quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.2% → 84.9%

Time: 10.5s

Alternatives: 8

Speedup: 15.9×

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: 51.2% 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: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-18}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+66}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e-18)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.45e+66)
     (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-18) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+66) {
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-8.5d-18)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.45d+66) then
        tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e-18) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+66) {
		tmp = (-b_2 / a) - (Math.sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8.5e-18:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.45e+66:
		tmp = (-b_2 / a) - (math.sqrt(((b_2 * b_2) - (c * a))) / a)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.5e-18)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.45e+66)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.5e-18)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.45e+66)
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.5e-18], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.45e+66], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.4999999999999995e-18

   1. Initial program 10.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -8.4999999999999995e-18 < b_2 < 1.44999999999999993e66

   1. Initial program 79.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. div-sub 79.9%

         \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]

      2. neg-mul-1 79.9%

         \[\leadsto \frac{\color{blue}{-1 \cdot b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      3. *-un-lft-identity 79.9%

         \[\leadsto \frac{-1 \cdot b_2}{\color{blue}{1 \cdot a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      4. times-frac 79.9%

         \[\leadsto \color{blue}{\frac{-1}{1} \cdot \frac{b_2}{a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      5. metadata-eval 79.9%

         \[\leadsto \color{blue}{-1} \cdot \frac{b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      6. add-sqr-sqrt 56.5%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      7. sqrt-prod 78.3%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      8. sqr-neg 78.3%

         \[\leadsto -1 \cdot \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      9. sqrt-unprod 22.4%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      10. add-sqr-sqrt 61.8%

          \[\leadsto -1 \cdot \frac{\color{blue}{-b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      11. fma-neg 61.8%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b_2}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

      12. add-sqr-sqrt 22.4%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      13. sqrt-unprod 78.3%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      14. sqr-neg 78.3%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      15. sqrt-prod 56.5%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      16. add-sqr-sqrt 79.9%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b_2}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   3. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 77.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}} \]

      2. associate-*r/ 77.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

      3. mul-1-neg 77.0%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

      4. distribute-rgt-neg-out 77.0%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{-c \cdot a}}\right)}{a} \]

      5. distribute-lft-neg-in 77.0%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{\left(-c\right) \cdot a}}\right)}{a} \]

      6. *-commutative 77.0%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{a \cdot \left(-c\right)}}\right)}{a} \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]
   6. Step-by-step derivation

      1. hypot-udef 74.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{\sqrt{b_2 \cdot b_2 + \sqrt{a \cdot \left(-c\right)} \cdot \sqrt{a \cdot \left(-c\right)}}}}{a} \]

      2. add-sqr-sqrt 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 + \color{blue}{a \cdot \left(-c\right)}}}{a} \]

      3. distribute-rgt-neg-out 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 + \color{blue}{\left(-a \cdot c\right)}}}{a} \]

      4. sub-neg 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{\color{blue}{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      5. pow1/2 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}{a} \]

      6. metadata-eval 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{\left(0.25 + 0.25\right)}}}{a} \]

      7. pow-prod-up 79.6%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}}}{a} \]

   7. Applied egg-rr 79.6%

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25} \cdot {\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}}}{a} \]
   8. Step-by-step derivation

      1. pow-sqr 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(2 \cdot 0.25\right)}}}{a} \]

      2. metadata-eval 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.5}}}{a} \]

      3. unpow1/2 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c}}}{a} \]

      4. *-commutative 79.9%

         \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - \color{blue}{c \cdot a}}}{a} \]

   9. Simplified 79.9%

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}}}{a} \]

   if 1.44999999999999993e66 < b_2

   1. Initial program 46.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 89.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.5 \cdot 10^{-18}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+66}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 2: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -4.15 \cdot 10^{-19}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.15e-19)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.45e+66)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.15e-19) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+66) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-4.15d-19)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 1.45d+66) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.15e-19) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+66) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.15e-19:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 1.45e+66:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.15e-19)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.45e+66)
		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(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.15e-19)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 1.45e+66)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.15e-19], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.45e+66], 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[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.1500000000000001e-19

   1. Initial program 10.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -4.1500000000000001e-19 < b_2 < 1.44999999999999993e66

   1. Initial program 79.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   if 1.44999999999999993e66 < b_2

   1. Initial program 46.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 89.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.15 \cdot 10^{-19}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{-19}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-19)
   (* -0.5 (/ c b_2))
   (if (<= b_2 9.2e-14)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-19) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 9.2e-14) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-7.2d-19)) then
        tmp = (-0.5d0) * (c / b_2)
    else if (b_2 <= 9.2d-14) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-19) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 9.2e-14) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2e-19:
		tmp = -0.5 * (c / b_2)
	elif b_2 <= 9.2e-14:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2e-19)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 9.2e-14)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e-19)
		tmp = -0.5 * (c / b_2);
	elseif (b_2 <= 9.2e-14)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e-19], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.2e-14], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.2000000000000002e-19

   1. Initial program 10.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 90.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -7.2000000000000002e-19 < b_2 < 9.19999999999999993e-14

   1. Initial program 77.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around 0 70.8%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   3. Step-by-step derivation

      1. associate-*r* 70.8%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. neg-mul-1 70.8%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{\left(-a\right)} \cdot c}}{a} \]

      3. *-commutative 70.8%

         \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   4. Simplified 70.8%

      \[\leadsto \frac{\left(-b_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   if 9.19999999999999993e-14 < b_2

   1. Initial program 59.7%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 84.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{-19}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 4: 68.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* -0.5 (/ c b_2))
   (+ (* -2.0 (/ b_2 a)) (* (/ c b_2) 0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	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 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = ((-2.0d0) * (b_2 / a)) + ((c / b_2) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(Float64(c / b_2) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = (-2.0 * (b_2 / a)) + ((c / b_2) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 29.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 67.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 70.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 54.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + \frac{c}{b_2} \cdot 0.5\\ \end{array} \]

Alternative 5: 43.5% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) 0.0 (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = -2.0 * (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 <= (-5d-310)) then
        tmp = 0.0d0
    else
        tmp = (-2.0d0) * (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 <= -5e-310) {
		tmp = 0.0;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = 0.0
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = 0.0;
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = 0.0;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], 0.0, N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 29.0%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. div-sub 28.6%

         \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]

      2. neg-mul-1 28.6%

         \[\leadsto \frac{\color{blue}{-1 \cdot b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      3. *-un-lft-identity 28.6%

         \[\leadsto \frac{-1 \cdot b_2}{\color{blue}{1 \cdot a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      4. times-frac 28.6%

         \[\leadsto \color{blue}{\frac{-1}{1} \cdot \frac{b_2}{a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      5. metadata-eval 28.6%

         \[\leadsto \color{blue}{-1} \cdot \frac{b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      7. sqrt-prod 25.8%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      8. sqr-neg 25.8%

         \[\leadsto -1 \cdot \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      9. sqrt-unprod 25.9%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      10. add-sqr-sqrt 25.9%

          \[\leadsto -1 \cdot \frac{\color{blue}{-b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      11. fma-neg 25.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b_2}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

      12. add-sqr-sqrt 25.9%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      13. sqrt-unprod 25.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      14. sqr-neg 25.8%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      15. sqrt-prod 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      16. add-sqr-sqrt 28.6%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b_2}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   3. Applied egg-rr 30.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 30.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}} \]

      2. associate-*r/ 30.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

      3. mul-1-neg 30.6%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

      4. distribute-rgt-neg-out 30.6%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{-c \cdot a}}\right)}{a} \]

      5. distribute-lft-neg-in 30.6%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{\left(-c\right) \cdot a}}\right)}{a} \]

      6. *-commutative 30.6%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{a \cdot \left(-c\right)}}\right)}{a} \]

   5. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

   6. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-1 \cdot \frac{b_2}{a} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\left(-\frac{b_2}{a}\right)} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right) \]

      2. distribute-frac-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\frac{-b_2}{a}} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right) \]

      3. +-commutative 0.0%

         \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + \frac{-b_2}{a}\right)} \]

      4. distribute-frac-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + \color{blue}{\left(-\frac{b_2}{a}\right)}\right) \]

      5. unsub-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} - \frac{b_2}{a}\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} - \frac{b_2}{a}\right) \]

      7. *-commutative 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} - \frac{b_2}{a}\right) \]

      8. unpow2 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} - \frac{b_2}{a}\right) \]

      9. rem-square-sqrt 11.6%

         \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} - \frac{b_2}{a}\right) \]

      10. neg-mul-1 11.6%

          \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \color{blue}{\left(-c\right)}}{b_2} - \frac{b_2}{a}\right) \]

   8. Simplified 11.6%

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(\frac{-0.5 \cdot \left(-c\right)}{b_2} - \frac{b_2}{a}\right)} \]

   9. Taylor expanded in a around 0 16.7%

      \[\leadsto \color{blue}{\frac{b_2 + -1 \cdot b_2}{a}} \]
   10. Step-by-step derivation

       1. distribute-rgt1-in 16.7%

          \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

       2. metadata-eval 16.7%

          \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

       3. associate-*r/ 10.5%

          \[\leadsto \color{blue}{0 \cdot \frac{b_2}{a}} \]

       4. mul0-lft 16.7%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 16.7%

       \[\leadsto \color{blue}{0} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 70.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 54.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 6: 68.1% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-302}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.4e-302) (* -0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.4e-302) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (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 <= (-1.4d-302)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (-2.0d0) * (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 <= -1.4e-302) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.4e-302:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.4e-302)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.4e-302)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.4e-302], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.4e-302

   1. Initial program 27.9%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around -inf 68.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if -1.4e-302 < b_2

   1. Initial program 71.3%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   2. Taylor expanded in b_2 around inf 53.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.4 \cdot 10^{-302}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array} \]

Alternative 7: 15.0% accurate, 22.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-302}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 -1.15e-302) 0.0 (/ c 0.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-302) {
		tmp = 0.0;
	} else {
		tmp = c / 0.0;
	}
	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 <= (-1.15d-302)) then
        tmp = 0.0d0
    else
        tmp = c / 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-302) {
		tmp = 0.0;
	} else {
		tmp = c / 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e-302:
		tmp = 0.0
	else:
		tmp = c / 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-302)
		tmp = 0.0;
	else
		tmp = Float64(c / 0.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15e-302)
		tmp = 0.0;
	else
		tmp = c / 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-302], 0.0, N[(c / 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.15000000000000001e-302

   1. Initial program 28.5%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. div-sub 28.0%

         \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]

      2. neg-mul-1 28.0%

         \[\leadsto \frac{\color{blue}{-1 \cdot b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      3. *-un-lft-identity 28.0%

         \[\leadsto \frac{-1 \cdot b_2}{\color{blue}{1 \cdot a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      4. times-frac 28.0%

         \[\leadsto \color{blue}{\frac{-1}{1} \cdot \frac{b_2}{a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      5. metadata-eval 28.0%

         \[\leadsto \color{blue}{-1} \cdot \frac{b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      6. add-sqr-sqrt 0.0%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      7. sqrt-prod 25.2%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      8. sqr-neg 25.2%

         \[\leadsto -1 \cdot \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      9. sqrt-unprod 25.3%

         \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      10. add-sqr-sqrt 25.3%

          \[\leadsto -1 \cdot \frac{\color{blue}{-b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

      11. fma-neg 25.3%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b_2}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

      12. add-sqr-sqrt 25.3%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      13. sqrt-unprod 25.2%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      14. sqr-neg 25.2%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      15. sqrt-prod 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

      16. add-sqr-sqrt 28.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b_2}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   3. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\right)} \]
   4. Step-by-step derivation

      1. fma-neg 30.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}} \]

      2. associate-*r/ 30.1%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

      3. mul-1-neg 30.1%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

      4. distribute-rgt-neg-out 30.1%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{-c \cdot a}}\right)}{a} \]

      5. distribute-lft-neg-in 30.1%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{\left(-c\right) \cdot a}}\right)}{a} \]

      6. *-commutative 30.1%

         \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{a \cdot \left(-c\right)}}\right)}{a} \]

   5. Simplified 30.1%

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

   6. Taylor expanded in b_2 around -inf 0.0%

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-1 \cdot \frac{b_2}{a} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\left(-\frac{b_2}{a}\right)} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right) \]

      2. distribute-frac-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\frac{-b_2}{a}} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right) \]

      3. +-commutative 0.0%

         \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + \frac{-b_2}{a}\right)} \]

      4. distribute-frac-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + \color{blue}{\left(-\frac{b_2}{a}\right)}\right) \]

      5. unsub-neg 0.0%

         \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} - \frac{b_2}{a}\right)} \]

      6. associate-*r/ 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} - \frac{b_2}{a}\right) \]

      7. *-commutative 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} - \frac{b_2}{a}\right) \]

      8. unpow2 0.0%

         \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} - \frac{b_2}{a}\right) \]

      9. rem-square-sqrt 11.6%

         \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} - \frac{b_2}{a}\right) \]

      10. neg-mul-1 11.6%

          \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \color{blue}{\left(-c\right)}}{b_2} - \frac{b_2}{a}\right) \]

   8. Simplified 11.6%

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(\frac{-0.5 \cdot \left(-c\right)}{b_2} - \frac{b_2}{a}\right)} \]

   9. Taylor expanded in a around 0 16.8%

      \[\leadsto \color{blue}{\frac{b_2 + -1 \cdot b_2}{a}} \]
   10. Step-by-step derivation

       1. distribute-rgt1-in 16.8%

          \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

       2. metadata-eval 16.8%

          \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

       3. associate-*r/ 10.5%

          \[\leadsto \color{blue}{0 \cdot \frac{b_2}{a}} \]

       4. mul0-lft 16.8%

          \[\leadsto \color{blue}{0} \]

   11. Simplified 16.8%

       \[\leadsto \color{blue}{0} \]

   if -1.15000000000000001e-302 < b_2

   1. Initial program 71.1%

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. add-sqr-sqrt 70.9%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}} \cdot \sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a} \]

      2. pow2 70.9%

         \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}^{2}}}{a} \]

      3. pow1/2 70.9%

         \[\leadsto \frac{\left(-b_2\right) - {\left(\sqrt{\color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5}}}\right)}^{2}}{a} \]

      4. sqrt-pow1 70.9%

         \[\leadsto \frac{\left(-b_2\right) - {\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{a} \]

      5. metadata-eval 70.9%

         \[\leadsto \frac{\left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2}}{a} \]

   3. Applied egg-rr 70.9%

      \[\leadsto \frac{\left(-b_2\right) - \color{blue}{{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]
   4. Step-by-step derivation

      1. flip-- 40.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}}{a} \]

      2. pow2 40.3%

         \[\leadsto \frac{\frac{\color{blue}{{\left(-b_2\right)}^{2}} - {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      3. pow-pow 39.9%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      4. metadata-eval 39.9%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - {\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.5}} \cdot {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      5. pow-pow 37.8%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - {\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} \cdot \color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      6. metadata-eval 37.8%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - {\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.5} \cdot {\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.5}}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      7. pow-prod-up 38.0%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(0.5 + 0.5\right)}}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      8. metadata-eval 38.0%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - {\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{1}}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      9. pow1 38.0%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}{\left(-b_2\right) + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      10. neg-mul-1 38.0%

          \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\color{blue}{-1 \cdot b_2} + {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}}}{a} \]

      11. fma-def 38.0%

          \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\color{blue}{\mathsf{fma}\left(-1, b_2, {\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}\right)}^{2}\right)}}}{a} \]

      12. pow-pow 37.9%

          \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \color{blue}{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(0.25 \cdot 2\right)}}\right)}}{a} \]

      13. metadata-eval 37.9%

          \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, {\left(b_2 \cdot b_2 - a \cdot c\right)}^{\color{blue}{0.5}}\right)}}{a} \]

      14. pow1/2 37.9%

          \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right)}}{a} \]

   5. Applied egg-rr 37.0%

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b_2\right)}^{2} - \left(b_2 \cdot b_2 - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}}{a} \]
   6. Step-by-step derivation

      1. unpow2 37.0%

         \[\leadsto \frac{\frac{{\left(-b_2\right)}^{2} - \left(\color{blue}{{b_2}^{2}} - a \cdot c\right)}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      2. associate--r- 38.2%

         \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b_2\right)}^{2} - {b_2}^{2}\right) + a \cdot c}}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      3. unpow2 38.2%

         \[\leadsto \frac{\frac{\left(\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)} - {b_2}^{2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      4. unpow2 38.2%

         \[\leadsto \frac{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \color{blue}{b_2 \cdot b_2}\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      5. difference-of-squares 47.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(-b_2\right) + b_2\right) \cdot \left(\left(-b_2\right) - b_2\right)} + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      6. +-commutative 47.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(b_2 + \left(-b_2\right)\right)} \cdot \left(\left(-b_2\right) - b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      7. neg-mul-1 47.8%

         \[\leadsto \frac{\frac{\left(b_2 + \color{blue}{-1 \cdot b_2}\right) \cdot \left(\left(-b_2\right) - b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      8. distribute-rgt1-in 47.8%

         \[\leadsto \frac{\frac{\color{blue}{\left(\left(-1 + 1\right) \cdot b_2\right)} \cdot \left(\left(-b_2\right) - b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      9. metadata-eval 47.8%

         \[\leadsto \frac{\frac{\left(\color{blue}{0} \cdot b_2\right) \cdot \left(\left(-b_2\right) - b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      10. mul0-lft 47.8%

          \[\leadsto \frac{\frac{\color{blue}{0} \cdot \left(\left(-b_2\right) - b_2\right) + a \cdot c}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      11. *-commutative 47.8%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b_2\right) - b_2\right) + \color{blue}{c \cdot a}}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)\right)}}{a} \]

      12. distribute-rgt-neg-out 47.8%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b_2\right) - b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{\color{blue}{-a \cdot c}}\right)\right)}}{a} \]

      13. *-commutative 47.8%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b_2\right) - b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{-\color{blue}{c \cdot a}}\right)\right)}}{a} \]

      14. distribute-rgt-neg-in 47.8%

          \[\leadsto \frac{\frac{0 \cdot \left(\left(-b_2\right) - b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right)\right)}}{a} \]

   7. Simplified 47.8%

      \[\leadsto \frac{\color{blue}{\frac{0 \cdot \left(\left(-b_2\right) - b_2\right) + c \cdot a}{\mathsf{fma}\left(-1, b_2, \mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)\right)}}}{a} \]

   8. Taylor expanded in c around 0 15.1%

      \[\leadsto \color{blue}{\frac{c}{b_2 + -1 \cdot b_2}} \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 15.1%

         \[\leadsto \frac{c}{\color{blue}{\left(-1 + 1\right) \cdot b_2}} \]

      2. metadata-eval 15.1%

         \[\leadsto \frac{c}{\color{blue}{0} \cdot b_2} \]

      3. mul0-lft 15.1%

         \[\leadsto \frac{c}{\color{blue}{0}} \]

   10. Simplified 15.1%

       \[\leadsto \color{blue}{\frac{c}{0}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.15 \cdot 10^{-302}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{0}\\ \end{array} \]

Alternative 8: 11.1% accurate, 112.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 0.0)

C

double code(double a, double b_2, double c) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double a, double b_2, double c) {
	return 0.0;
}

Python

def code(a, b_2, c):
	return 0.0

Julia

function code(a, b_2, c)
	return 0.0
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = 0.0;
end

Wolfram

code[a_, b$95$2_, c_] := 0.0

Derivation

1. Initial program 50.3%

   \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. div-sub 50.1%

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]

   2. neg-mul-1 50.1%

      \[\leadsto \frac{\color{blue}{-1 \cdot b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   3. *-un-lft-identity 50.1%

      \[\leadsto \frac{-1 \cdot b_2}{\color{blue}{1 \cdot a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   4. times-frac 50.1%

      \[\leadsto \color{blue}{\frac{-1}{1} \cdot \frac{b_2}{a}} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   5. metadata-eval 50.1%

      \[\leadsto \color{blue}{-1} \cdot \frac{b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   6. add-sqr-sqrt 36.0%

      \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   7. sqrt-prod 48.4%

      \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{b_2 \cdot b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   8. sqr-neg 48.4%

      \[\leadsto -1 \cdot \frac{\sqrt{\color{blue}{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   9. sqrt-unprod 12.8%

      \[\leadsto -1 \cdot \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   10. add-sqr-sqrt 31.8%

       \[\leadsto -1 \cdot \frac{\color{blue}{-b_2}}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

   11. fma-neg 31.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{-b_2}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)} \]

   12. add-sqr-sqrt 12.8%

       \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{-b_2} \cdot \sqrt{-b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   13. sqrt-unprod 48.4%

       \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{\left(-b_2\right) \cdot \left(-b_2\right)}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   14. sqr-neg 48.4%

       \[\leadsto \mathsf{fma}\left(-1, \frac{\sqrt{\color{blue}{b_2 \cdot b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   15. sqrt-prod 36.0%

       \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{\sqrt{b_2} \cdot \sqrt{b_2}}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

   16. add-sqr-sqrt 50.1%

       \[\leadsto \mathsf{fma}\left(-1, \frac{\color{blue}{b_2}}{a}, -\frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right) \]

3. Applied egg-rr 52.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{b_2}{a}, -\frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\right)} \]
4. Step-by-step derivation

   1. fma-neg 52.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}} \]

   2. associate-*r/ 52.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

   3. mul-1-neg 52.0%

      \[\leadsto \frac{\color{blue}{-b_2}}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)}{a} \]

   4. distribute-rgt-neg-out 52.0%

      \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{-c \cdot a}}\right)}{a} \]

   5. distribute-lft-neg-in 52.0%

      \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{\left(-c\right) \cdot a}}\right)}{a} \]

   6. *-commutative 52.0%

      \[\leadsto \frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{\color{blue}{a \cdot \left(-c\right)}}\right)}{a} \]

5. Simplified 52.0%

   \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\mathsf{hypot}\left(b_2, \sqrt{a \cdot \left(-c\right)}\right)}{a}} \]

6. Taylor expanded in b_2 around -inf 0.0%

   \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-1 \cdot \frac{b_2}{a} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 0.0%

      \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\left(-\frac{b_2}{a}\right)} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right) \]

   2. distribute-frac-neg 0.0%

      \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\frac{-b_2}{a}} + -0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2}\right) \]

   3. +-commutative 0.0%

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + \frac{-b_2}{a}\right)} \]

   4. distribute-frac-neg 0.0%

      \[\leadsto \frac{-b_2}{a} - \left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} + \color{blue}{\left(-\frac{b_2}{a}\right)}\right) \]

   5. unsub-neg 0.0%

      \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(-0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b_2} - \frac{b_2}{a}\right)} \]

   6. associate-*r/ 0.0%

      \[\leadsto \frac{-b_2}{a} - \left(\color{blue}{\frac{-0.5 \cdot \left(c \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{b_2}} - \frac{b_2}{a}\right) \]

   7. *-commutative 0.0%

      \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot c\right)}}{b_2} - \frac{b_2}{a}\right) \]

   8. unpow2 0.0%

      \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c\right)}{b_2} - \frac{b_2}{a}\right) \]

   9. rem-square-sqrt 6.5%

      \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \left(\color{blue}{-1} \cdot c\right)}{b_2} - \frac{b_2}{a}\right) \]

   10. neg-mul-1 6.5%

       \[\leadsto \frac{-b_2}{a} - \left(\frac{-0.5 \cdot \color{blue}{\left(-c\right)}}{b_2} - \frac{b_2}{a}\right) \]

8. Simplified 6.5%

   \[\leadsto \frac{-b_2}{a} - \color{blue}{\left(\frac{-0.5 \cdot \left(-c\right)}{b_2} - \frac{b_2}{a}\right)} \]

9. Taylor expanded in a around 0 9.6%

   \[\leadsto \color{blue}{\frac{b_2 + -1 \cdot b_2}{a}} \]
10. Step-by-step derivation

    1. distribute-rgt1-in 9.6%

       \[\leadsto \frac{\color{blue}{\left(-1 + 1\right) \cdot b_2}}{a} \]

    2. metadata-eval 9.6%

       \[\leadsto \frac{\color{blue}{0} \cdot b_2}{a} \]

    3. associate-*r/ 6.4%

       \[\leadsto \color{blue}{0 \cdot \frac{b_2}{a}} \]

    4. mul0-lft 9.6%

       \[\leadsto \color{blue}{0} \]

11. Simplified 9.6%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 9.6%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))