quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.5% → 84.8%

Time: 3.7s

Alternatives: 10

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a \cdot \frac{\frac{c}{b\_2}}{b\_2}, 0.125, 0.5\right) \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.5e-48)
   (- (/ (* (fma (* a (/ (/ c b_2) b_2)) 0.125 0.5) c) b_2))
   (if (<= b_2 6.2e+65)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-48) {
		tmp = -((fma((a * ((c / b_2) / b_2)), 0.125, 0.5) * c) / b_2);
	} else if (b_2 <= 6.2e+65) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.5e-48)
		tmp = Float64(-Float64(Float64(fma(Float64(a * Float64(Float64(c / b_2) / b_2)), 0.125, 0.5) * c) / b_2));
	elseif (b_2 <= 6.2e+65)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.5e-48], (-N[(N[(N[(N[(a * N[(N[(c / b$95$2), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision] * 0.125 + 0.5), $MachinePrecision] * c), $MachinePrecision] / b$95$2), $MachinePrecision]), If[LessEqual[b$95$2, 6.2e+65], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.49999999999999991e-48

   1. Initial program 15.8%

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

   2. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{1}{2} \cdot c}{b\_2}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{\frac{1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{1}{2} \cdot c}{b\_2} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{1}{8} + \frac{1}{2} \cdot c}{b\_2} \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 87.4

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

   7. Applied rewrites 87.4%

      \[\leadsto -\frac{\mathsf{fma}\left(a \cdot \frac{c}{b\_2 \cdot b\_2}, 0.125, 0.5\right) \cdot c}{b\_2} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 87.3

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

   9. Applied rewrites 87.3%

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

   if -3.49999999999999991e-48 < b_2 < 6.19999999999999981e65

   1. Initial program 76.8%

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

   if 6.19999999999999981e65 < b_2

   1. Initial program 58.0%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

Alternative 2: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.5 \cdot 10^{-48}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(a \cdot \frac{c}{b\_2 \cdot b\_2}, 0.125, 0.5\right) \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.5e-48)
   (- (/ (* (fma (* a (/ c (* b_2 b_2))) 0.125 0.5) c) b_2))
   (if (<= b_2 6.2e+65)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.5e-48) {
		tmp = -((fma((a * (c / (b_2 * b_2))), 0.125, 0.5) * c) / b_2);
	} else if (b_2 <= 6.2e+65) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.5e-48)
		tmp = Float64(-Float64(Float64(fma(Float64(a * Float64(c / Float64(b_2 * b_2))), 0.125, 0.5) * c) / b_2));
	elseif (b_2 <= 6.2e+65)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.5e-48], (-N[(N[(N[(N[(a * N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.125 + 0.5), $MachinePrecision] * c), $MachinePrecision] / b$95$2), $MachinePrecision]), If[LessEqual[b$95$2, 6.2e+65], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.49999999999999991e-48

   1. Initial program 15.8%

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

   2. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\frac{1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{1}{2} \cdot c}{b\_2}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{\frac{1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{1}{2} \cdot c}{b\_2} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{\frac{1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{1}{2} \cdot c}{b\_2} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\frac{a \cdot {c}^{2}}{{b\_2}^{2}} \cdot \frac{1}{8} + \frac{1}{2} \cdot c}{b\_2} \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if -3.49999999999999991e-48 < b_2 < 6.19999999999999981e65

   1. Initial program 76.8%

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

   if 6.19999999999999981e65 < b_2

   1. Initial program 58.0%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

Alternative 3: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.2 \cdot 10^{+65}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-63)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.2e+65)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.2e+65) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.6d-63)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 6.2d+65) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.2e+65) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.6e-63:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 6.2e+65:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.6e-63)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.2e+65)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.6e-63)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 6.2e+65)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.6e-63], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.2e+65], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.6e-63

   1. Initial program 16.9%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -4.6e-63 < b_2 < 6.19999999999999981e65

   1. Initial program 77.7%

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

   if 6.19999999999999981e65 < b_2

   1. Initial program 58.0%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.7

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

   4. Applied rewrites 94.7%

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

Alternative 4: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-63)
   (* -0.5 (/ c b_2))
   (if (<= b_2 7e+22)
     (/ (- (- b_2) (sqrt (* (- a) c))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 7e+22) {
		tmp = (-b_2 - sqrt((-a * c))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.6d-63)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 7d+22) then
        tmp = (-b_2 - sqrt((-a * c))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 7e+22) {
		tmp = (-b_2 - Math.sqrt((-a * c))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.6e-63:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 7e+22:
		tmp = (-b_2 - math.sqrt((-a * c))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.6e-63)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 7e+22)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.6e-63)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 7e+22)
		tmp = (-b_2 - sqrt((-a * c))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.6e-63], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7e+22], N[(N[((-b$95$2) - N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.6e-63

   1. Initial program 16.9%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -4.6e-63 < b_2 < 7e22

   1. Initial program 76.3%

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

   2. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 62.1

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

   4. Applied rewrites 62.1%

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

   if 7e22 < b_2

   1. Initial program 62.7%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

Alternative 5: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-63}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.1 \cdot 10^{-57}:\\ \;\;\;\;\frac{-\sqrt{\left(-a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-63)
   (* -0.5 (/ c b_2))
   (if (<= b_2 2.1e-57) (/ (- (sqrt (* (- a) c))) a) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.1e-57) {
		tmp = -sqrt((-a * c)) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.6d-63)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 2.1d-57) then
        tmp = -sqrt((-a * c)) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-63) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 2.1e-57) {
		tmp = -Math.sqrt((-a * c)) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.6e-63:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 2.1e-57:
		tmp = -math.sqrt((-a * c)) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.6e-63)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 2.1e-57)
		tmp = Float64(Float64(-sqrt(Float64(Float64(-a) * c))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.6e-63)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 2.1e-57)
		tmp = -sqrt((-a * c)) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.6e-63], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.1e-57], N[((-N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.6e-63

   1. Initial program 16.9%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -4.6e-63 < b_2 < 2.0999999999999999e-57

   1. Initial program 73.3%

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

   2. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. *-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 65.3

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

   4. Applied rewrites 65.3%

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

   if 2.0999999999999999e-57 < b_2

   1. Initial program 67.9%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 87.0

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

   4. Applied rewrites 87.0%

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

Alternative 6: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{-c}{a}}\\ \mathbf{if}\;b\_2 \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -4.5 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq 1.15 \cdot 10^{-285}:\\ \;\;\;\;-t\_0\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (sqrt (/ (- c) a))))
   (if (<= b_2 -1.5e-64)
     (* -0.5 (/ c b_2))
     (if (<= b_2 -4.5e-177)
       t_0
       (if (<= b_2 1.15e-285)
         (- t_0)
         (if (<= b_2 3.5e-55) t_0 (* (/ b_2 a) -2.0)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt((-c / a));
	double tmp;
	if (b_2 <= -1.5e-64) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -4.5e-177) {
		tmp = t_0;
	} else if (b_2 <= 1.15e-285) {
		tmp = -t_0;
	} else if (b_2 <= 3.5e-55) {
		tmp = t_0;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((-c / a))
    if (b_2 <= (-1.5d-64)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= (-4.5d-177)) then
        tmp = t_0
    else if (b_2 <= 1.15d-285) then
        tmp = -t_0
    else if (b_2 <= 3.5d-55) then
        tmp = t_0
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt((-c / a));
	double tmp;
	if (b_2 <= -1.5e-64) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= -4.5e-177) {
		tmp = t_0;
	} else if (b_2 <= 1.15e-285) {
		tmp = -t_0;
	} else if (b_2 <= 3.5e-55) {
		tmp = t_0;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt((-c / a))
	tmp = 0
	if b_2 <= -1.5e-64:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= -4.5e-177:
		tmp = t_0
	elif b_2 <= 1.15e-285:
		tmp = -t_0
	elif b_2 <= 3.5e-55:
		tmp = t_0
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	t_0 = sqrt(Float64(Float64(-c) / a))
	tmp = 0.0
	if (b_2 <= -1.5e-64)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= -4.5e-177)
		tmp = t_0;
	elseif (b_2 <= 1.15e-285)
		tmp = Float64(-t_0);
	elseif (b_2 <= 3.5e-55)
		tmp = t_0;
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = sqrt((-c / a));
	tmp = 0.0;
	if (b_2 <= -1.5e-64)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= -4.5e-177)
		tmp = t_0;
	elseif (b_2 <= 1.15e-285)
		tmp = -t_0;
	elseif (b_2 <= 3.5e-55)
		tmp = t_0;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$2, -1.5e-64], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -4.5e-177], t$95$0, If[LessEqual[b$95$2, 1.15e-285], (-t$95$0), If[LessEqual[b$95$2, 3.5e-55], t$95$0, N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.5e-64

   1. Initial program 17.0%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -1.5e-64 < b_2 < -4.5000000000000003e-177 or 1.14999999999999998e-285 < b_2 < 3.50000000000000025e-55

   1. Initial program 73.8%

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

   2. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 28.5

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

   4. Applied rewrites 28.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 28.5

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

   6. Applied rewrites 28.5%

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

   if -4.5000000000000003e-177 < b_2 < 1.14999999999999998e-285

   1. Initial program 72.5%

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

   2. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 72.5

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

   4. Applied rewrites 72.5%

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

   5. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sqrt-prod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. mul-1-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 40.5

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

   7. Applied rewrites 40.5%

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

   if 3.50000000000000025e-55 < b_2

   1. Initial program 67.8%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 7: 68.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.5 \cdot 10^{-55}:\\ \;\;\;\;\sqrt{\frac{-c}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.5e-64)
   (* -0.5 (/ c b_2))
   (if (<= b_2 3.5e-55) (sqrt (/ (- c) a)) (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.5e-64) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.5e-55) {
		tmp = sqrt((-c / a));
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.5d-64)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 3.5d-55) then
        tmp = sqrt((-c / a))
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.5e-64) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.5e-55) {
		tmp = Math.sqrt((-c / a));
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.5e-64:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 3.5e-55:
		tmp = math.sqrt((-c / a))
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.5e-64)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3.5e-55)
		tmp = sqrt(Float64(Float64(-c) / a));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.5e-64)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 3.5e-55)
		tmp = sqrt((-c / a));
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.5e-64], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5e-55], N[Sqrt[N[((-c) / a), $MachinePrecision]], $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.5e-64

   1. Initial program 17.0%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -1.5e-64 < b_2 < 3.50000000000000025e-55

   1. Initial program 73.4%

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

   2. Taylor expanded in a around -inf

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

      1. sqrt-unprod N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 30.1

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

   4. Applied rewrites 30.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. mul-1-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 30.1

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

   6. Applied rewrites 30.1%

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

   if 3.50000000000000025e-55 < b_2

   1. Initial program 67.8%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 87.2

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

   4. Applied rewrites 87.2%

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

Alternative 8: 66.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -0.5 (/ c b_2)) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 31.8%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 72.0%

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

   2. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 65.9

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

   4. Applied rewrites 65.9%

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

Alternative 9: 47.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311) (* -0.5 (/ c b_2)) (/ (- b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-311)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = -b_2 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -b_2 / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-311:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -b_2 / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-311)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -b_2 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[((-b$95$2) / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 31.8%

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

   2. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 72.0%

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

   2. Taylor expanded in c around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. sqrt-unprod N/A

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

      13. lower-sqrt.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 37.5

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

   4. Applied rewrites 37.5%

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

   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 26.8

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

   7. Applied rewrites 26.8%

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

Alternative 10: 14.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.5%

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

2. Taylor expanded in c around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 N/A

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

   12. sqrt-unprod N/A

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

   13. lower-sqrt.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 31.5

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

4. Applied rewrites 31.5%

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

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 14.6

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

7. Applied rewrites 14.6%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))