|}, c, \frac{\left|b\_2\right| - b\_2}{a}\right)} \]

   5. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift-neg.f64 15.8

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

   7. Applied rewrites 15.8%

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

   if -5.39999999999999982e-43 < b_2

   1. Initial program 53.3%

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

   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 17.1

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

   4. Applied rewrites 17.1%

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

Alternative 11: 15.8% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

2. Taylor expanded in c around 0

   \[\leadsto \color{blue}{\left(c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) + \frac{\sqrt{{b\_2}^{2}}}{a}\right) - \frac{b\_2}{a}} \]
3. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) + \color{blue}{\left(\frac{\sqrt{{b\_2}^{2}}}{a} - \frac{b\_2}{a}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) \cdot c + \left(\color{blue}{\frac{\sqrt{{b\_2}^{2}}}{a}} - \frac{b\_2}{a}\right) \]

   3. div-sub N/A

      \[\leadsto \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}\right) \cdot c + \frac{\sqrt{{b\_2}^{2}} - b\_2}{\color{blue}{a}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{\left(\sqrt{{b\_2}^{2}}\right)}^{3}} - \frac{1}{2} \cdot \frac{1}{\sqrt{{b\_2}^{2}}}, \color{blue}{c}, \frac{\sqrt{{b\_2}^{2}} - b\_2}{a}\right) \]

4. Applied rewrites 60.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c \cdot a}{\left|b\_2\right| \cdot \left(b\_2 \cdot b\_2\right)} \cdot -0.125 - \frac{0.5}{\left|b\_2\right|}, c, \frac{\left|b\_2\right| - b\_2}{a}\right)} \]

5. Taylor expanded in b_2 around inf

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

   1. associate-*r/ N/A

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

   2. mul-1-neg N/A

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

   3. lower-/.f64 N/A

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

   4. lift-neg.f64 15.8

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

7. Applied rewrites 15.8%

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

Developer Target 1: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2025139 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :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) (/ (- sqtD b_2) a) (/ (- c) (+ b_2 sqtD)))))

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