jeff quadratic root 1

Percentage Accurate: 72.1% → 89.7%

Time: 10.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\\ \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (- (* b b) (* (* 4.0 a) c)))))
   (if (>= b 0.0) (/ (- (- b) t_0) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) t_0)))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((b * b) - ((4.0d0 * a) * c)))
    if (b >= 0.0d0) then
        tmp = (-b - t_0) / (2.0d0 * a)
    else
        tmp = (2.0d0 * c) / (-b + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt(((b * b) - ((4.0 * a) * c)));
	double tmp;
	if (b >= 0.0) {
		tmp = (-b - t_0) / (2.0 * a);
	} else {
		tmp = (2.0 * c) / (-b + t_0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt(((b * b) - ((4.0 * a) * c)))
	tmp = 0
	if b >= 0.0:
		tmp = (-b - t_0) / (2.0 * a)
	else:
		tmp = (2.0 * c) / (-b + t_0)
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))
	tmp = 0.0
	if (b >= 0.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / Float64(2.0 * a));
	else
		tmp = Float64(Float64(2.0 * c) / Float64(Float64(-b) + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt(((b * b) - ((4.0 * a) * c)));
	tmp = 0.0;
	if (b >= 0.0)
		tmp = (-b - t_0) / (2.0 * a);
	else
		tmp = (2.0 * c) / (-b + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[GreaterEqual[b, 0.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * c), $MachinePrecision] / N[((-b) + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-304}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+136)
   (/ c (- b))
   (if (<= b 3.05e-304)
     (/ (+ c c) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
     (if (<= b 2.45e+39)
       (* (/ (+ (sqrt (fma (* c a) -4.0 (* b b))) b) a) -0.5)
       (if (>= b 0.0)
         (/ (fma a (/ c b) (- b)) a)
         (* (/ 2.0 (- (- b) b)) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+136) {
		tmp = c / -b;
	} else if (b <= 3.05e-304) {
		tmp = (c + c) / (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else if (b <= 2.45e+39) {
		tmp = ((sqrt(fma((c * a), -4.0, (b * b))) + b) / a) * -0.5;
	} else if (b >= 0.0) {
		tmp = fma(a, (c / b), -b) / a;
	} else {
		tmp = (2.0 / (-b - b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+136)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 3.05e-304)
		tmp = Float64(Float64(c + c) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	elseif (b <= 2.45e+39)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) + b) / a) * -0.5);
	elseif (b >= 0.0)
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(-b) - b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+136], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 3.05e-304], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e+39], N[(N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], If[GreaterEqual[b, 0.0], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision], N[(N[(2.0 / N[((-b) - b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.49999999999999989e136

   1. Initial program 39.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 39.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 93.5%

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

   if -1.49999999999999989e136 < b < 3.0500000000000002e-304

   1. Initial program 89.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 89.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.8%

      \[\leadsto \frac{c + c}{\color{blue}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} - b} \]

   if 3.0500000000000002e-304 < b < 2.44999999999999994e39

   1. Initial program 84.6%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 84.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}}\\ \end{array} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a} \cdot \frac{-1}{2}} \]

   7. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5} \]

   if 2.44999999999999994e39 < b

   1. Initial program 62.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{t\_0 + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{t\_0 - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (fma (* c a) -4.0 (* b b)))))
   (if (<= b -1.5e+136)
     (/ c (- b))
     (if (<= b 2.45e+39)
       (if (>= b 0.0) (* (/ (+ t_0 b) a) -0.5) (/ (* c 2.0) (- t_0 b)))
       (if (>= b 0.0)
         (/ (fma a (/ c b) (- b)) a)
         (* (/ 2.0 (- (- b) b)) c))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt(fma((c * a), -4.0, (b * b)));
	double tmp;
	if (b <= -1.5e+136) {
		tmp = c / -b;
	} else if (b <= 2.45e+39) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = ((t_0 + b) / a) * -0.5;
		} else {
			tmp_1 = (c * 2.0) / (t_0 - b);
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = fma(a, (c / b), -b) / a;
	} else {
		tmp = (2.0 / (-b - b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = sqrt(fma(Float64(c * a), -4.0, Float64(b * b)))
	tmp = 0.0
	if (b <= -1.5e+136)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.45e+39)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(Float64(t_0 + b) / a) * -0.5);
		else
			tmp_1 = Float64(Float64(c * 2.0) / Float64(t_0 - b));
		end
		tmp = tmp_1;
	elseif (b >= 0.0)
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(-b) - b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.5e+136], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.45e+39], If[GreaterEqual[b, 0.0], N[(N[(N[(t$95$0 + b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(t$95$0 - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision], N[(N[(2.0 / N[((-b) - b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999989e136

   1. Initial program 39.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 39.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 93.5%

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

   if -1.49999999999999989e136 < b < 2.44999999999999994e39

   1. Initial program 87.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   if 2.44999999999999994e39 < b

   1. Initial program 62.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+136)
   (/ c (- b))
   (if (<= b 2.45e+39)
     (if (>= b 0.0)
       (* (+ (sqrt (fma -4.0 (* c a) (* b b))) b) (/ -0.5 a))
       (/ (* c 2.0) (- (sqrt (fma (* c a) -4.0 (* b b))) b)))
     (if (>= b 0.0) (/ (fma a (/ c b) (- b)) a) (* (/ 2.0 (- (- b) b)) c)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+136) {
		tmp = c / -b;
	} else if (b <= 2.45e+39) {
		double tmp_1;
		if (b >= 0.0) {
			tmp_1 = (sqrt(fma(-4.0, (c * a), (b * b))) + b) * (-0.5 / a);
		} else {
			tmp_1 = (c * 2.0) / (sqrt(fma((c * a), -4.0, (b * b))) - b);
		}
		tmp = tmp_1;
	} else if (b >= 0.0) {
		tmp = fma(a, (c / b), -b) / a;
	} else {
		tmp = (2.0 / (-b - b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+136)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 2.45e+39)
		tmp_1 = 0.0
		if (b >= 0.0)
			tmp_1 = Float64(Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) + b) * Float64(-0.5 / a));
		else
			tmp_1 = Float64(Float64(c * 2.0) / Float64(sqrt(fma(Float64(c * a), -4.0, Float64(b * b))) - b));
		end
		tmp = tmp_1;
	elseif (b >= 0.0)
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(-b) - b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+136], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 2.45e+39], If[GreaterEqual[b, 0.0], N[(N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -4.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], If[GreaterEqual[b, 0.0], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision], N[(N[(2.0 / N[((-b) - b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999989e136

   1. Initial program 39.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 39.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 93.5%

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

   if -1.49999999999999989e136 < b < 2.44999999999999994e39

   1. Initial program 87.9%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

   7. Applied rewrites 87.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b\right) \cdot \color{blue}{\frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   if 2.44999999999999994e39 < b

   1. Initial program 62.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-218}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \frac{-0.5}{a}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+136)
   (/ c (- b))
   (if (<= b 6.6e-218)
     (/ (+ c c) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
     (if (<= b 2.45e+39)
       (* (+ (sqrt (fma (* -4.0 c) a (* b b))) b) (/ -0.5 a))
       (if (>= b 0.0)
         (/ (fma a (/ c b) (- b)) a)
         (* (/ 2.0 (- (- b) b)) c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+136) {
		tmp = c / -b;
	} else if (b <= 6.6e-218) {
		tmp = (c + c) / (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else if (b <= 2.45e+39) {
		tmp = (sqrt(fma((-4.0 * c), a, (b * b))) + b) * (-0.5 / a);
	} else if (b >= 0.0) {
		tmp = fma(a, (c / b), -b) / a;
	} else {
		tmp = (2.0 / (-b - b)) * c;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+136)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 6.6e-218)
		tmp = Float64(Float64(c + c) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	elseif (b <= 2.45e+39)
		tmp = Float64(Float64(sqrt(fma(Float64(-4.0 * c), a, Float64(b * b))) + b) * Float64(-0.5 / a));
	elseif (b >= 0.0)
		tmp = Float64(fma(a, Float64(c / b), Float64(-b)) / a);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(-b) - b)) * c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+136], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 6.6e-218], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e+39], N[(N[(N[Sqrt[N[(N[(-4.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], If[GreaterEqual[b, 0.0], N[(N[(a * N[(c / b), $MachinePrecision] + (-b)), $MachinePrecision] / a), $MachinePrecision], N[(N[(2.0 / N[((-b) - b), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.49999999999999989e136

   1. Initial program 39.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 39.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 93.5%

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

   if -1.49999999999999989e136 < b < 6.60000000000000046e-218

   1. Initial program 90.0%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 88.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 89.9%

      \[\leadsto \frac{c + c}{\color{blue}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} - b} \]

   if 6.60000000000000046e-218 < b < 2.44999999999999994e39

   1. Initial program 82.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 82.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}}\\ \end{array} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a} \cdot \frac{-1}{2}} \]

   7. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5} \]
   8. Step-by-step derivation

   9. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b\right) \cdot \frac{-0.5}{a}} \]

   if 2.44999999999999994e39 < b

   1. Initial program 62.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.7%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(-b\right) - b} \cdot c\\ \end{array} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.5 \cdot 10^{+136}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{c + c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.5e+136)
   (/ c (- b))
   (if (<= b 7.2e-38)
     (/ (+ c c) (- (sqrt (fma -4.0 (* c a) (* b b))) b))
     (if (>= b 0.0) (fma (/ b a) -1.0 (/ c b)) (/ (* c 2.0) (- (- b) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.5e+136) {
		tmp = c / -b;
	} else if (b <= 7.2e-38) {
		tmp = (c + c) / (sqrt(fma(-4.0, (c * a), (b * b))) - b);
	} else if (b >= 0.0) {
		tmp = fma((b / a), -1.0, (c / b));
	} else {
		tmp = (c * 2.0) / (-b - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.5e+136)
		tmp = Float64(c / Float64(-b));
	elseif (b <= 7.2e-38)
		tmp = Float64(Float64(c + c) / Float64(sqrt(fma(-4.0, Float64(c * a), Float64(b * b))) - b));
	elseif (b >= 0.0)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.5e+136], N[(c / (-b)), $MachinePrecision], If[LessEqual[b, 7.2e-38], N[(N[(c + c), $MachinePrecision] / N[(N[Sqrt[N[(-4.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[GreaterEqual[b, 0.0], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.49999999999999989e136

   1. Initial program 39.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 39.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 39.1%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 93.5%

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

   if -1.49999999999999989e136 < b < 7.2000000000000001e-38

   1. Initial program 87.1%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 80.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 83.4%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]
   8. Step-by-step derivation

   9. Applied rewrites 83.4%

      \[\leadsto \frac{c + c}{\color{blue}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)}} - b} \]

   if 7.2000000000000001e-38 < b

   1. Initial program 69.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.3%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]

   12. Taylor expanded in a around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{\color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
   13. Step-by-step derivation

   14. Applied rewrites 92.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (>= b 0.0) (fma (/ b a) -1.0 (/ c b)) (/ (* c 2.0) (- (- b) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b >= 0.0) {
		tmp = fma((b / a), -1.0, (c / b));
	} else {
		tmp = (c * 2.0) / (-b - b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b >= 0.0)
		tmp = fma(Float64(b / a), -1.0, Float64(c / b));
	else
		tmp = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[GreaterEqual[b, 0.0], N[(N[(b / a), $MachinePrecision] * -1.0 + N[(c / b), $MachinePrecision]), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 71.0%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
4. Step-by-step derivation

5. Applied rewrites 71.0%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

6. Taylor expanded in a around 0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
7. Step-by-step derivation

8. Applied rewrites 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

9. Taylor expanded in b around -inf

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
10. Step-by-step derivation

11. Applied rewrites 69.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]

12. Taylor expanded in a around inf

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-1 \cdot \frac{b}{a} + \frac{c}{\color{blue}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
13. Step-by-step derivation

14. Applied rewrites 69.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{a}, -1, \frac{c}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
15. Add Preprocessing

Alternative 7: 68.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.8e-299)
   (if (>= b 0.0) (/ c b) (/ (* c 2.0) (- (- b) b)))
   (* (/ (* 2.0 b) a) -0.5)))

C

double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.8e-299) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = ((2.0 * b) / a) * -0.5;
	}
	return tmp_1;
}

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, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    real(8) :: tmp_1
    real(8) :: tmp_2
    if (b <= 2.8d-299) then
        if (b >= 0.0d0) then
            tmp_2 = c / b
        else
            tmp_2 = (c * 2.0d0) / (-b - b)
        end if
        tmp_1 = tmp_2
    else
        tmp_1 = ((2.0d0 * b) / a) * (-0.5d0)
    end if
    code = tmp_1
end function

Java

public static double code(double a, double b, double c) {
	double tmp_1;
	if (b <= 2.8e-299) {
		double tmp_2;
		if (b >= 0.0) {
			tmp_2 = c / b;
		} else {
			tmp_2 = (c * 2.0) / (-b - b);
		}
		tmp_1 = tmp_2;
	} else {
		tmp_1 = ((2.0 * b) / a) * -0.5;
	}
	return tmp_1;
}

Python

def code(a, b, c):
	tmp_1 = 0
	if b <= 2.8e-299:
		tmp_2 = 0
		if b >= 0.0:
			tmp_2 = c / b
		else:
			tmp_2 = (c * 2.0) / (-b - b)
		tmp_1 = tmp_2
	else:
		tmp_1 = ((2.0 * b) / a) * -0.5
	return tmp_1

Julia

function code(a, b, c)
	tmp_1 = 0.0
	if (b <= 2.8e-299)
		tmp_2 = 0.0
		if (b >= 0.0)
			tmp_2 = Float64(c / b);
		else
			tmp_2 = Float64(Float64(c * 2.0) / Float64(Float64(-b) - b));
		end
		tmp_1 = tmp_2;
	else
		tmp_1 = Float64(Float64(Float64(2.0 * b) / a) * -0.5);
	end
	return tmp_1
end

MATLAB

function tmp_4 = code(a, b, c)
	tmp_2 = 0.0;
	if (b <= 2.8e-299)
		tmp_3 = 0.0;
		if (b >= 0.0)
			tmp_3 = c / b;
		else
			tmp_3 = (c * 2.0) / (-b - b);
		end
		tmp_2 = tmp_3;
	else
		tmp_2 = ((2.0 * b) / a) * -0.5;
	end
	tmp_4 = tmp_2;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.8e-299], If[GreaterEqual[b, 0.0], N[(c / b), $MachinePrecision], N[(N[(c * 2.0), $MachinePrecision] / N[((-b) - b), $MachinePrecision]), $MachinePrecision]], N[(N[(N[(2.0 * b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.8000000000000001e-299

   1. Initial program 70.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ } \end{array}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1 \cdot b + \frac{a \cdot c}{b}}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{\color{blue}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} - b}\\ \end{array} \]

   9. Taylor expanded in b around -inf

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{-1 \cdot b - b}\\ \end{array} \]
   10. Step-by-step derivation

   11. Applied rewrites 66.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{b}, -b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]

   12. Taylor expanded in a around inf

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]
   13. Step-by-step derivation

   14. Applied rewrites 66.8%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array} \]

   if 2.8000000000000001e-299 < b

   1. Initial program 72.2%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 72.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}}\\ \end{array} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a} \cdot \frac{-1}{2}} \]

   7. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{2 \cdot b}{a} \cdot \frac{-1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 72.4%

       \[\leadsto \frac{2 \cdot b}{a} \cdot -0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{-b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot b}{a} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ c (- b)) (* (/ (* 2.0 b) a) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = c / -b;
	} else {
		tmp = ((2.0 * b) / a) * -0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d-310)) then
        tmp = c / -b
    else
        tmp = ((2.0d0 * b) / a) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = c / -b;
	} else {
		tmp = ((2.0 * b) / a) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = c / -b
	else:
		tmp = ((2.0 * b) / a) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(c / Float64(-b));
	else
		tmp = Float64(Float64(Float64(2.0 * b) / a) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = c / -b;
	else
		tmp = ((2.0 * b) / a) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(c / (-b)), $MachinePrecision], N[(N[(N[(2.0 * b), $MachinePrecision] / a), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 69.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 69.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

      3. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 2.4%

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

   11. Taylor expanded in b around -inf

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

   13. Applied rewrites 67.7%

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

   if -9.999999999999969e-311 < b

   1. Initial program 72.7%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
   2. Add Preprocessing

   4. Applied rewrites 72.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\frac{\mathsf{fma}\left(-b, b, \mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)\right)}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}}\\ \end{array} \]

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}}}{a}\\ } \end{array}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      2. metadata-eval N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}\\ \end{array} \]

      4. if-same N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b + \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{a} \cdot \frac{-1}{2}} \]

   7. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)} + b}{a} \cdot -0.5} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{2 \cdot b}{a} \cdot \frac{-1}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.1%

       \[\leadsto \frac{2 \cdot b}{a} \cdot -0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 36.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 71.0%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]
2. Add Preprocessing

4. Applied rewrites 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{\mathsf{fma}\left(c, a \cdot 4, b \cdot b\right) - b \cdot b}{\left(\left(\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + \left(-b\right)\right) \cdot 2\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array} \]

5. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}\\ } \end{array}} \]
6. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

   2. metadata-eval N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{{b}^{2} + -4 \cdot \left(a \cdot c\right)} - b}\\ \end{array} \]

   3. +-commutative N/A

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}\\ \end{array} \]

   4. if-same N/A

      \[\leadsto \color{blue}{2 \cdot \frac{c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{2 \cdot c}}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b} \]

   8. lower--.f64 N/A

      \[\leadsto \frac{2 \cdot c}{\color{blue}{\sqrt{-4 \cdot \left(a \cdot c\right) + {b}^{2}} - b}} \]

7. Applied rewrites 52.6%

   \[\leadsto \color{blue}{\frac{2 \cdot c}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}} \]

8. Taylor expanded in a around 0

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

10. Applied rewrites 25.6%

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

11. Taylor expanded in b around -inf

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

13. Applied rewrites 39.4%

    \[\leadsto \frac{c}{\color{blue}{-b}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024354 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))