quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.7% → 90.8%

Time: 6.5s

Alternatives: 9

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: 51.7% 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: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.25 \cdot 10^{-267}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{+89}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.1e+142)
   (* -0.5 (/ c b_2))
   (if (<= b_2 3.25e-267)
     (/ c (+ (- b_2) (sqrt (fma (- c) a (* b_2 b_2)))))
     (if (<= b_2 1.9e+89)
       (/ (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- a))
       (/ (+ b_2 b_2) (- a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e+142) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.25e-267) {
		tmp = c / (-b_2 + sqrt(fma(-c, a, (b_2 * b_2))));
	} else if (b_2 <= 1.9e+89) {
		tmp = (b_2 + sqrt(((b_2 * b_2) - (a * c)))) / -a;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.1e+142)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3.25e-267)
		tmp = Float64(c / Float64(Float64(-b_2) + sqrt(fma(Float64(-c), a, Float64(b_2 * b_2)))));
	elseif (b_2 <= 1.9e+89)
		tmp = Float64(Float64(b_2 + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-a));
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.1e+142], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.25e-267], N[(c / N[((-b$95$2) + N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.9e+89], 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 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.09999999999999993e142

   1. Initial program 6.2%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 98.1%

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

   if -1.09999999999999993e142 < b_2 < 3.25e-267

   1. Initial program 49.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 49.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.3

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 75.3

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

   9. Applied rewrites 75.3%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 87.7%

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

   if 3.25e-267 < b_2 < 1.90000000000000012e89

   1. Initial program 79.1%

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

   if 1.90000000000000012e89 < b_2

   1. Initial program 50.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 96.6%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.25 \cdot 10^{-267}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{+89}:\\ \;\;\;\;\frac{b\_2 + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.1 \cdot 10^{+142}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{c}{\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.1e+142)
   (* -0.5 (/ c b_2))
   (if (<= b_2 6.8e-91)
     (/ c (+ (- b_2) (sqrt (fma (- c) a (* b_2 b_2)))))
     (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.1e+142) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 6.8e-91) {
		tmp = c / (-b_2 + sqrt(fma(-c, a, (b_2 * b_2))));
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.1e+142)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 6.8e-91)
		tmp = Float64(c / Float64(Float64(-b_2) + sqrt(fma(Float64(-c), a, Float64(b_2 * b_2)))));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.1e+142], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6.8e-91], N[(c / N[((-b$95$2) + N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.09999999999999993e142

   1. Initial program 6.2%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 98.1%

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

   if -1.09999999999999993e142 < b_2 < 6.80000000000000053e-91

   1. Initial program 54.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 53.1%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 73.8

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 73.8

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

   9. Applied rewrites 73.8%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 83.9%

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

   if 6.80000000000000053e-91 < b_2

   1. Initial program 62.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 84.7%

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

Alternative 3: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2 + \sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-54)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6.8e-91)
     (/ (+ b_2 (sqrt (* c (- a)))) (- a))
     (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-54) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6.8e-91) {
		tmp = (b_2 + sqrt((c * -a))) / -a;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-54)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6.8e-91)
		tmp = Float64(Float64(b_2 + sqrt(Float64(c * Float64(-a)))) / Float64(-a));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-54], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6.8e-91], N[(N[(b$95$2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.9999999999999997e-54

   1. Initial program 23.1%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 85.5%

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

   if -8.9999999999999997e-54 < b_2 < 6.80000000000000053e-91

   1. Initial program 68.1%

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

   3. Taylor expanded in a around inf

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

   5. Applied rewrites 65.2%

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

   if 6.80000000000000053e-91 < b_2

   1. Initial program 62.3%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 84.7%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-54}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{b\_2 + \sqrt{c \cdot \left(-a\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, \frac{b\_2}{a} \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (/ (* -0.5 c) b_2)
   (fma (/ 0.5 b_2) c (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = fma((0.5 / b_2), c, ((b_2 / a) * -2.0));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(Float64(b_2 / a) * -2.0));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 32.0%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 70.6%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 66.2%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 68.3%

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

Alternative 5: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b\_2, \frac{-2}{a}, \frac{0.5}{b\_2} \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309)
   (/ (* -0.5 c) b_2)
   (fma b_2 (/ -2.0 a) (* (/ 0.5 b_2) c))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = fma(b_2, (-2.0 / a), ((0.5 / b_2) * c));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = fma(b_2, Float64(-2.0 / a), Float64(Float64(0.5 / b_2) * c));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision] + N[(N[(0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 32.0%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 70.6%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 66.2%

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

   3. Taylor expanded in c around 0

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

   5. Applied rewrites 68.3%

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

   7. Applied rewrites 68.0%

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

Alternative 6: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ (* -0.5 c) b_2) (/ (+ b_2 b_2) (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 + 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 <= (-1d-309)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 + 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 <= -1e-309) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 + b_2) / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 + b_2) / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 + b_2) / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 + b_2) / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 32.0%

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

   3. Taylor expanded in b_2 around -inf

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

   5. Applied rewrites 70.6%

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

   7. Applied rewrites 70.6%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 66.2%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 67.7%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5 \cdot c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (* -0.5 c) b_2))

C

double code(double a, double b_2, double c) {
	return (-0.5 * c) / b_2;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = ((-0.5d0) * c) / b_2
end function

Java

public static double code(double a, double b_2, double c) {
	return (-0.5 * c) / b_2;
}

Python

def code(a, b_2, c):
	return (-0.5 * c) / b_2

Julia

function code(a, b_2, c)
	return Float64(Float64(-0.5 * c) / b_2)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-0.5 * c) / b_2;
end

Wolfram

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

Derivation

1. Initial program 48.4%

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

3. Taylor expanded in b_2 around -inf

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

5. Applied rewrites 37.8%

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

7. Applied rewrites 37.8%

   \[\leadsto \frac{-0.5 \cdot c}{\color{blue}{b\_2}} \]
8. Add Preprocessing

Alternative 8: 35.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b\_2} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* -0.5 (/ c b_2)))

C

double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b_2, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b_2)
end function

Java

public static double code(double a, double b_2, double c) {
	return -0.5 * (c / b_2);
}

Python

def code(a, b_2, c):
	return -0.5 * (c / b_2)

Julia

function code(a, b_2, c)
	return Float64(-0.5 * Float64(c / b_2))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = -0.5 * (c / b_2);
end

Wolfram

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

Derivation

1. Initial program 48.4%

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

3. Taylor expanded in b_2 around -inf

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

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
6. Add Preprocessing

Alternative 9: 11.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b\_2} \cdot 0.5 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* (/ c b_2) 0.5))

C

double code(double a, double b_2, double c) {
	return (c / b_2) * 0.5;
}

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 = (c / b_2) * 0.5d0
end function

Java

public static double code(double a, double b_2, double c) {
	return (c / b_2) * 0.5;
}

Python

def code(a, b_2, c):
	return (c / b_2) * 0.5

Julia

function code(a, b_2, c)
	return Float64(Float64(c / b_2) * 0.5)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (c / b_2) * 0.5;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 48.4%

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

3. Taylor expanded in c around 0

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

5. Applied rewrites 34.0%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 13.5%

   \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025019 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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