quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.3% → 83.2%

Time: 12.2s

Alternatives: 10

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{b\_2}{\sqrt[3]{a}} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.02e-131)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.65e-30)
     (-
      (* (/ -1.0 (pow (cbrt a) 2.0)) (/ b_2 (cbrt a)))
      (/ (hypot b_2 (sqrt (* c (- a)))) a))
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.02e-131) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.65e-30) {
		tmp = ((-1.0 / pow(cbrt(a), 2.0)) * (b_2 / cbrt(a))) - (hypot(b_2, sqrt((c * -a))) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.02e-131) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.65e-30) {
		tmp = ((-1.0 / Math.pow(Math.cbrt(a), 2.0)) * (b_2 / Math.cbrt(a))) - (Math.hypot(b_2, Math.sqrt((c * -a))) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.02e-131)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.65e-30)
		tmp = Float64(Float64(Float64(-1.0 / (cbrt(a) ^ 2.0)) * Float64(b_2 / cbrt(a))) - Float64(hypot(b_2, sqrt(Float64(c * Float64(-a)))) / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.02e-131], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.65e-30], N[(N[(N[(-1.0 / N[Power[N[Power[a, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(b$95$2 / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.02000000000000001e-131

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf 81.8%

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

      1. associate-*r/ 81.8%

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

   5. Simplified 81.8%

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

   if -1.02000000000000001e-131 < b_2 < 2.64999999999999987e-30

   1. Initial program 76.9%

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

      1. div-sub 76.9%

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

      2. neg-mul-1 76.9%

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

      3. add-cube-cbrt 76.7%

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

      4. times-frac 76.7%

         \[\leadsto \color{blue}{\frac{-1}{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \frac{b\_2}{\sqrt[3]{a}}} - \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]

      5. fma-neg 76.8%

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

      6. pow2 76.8%

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

      7. sub-neg 76.8%

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

      8. add-sqr-sqrt 75.3%

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

      9. hypot-define 81.3%

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

      10. distribute-rgt-neg-in 81.3%

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

   4. Applied egg-rr 81.3%

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

      1. fma-undefine 81.2%

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

      2. unsub-neg 81.2%

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

   6. Simplified 81.2%

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

   if 2.64999999999999987e-30 < b_2

   1. Initial program 64.3%

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

   3. Taylor expanded in c around 0 93.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.02 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1}{{\left(\sqrt[3]{a}\right)}^{2}} \cdot \frac{b\_2}{\sqrt[3]{a}} - \frac{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.2e-131)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 6e+76)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e-131) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e+76) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.2d-131)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 6d+76) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.2e-131) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 6e+76) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.2e-131:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 6e+76:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.2e-131)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 6e+76)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.2e-131)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 6e+76)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.2e-131], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e+76], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.19999999999999993e-131

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf 81.8%

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

      1. associate-*r/ 81.8%

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

   5. Simplified 81.8%

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

   if -5.19999999999999993e-131 < b_2 < 5.9999999999999996e76

   1. Initial program 81.6%

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

   if 5.9999999999999996e76 < b_2

   1. Initial program 52.3%

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

   3. Taylor expanded in c around 0 95.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5.3e-131)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 7.5e-77)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.3e-131) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7.5e-77) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5.3d-131)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 7.5d-77) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5.3e-131) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 7.5e-77) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5.3e-131:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 7.5e-77:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5.3e-131)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 7.5e-77)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5.3e-131)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 7.5e-77)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5.3e-131], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-77], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -5.30000000000000018e-131

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf 81.8%

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

      1. associate-*r/ 81.8%

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

   5. Simplified 81.8%

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

   if -5.30000000000000018e-131 < b_2 < 7.5000000000000006e-77

   1. Initial program 75.8%

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

   3. Taylor expanded in b_2 around 0 71.8%

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

      1. mul-1-neg 71.8%

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

      2. distribute-rgt-neg-out 71.8%

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

   5. Simplified 71.8%

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

   if 7.5000000000000006e-77 < b_2

   1. Initial program 66.0%

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

   3. Taylor expanded in b_2 around inf 90.8%

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

      1. *-commutative 90.8%

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

   5. Simplified 90.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4e-310)
   (/ (* -0.5 c) b_2)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4e-310], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.999999999999988e-310

   1. Initial program 34.1%

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

   3. Taylor expanded in b_2 around -inf 67.9%

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

      1. associate-*r/ 67.9%

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

   5. Simplified 67.9%

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

   if -3.999999999999988e-310 < b_2

   1. Initial program 66.9%

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

   3. Taylor expanded in c around 0 72.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.45 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.45e-296) (/ (* -0.5 c) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.45e-296) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.45d-296)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.45e-296) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.45e-296:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.45e-296)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.45e-296)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.45e-296], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.44999999999999991e-296

   1. Initial program 32.1%

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

   3. Taylor expanded in b_2 around -inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

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

   if -1.44999999999999991e-296 < b_2

   1. Initial program 67.9%

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

   3. Taylor expanded in b_2 around inf 70.7%

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

      1. *-commutative 70.7%

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

   5. Simplified 70.7%

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

Alternative 6: 47.6% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.45 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.45e-296) (/ (* -0.5 c) b_2) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.45e-296) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.45d-296)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.45e-296) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.45e-296:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.45e-296)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.45e-296)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.45e-296], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.44999999999999991e-296

   1. Initial program 32.1%

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

   3. Taylor expanded in b_2 around -inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

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

   if -1.44999999999999991e-296 < b_2

   1. Initial program 67.9%

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

      1. prod-diff 67.7%

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

      2. *-commutative 67.7%

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

      3. fma-neg 67.7%

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

      4. prod-diff 67.7%

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

      5. *-commutative 67.7%

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

      6. fma-neg 67.7%

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

      7. associate-+l+ 67.7%

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

      8. pow2 67.7%

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

      9. *-commutative 67.7%

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

      10. fma-undefine 67.7%

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

      11. distribute-lft-neg-in 67.7%

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

      12. *-commutative 67.7%

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

      13. distribute-rgt-neg-in 67.7%

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

      14. fma-define 67.7%

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

      15. *-commutative 67.7%

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

      16. fma-undefine 67.7%

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

      17. distribute-lft-neg-in 67.7%

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

      18. *-commutative 67.7%

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

      19. distribute-rgt-neg-in 67.7%

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

   4. Applied egg-rr 67.7%

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

      1. count-2 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in c around inf 24.1%

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

      1. mul-1-neg 24.1%

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

      2. *-commutative 24.1%

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

      3. distribute-rgt1-in 24.1%

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

      4. metadata-eval 24.1%

         \[\leadsto \frac{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(\color{blue}{0} \cdot a\right) \cdot 2 - a}{c}}\right)}{a} \]

   9. Simplified 24.1%

      \[\leadsto \frac{\color{blue}{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(0 \cdot a\right) \cdot 2 - a}{c}}\right)}}{a} \]

   10. Taylor expanded in b_2 around inf 25.3%

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

       1. mul-1-neg 25.3%

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

   12. Simplified 25.3%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.45 \cdot 10^{-296}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{-0.5}{\frac{b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e-292) (/ -0.5 (/ b_2 c)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-292) {
		tmp = -0.5 / (b_2 / c);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.5d-292)) then
        tmp = (-0.5d0) / (b_2 / c)
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e-292) {
		tmp = -0.5 / (b_2 / c);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.5e-292:
		tmp = -0.5 / (b_2 / c)
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.5e-292)
		tmp = Float64(-0.5 / Float64(b_2 / c));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.5e-292)
		tmp = -0.5 / (b_2 / c);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.5e-292], N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.49999999999999956e-292

   1. Initial program 32.1%

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

   3. Taylor expanded in b_2 around -inf 70.0%

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

      1. associate-*r/ 70.0%

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

   5. Simplified 70.0%

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

      1. add-cbrt-cube 39.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{-0.5 \cdot c}{b\_2} \cdot \frac{-0.5 \cdot c}{b\_2}\right) \cdot \frac{-0.5 \cdot c}{b\_2}}} \]

      2. pow1/3 32.6%

         \[\leadsto \color{blue}{{\left(\left(\frac{-0.5 \cdot c}{b\_2} \cdot \frac{-0.5 \cdot c}{b\_2}\right) \cdot \frac{-0.5 \cdot c}{b\_2}\right)}^{0.3333333333333333}} \]

      3. pow3 32.6%

         \[\leadsto {\color{blue}{\left({\left(\frac{-0.5 \cdot c}{b\_2}\right)}^{3}\right)}}^{0.3333333333333333} \]

      4. associate-/l* 32.6%

         \[\leadsto {\left({\color{blue}{\left(-0.5 \cdot \frac{c}{b\_2}\right)}}^{3}\right)}^{0.3333333333333333} \]

   7. Applied egg-rr 32.6%

      \[\leadsto \color{blue}{{\left({\left(-0.5 \cdot \frac{c}{b\_2}\right)}^{3}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 39.7%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(-0.5 \cdot \frac{c}{b\_2}\right)}^{3}}} \]

      2. rem-cbrt-cube 70.0%

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

      3. clear-num 69.5%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{b\_2}{c}}} \]

      4. un-div-inv 69.5%

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

   9. Applied egg-rr 69.5%

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

   if -4.49999999999999956e-292 < b_2

   1. Initial program 67.9%

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

      1. prod-diff 67.7%

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

      2. *-commutative 67.7%

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

      3. fma-neg 67.7%

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

      4. prod-diff 67.7%

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

      5. *-commutative 67.7%

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

      6. fma-neg 67.7%

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

      7. associate-+l+ 67.7%

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

      8. pow2 67.7%

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

      9. *-commutative 67.7%

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

      10. fma-undefine 67.7%

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

      11. distribute-lft-neg-in 67.7%

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

      12. *-commutative 67.7%

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

      13. distribute-rgt-neg-in 67.7%

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

      14. fma-define 67.7%

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

      15. *-commutative 67.7%

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

      16. fma-undefine 67.7%

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

      17. distribute-lft-neg-in 67.7%

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

      18. *-commutative 67.7%

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

      19. distribute-rgt-neg-in 67.7%

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

   4. Applied egg-rr 67.7%

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

      1. count-2 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in c around inf 24.1%

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

      1. mul-1-neg 24.1%

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

      2. *-commutative 24.1%

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

      3. distribute-rgt1-in 24.1%

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

      4. metadata-eval 24.1%

         \[\leadsto \frac{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(\color{blue}{0} \cdot a\right) \cdot 2 - a}{c}}\right)}{a} \]

   9. Simplified 24.1%

      \[\leadsto \frac{\color{blue}{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(0 \cdot a\right) \cdot 2 - a}{c}}\right)}}{a} \]

   10. Taylor expanded in b_2 around inf 25.3%

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

       1. mul-1-neg 25.3%

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

   12. Simplified 25.3%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{-0.5}{\frac{b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 23.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e+39) (* 0.5 (/ c b_2)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e+39) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.8d+39)) then
        tmp = 0.5d0 * (c / b_2)
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e+39) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e+39:
		tmp = 0.5 * (c / b_2)
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e+39)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e+39)
		tmp = 0.5 * (c / b_2);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.8e+39], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -3.7999999999999998e39

   1. Initial program 12.7%

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

   3. Taylor expanded in c around 0 2.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}} \]

   4. Taylor expanded in b_2 around 0 35.3%

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

   if -3.7999999999999998e39 < b_2

   1. Initial program 65.4%

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

      1. prod-diff 65.2%

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

      2. *-commutative 65.2%

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

      3. fma-neg 65.2%

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

      4. prod-diff 65.2%

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

      5. *-commutative 65.2%

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

      6. fma-neg 65.2%

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

      7. associate-+l+ 65.1%

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

      8. pow2 65.1%

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

      9. *-commutative 65.1%

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

      10. fma-undefine 65.2%

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

      11. distribute-lft-neg-in 65.2%

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

      12. *-commutative 65.2%

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

      13. distribute-rgt-neg-in 65.2%

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

      14. fma-define 65.1%

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

      15. *-commutative 65.1%

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

      16. fma-undefine 65.2%

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

      17. distribute-lft-neg-in 65.2%

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

      18. *-commutative 65.2%

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

      19. distribute-rgt-neg-in 65.2%

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

   4. Applied egg-rr 65.1%

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

      1. count-2 65.1%

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

   6. Simplified 65.1%

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

   7. Taylor expanded in c around inf 22.3%

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

      1. mul-1-neg 22.3%

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

      2. *-commutative 22.3%

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

      3. distribute-rgt1-in 22.3%

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

      4. metadata-eval 22.3%

         \[\leadsto \frac{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(\color{blue}{0} \cdot a\right) \cdot 2 - a}{c}}\right)}{a} \]

   9. Simplified 22.3%

      \[\leadsto \frac{\color{blue}{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(0 \cdot a\right) \cdot 2 - a}{c}}\right)}}{a} \]

   10. Taylor expanded in b_2 around inf 18.8%

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

       1. mul-1-neg 18.8%

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

   12. Simplified 18.8%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 15.3% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

   1. prod-diff 49.9%

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

   2. *-commutative 49.9%

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

   3. fma-neg 49.9%

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

   4. prod-diff 49.9%

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

   5. *-commutative 49.9%

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

   6. fma-neg 49.9%

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

   7. associate-+l+ 49.9%

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

   8. pow2 49.9%

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

   9. *-commutative 49.9%

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

   10. fma-undefine 49.9%

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

   11. distribute-lft-neg-in 49.9%

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

   12. *-commutative 49.9%

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

   13. distribute-rgt-neg-in 49.9%

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

   14. fma-define 49.9%

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

   15. *-commutative 49.9%

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

   16. fma-undefine 49.9%

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

   17. distribute-lft-neg-in 49.9%

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

   18. *-commutative 49.9%

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

   19. distribute-rgt-neg-in 49.9%

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

4. Applied egg-rr 49.9%

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

   1. count-2 49.9%

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

6. Simplified 49.9%

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

7. Taylor expanded in c around inf 17.1%

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

   1. mul-1-neg 17.1%

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

   2. *-commutative 17.1%

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

   3. distribute-rgt1-in 17.1%

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

   4. metadata-eval 17.1%

      \[\leadsto \frac{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(\color{blue}{0} \cdot a\right) \cdot 2 - a}{c}}\right)}{a} \]

9. Simplified 17.1%

   \[\leadsto \frac{\color{blue}{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(0 \cdot a\right) \cdot 2 - a}{c}}\right)}}{a} \]

10. Taylor expanded in b_2 around inf 14.1%

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

    1. mul-1-neg 14.1%

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

12. Simplified 14.1%

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

13. Final simplification 14.1%

    \[\leadsto \frac{b\_2}{-a} \]
14. Add Preprocessing

Alternative 10: 2.6% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

   1. prod-diff 49.9%

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

   2. *-commutative 49.9%

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

   3. fma-neg 49.9%

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

   4. prod-diff 49.9%

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

   5. *-commutative 49.9%

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

   6. fma-neg 49.9%

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

   7. associate-+l+ 49.9%

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

   8. pow2 49.9%

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

   9. *-commutative 49.9%

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

   10. fma-undefine 49.9%

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

   11. distribute-lft-neg-in 49.9%

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

   12. *-commutative 49.9%

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

   13. distribute-rgt-neg-in 49.9%

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

   14. fma-define 49.9%

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

   15. *-commutative 49.9%

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

   16. fma-undefine 49.9%

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

   17. distribute-lft-neg-in 49.9%

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

   18. *-commutative 49.9%

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

   19. distribute-rgt-neg-in 49.9%

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

4. Applied egg-rr 49.9%

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

   1. count-2 49.9%

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

6. Simplified 49.9%

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

7. Taylor expanded in c around inf 17.1%

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

   1. mul-1-neg 17.1%

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

   2. *-commutative 17.1%

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

   3. distribute-rgt1-in 17.1%

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

   4. metadata-eval 17.1%

      \[\leadsto \frac{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(\color{blue}{0} \cdot a\right) \cdot 2 - a}{c}}\right)}{a} \]

9. Simplified 17.1%

   \[\leadsto \frac{\color{blue}{c \cdot \left(\left(-\frac{b\_2}{c}\right) - \sqrt{\frac{\left(0 \cdot a\right) \cdot 2 - a}{c}}\right)}}{a} \]

10. Taylor expanded in b_2 around inf 14.1%

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

    1. mul-1-neg 14.1%

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

12. Simplified 14.1%

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

    1. add-sqr-sqrt 1.4%

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

    2. sqrt-unprod 2.0%

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

    3. sqr-neg 2.0%

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

    4. sqrt-unprod 0.6%

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

    5. add-sqr-sqrt 2.4%

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

    6. *-un-lft-identity 2.4%

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

14. Applied egg-rr 2.4%

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

    1. *-lft-identity 2.4%

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

16. Simplified 2.4%

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

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024116 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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