quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.4% → 83.6%

Time: 10.1s

Alternatives: 9

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: 51.4% 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: 83.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.8e-76)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.05e-17)
     (- (/ (- b_2) a) (/ (sqrt (- (* b_2 b_2) (* c a))) a))
     (/ (- (- b_2) (+ b_2 (* -0.5 (/ c (/ b_2 a))))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-76) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.05e-17) {
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / 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 <= (-6.8d-76)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.05d-17) then
        tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a)
    else
        tmp = (-b_2 - (b_2 + ((-0.5d0) * (c / (b_2 / a))))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-76) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.05e-17) {
		tmp = (-b_2 / a) - (Math.sqrt(((b_2 * b_2) - (c * a))) / a);
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.8e-76:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.05e-17:
		tmp = (-b_2 / a) - (math.sqrt(((b_2 * b_2) - (c * a))) / a)
	else:
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.8e-76)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.05e-17)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a));
	else
		tmp = Float64(Float64(Float64(-b_2) - Float64(b_2 + Float64(-0.5 * Float64(c / Float64(b_2 / a))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.8e-76)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.05e-17)
		tmp = (-b_2 / a) - (sqrt(((b_2 * b_2) - (c * a))) / a);
	else
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.8e-76], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-17], 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[((-b$95$2) - N[(b$95$2 + N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.7999999999999998e-76

   1. Initial program 16.3%

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

   2. Taylor expanded in b_2 around -inf 87.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.5%

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

   4. Simplified 87.5%

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

   if -6.7999999999999998e-76 < b_2 < 1.04999999999999996e-17

   1. Initial program 81.6%

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

      1. div-sub 81.6%

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

      2. neg-mul-1 81.6%

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

      3. *-un-lft-identity 81.6%

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

      4. times-frac 81.6%

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

      5. metadata-eval 81.6%

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

      6. add-sqr-sqrt 50.9%

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

      7. sqrt-prod 80.6%

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

      8. sqr-neg 80.6%

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

      9. sqrt-unprod 30.5%

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

      10. add-sqr-sqrt 68.6%

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

      11. fma-neg 68.6%

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

      12. add-sqr-sqrt 30.5%

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

      13. sqrt-unprod 80.6%

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

      14. sqr-neg 80.6%

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

      15. sqrt-prod 50.9%

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

      16. add-sqr-sqrt 81.6%

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

   3. Applied egg-rr 78.6%

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

      1. fma-neg 78.6%

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

      2. associate-*r/ 78.6%

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

      3. mul-1-neg 78.6%

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

   5. Simplified 78.6%

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

      1. hypot-udef 75.9%

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

      2. add-sqr-sqrt 75.9%

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

      3. hypot-udef 75.9%

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

      4. hypot-udef 75.9%

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

      5. pow1/2 75.9%

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

      6. hypot-udef 75.9%

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

      7. hypot-udef 75.9%

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

      8. add-sqr-sqrt 75.9%

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

      9. add-sqr-sqrt 81.6%

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

      10. fma-def 81.6%

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

      11. *-commutative 81.6%

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

   7. Applied egg-rr 81.6%

      \[\leadsto \frac{-b_2}{a} - \frac{\color{blue}{{\left(\mathsf{fma}\left(b_2, b_2, \left(-a\right) \cdot c\right)\right)}^{0.5}}}{a} \]
   8. Step-by-step derivation

      1. unpow1/2 81.6%

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

      2. fma-udef 81.6%

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

      3. distribute-lft-neg-out 81.6%

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

      4. unsub-neg 81.6%

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

      5. *-commutative 81.6%

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

   9. Simplified 81.6%

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

   if 1.04999999999999996e-17 < b_2

   1. Initial program 67.4%

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

   2. Taylor expanded in b_2 around inf 84.2%

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

      1. associate-/l* 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \]

Alternative 2: 83.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.8e-76)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.05e-17)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (- (- b_2) (+ b_2 (* -0.5 (/ c (/ b_2 a))))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-76) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.05e-17) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / 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 <= (-6.8d-76)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.05d-17) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (-b_2 - (b_2 + ((-0.5d0) * (c / (b_2 / a))))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-76) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.05e-17) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.8e-76:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.05e-17:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.8e-76)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.05e-17)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - Float64(b_2 + Float64(-0.5 * Float64(c / Float64(b_2 / a))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.8e-76)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.05e-17)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.8e-76], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.05e-17], 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[((-b$95$2) - N[(b$95$2 + N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.7999999999999998e-76

   1. Initial program 16.3%

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

   2. Taylor expanded in b_2 around -inf 87.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.5%

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

   4. Simplified 87.5%

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

   if -6.7999999999999998e-76 < b_2 < 1.04999999999999996e-17

   1. Initial program 81.6%

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

   if 1.04999999999999996e-17 < b_2

   1. Initial program 67.4%

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

   2. Taylor expanded in b_2 around inf 84.2%

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

      1. associate-/l* 94.6%

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

   4. Simplified 94.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \]

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.8e-76)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 2.2e-95)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (/ (- (- b_2) (+ b_2 (* -0.5 (/ c (/ b_2 a))))) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-76) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.2e-95) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / 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 <= (-6.8d-76)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 2.2d-95) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = (-b_2 - (b_2 + ((-0.5d0) * (c / (b_2 / a))))) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.8e-76) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 2.2e-95) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.8e-76:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 2.2e-95:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.8e-76)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 2.2e-95)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(Float64(-b_2) - Float64(b_2 + Float64(-0.5 * Float64(c / Float64(b_2 / a))))) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.8e-76)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 2.2e-95)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-b_2 - (b_2 + (-0.5 * (c / (b_2 / a))))) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.8e-76], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.2e-95], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[((-b$95$2) - N[(b$95$2 + N[(-0.5 * N[(c / N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.7999999999999998e-76

   1. Initial program 16.3%

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

   2. Taylor expanded in b_2 around -inf 87.5%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 87.5%

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

   4. Simplified 87.5%

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

   if -6.7999999999999998e-76 < b_2 < 2.1999999999999999e-95

   1. Initial program 77.2%

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

   2. Taylor expanded in b_2 around 0 75.5%

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

      1. mul-1-neg 75.5%

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

      2. distribute-rgt-neg-out 75.5%

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

   4. Simplified 75.5%

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

   if 2.1999999999999999e-95 < b_2

   1. Initial program 73.8%

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

   2. Taylor expanded in b_2 around inf 79.1%

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

      1. associate-/l* 87.5%

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

   4. Simplified 87.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 2.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) - \left(b_2 + -0.5 \cdot \frac{c}{\frac{b_2}{a}}\right)}{a}\\ \end{array} \]

Alternative 4: 68.1% accurate, 8.6× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (/ (* -0.5 c) b_2)
   (+ (* (/ b_2 a) -2.0) (* 0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((b_2 / a) * -2.0) + (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 <= (-2d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((b_2 / a) * (-2.0d0)) + (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 <= -2e-310) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(Float64(Float64(b_2 / a) * -2.0) + 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 <= -2e-310)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = ((b_2 / a) * -2.0) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 33.8%

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

   2. Taylor expanded in b_2 around -inf 67.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 67.2%

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

   4. Simplified 67.2%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 75.4%

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

   2. Taylor expanded in b_2 around inf 66.9%

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

4. Final simplification 67.0%

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

Alternative 5: 42.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -10500000000000:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -10500000000000.0) (* 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 <= -10500000000000.0) {
		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 <= (-10500000000000.0d0)) 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 <= -10500000000000.0) {
		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 <= -10500000000000.0:
		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 <= -10500000000000.0)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -10500000000000.0)
		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, -10500000000000.0], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.05e13

   1. Initial program 14.3%

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

   2. Taylor expanded in b_2 around inf 2.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}}}{a} \]

   3. Taylor expanded in b_2 around 0 36.3%

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

   if -1.05e13 < b_2

   1. Initial program 72.5%

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

   2. Taylor expanded in b_2 around inf 49.0%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   3. Taylor expanded in b_2 around inf 49.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ 49.3%

         \[\leadsto \color{blue}{\frac{-2 \cdot b_2}{a}} \]

      2. associate-*l/ 49.2%

         \[\leadsto \color{blue}{\frac{-2}{a} \cdot b_2} \]

      3. *-commutative 49.2%

         \[\leadsto \color{blue}{b_2 \cdot \frac{-2}{a}} \]

   5. Simplified 49.2%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -10500000000000:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \end{array} \]

Alternative 6: 42.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -72000000:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -72000000.0) (* 0.5 (/ c b_2)) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -72000000.0) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = -2.0 / (a / 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 <= (-72000000.0d0)) then
        tmp = 0.5d0 * (c / b_2)
    else
        tmp = (-2.0d0) / (a / 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 <= -72000000.0) {
		tmp = 0.5 * (c / b_2);
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -72000000.0:
		tmp = 0.5 * (c / b_2)
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -72000000.0)
		tmp = Float64(0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -72000000.0)
		tmp = 0.5 * (c / b_2);
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -72000000.0], N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -7.2e7

   1. Initial program 14.3%

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

   2. Taylor expanded in b_2 around inf 2.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}}}{a} \]

   3. Taylor expanded in b_2 around 0 36.3%

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

   if -7.2e7 < b_2

   1. Initial program 72.5%

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

      1. add-exp-log 40.9%

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

      2. add-sqr-sqrt 9.5%

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

      3. sqrt-unprod 21.9%

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

      4. sqr-neg 21.9%

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

      5. sqrt-prod 17.0%

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

      6. add-sqr-sqrt 25.9%

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

      7. sub-neg 25.9%

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

      8. add-sqr-sqrt 23.8%

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

      9. hypot-def 21.7%

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

      10. *-commutative 21.7%

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

      11. distribute-rgt-neg-in 21.7%

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

   3. Applied egg-rr 21.7%

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

   4. Taylor expanded in b_2 around -inf 2.5%

      \[\leadsto \color{blue}{2 \cdot \frac{b_2}{a}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 1.6%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{b_2}{a}} \cdot \sqrt{2 \cdot \frac{b_2}{a}}} \]

      2. sqrt-unprod 21.5%

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

      3. swap-sqr 21.5%

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

      4. metadata-eval 21.5%

         \[\leadsto \sqrt{\color{blue}{4} \cdot \left(\frac{b_2}{a} \cdot \frac{b_2}{a}\right)} \]

      5. metadata-eval 21.5%

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

      6. swap-sqr 21.5%

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

      7. sqrt-unprod 27.4%

         \[\leadsto \color{blue}{\sqrt{-2 \cdot \frac{b_2}{a}} \cdot \sqrt{-2 \cdot \frac{b_2}{a}}} \]

      8. add-sqr-sqrt 49.3%

         \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

      9. clear-num 49.2%

         \[\leadsto -2 \cdot \color{blue}{\frac{1}{\frac{a}{b_2}}} \]

      10. un-div-inv 49.2%

          \[\leadsto \color{blue}{\frac{-2}{\frac{a}{b_2}}} \]

   6. Applied egg-rr 49.2%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -72000000:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 7: 67.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-275) (/ (* -0.5 c) b_2) (/ -2.0 (/ a b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-275) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 / (a / 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 <= (-2.8d-275)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = (-2.0d0) / (a / 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 <= -2.8e-275) {
		tmp = (-0.5 * c) / b_2;
	} else {
		tmp = -2.0 / (a / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-275:
		tmp = (-0.5 * c) / b_2
	else:
		tmp = -2.0 / (a / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-275)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	else
		tmp = Float64(-2.0 / Float64(a / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-275)
		tmp = (-0.5 * c) / b_2;
	else
		tmp = -2.0 / (a / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-275], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(-2.0 / N[(a / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.79999999999999994e-275

   1. Initial program 31.7%

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

   2. Taylor expanded in b_2 around -inf 69.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.3%

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

   4. Simplified 69.3%

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

   if -2.79999999999999994e-275 < b_2

   1. Initial program 76.1%

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

      1. add-exp-log 44.1%

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

      2. add-sqr-sqrt 2.1%

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

      3. sqrt-unprod 18.6%

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

      4. sqr-neg 18.6%

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

      5. sqrt-prod 22.7%

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

      6. add-sqr-sqrt 24.0%

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

      7. sub-neg 24.0%

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

      8. add-sqr-sqrt 21.2%

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

      9. hypot-def 18.5%

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

      10. *-commutative 18.5%

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

      11. distribute-rgt-neg-in 18.5%

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

   3. Applied egg-rr 18.5%

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

   4. Taylor expanded in b_2 around -inf 1.8%

      \[\leadsto \color{blue}{2 \cdot \frac{b_2}{a}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 1.0%

         \[\leadsto \color{blue}{\sqrt{2 \cdot \frac{b_2}{a}} \cdot \sqrt{2 \cdot \frac{b_2}{a}}} \]

      2. sqrt-unprod 27.4%

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

      3. swap-sqr 27.4%

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

      4. metadata-eval 27.4%

         \[\leadsto \sqrt{\color{blue}{4} \cdot \left(\frac{b_2}{a} \cdot \frac{b_2}{a}\right)} \]

      5. metadata-eval 27.4%

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

      6. swap-sqr 27.4%

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

      7. sqrt-unprod 35.9%

         \[\leadsto \color{blue}{\sqrt{-2 \cdot \frac{b_2}{a}} \cdot \sqrt{-2 \cdot \frac{b_2}{a}}} \]

      8. add-sqr-sqrt 64.8%

         \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]

      9. clear-num 64.6%

         \[\leadsto -2 \cdot \color{blue}{\frac{1}{\frac{a}{b_2}}} \]

      10. un-div-inv 64.6%

          \[\leadsto \color{blue}{\frac{-2}{\frac{a}{b_2}}} \]

   6. Applied egg-rr 64.6%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.8 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{\frac{a}{b_2}}\\ \end{array} \]

Alternative 8: 67.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.9e-275) (/ (* -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 <= -2.9e-275) {
		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 <= (-2.9d-275)) 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 <= -2.9e-275) {
		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 <= -2.9e-275:
		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 <= -2.9e-275)
		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 <= -2.9e-275)
		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, -2.9e-275], 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 < -2.9e-275

   1. Initial program 31.7%

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

   2. Taylor expanded in b_2 around -inf 69.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.3%

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

   4. Simplified 69.3%

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

   if -2.9e-275 < b_2

   1. Initial program 76.1%

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

   2. Taylor expanded in b_2 around inf 64.8%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   3. Step-by-step derivation

      1. *-commutative 64.8%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   4. Simplified 64.8%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{-275}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \end{array} \]

Alternative 9: 10.4% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{c}{b_2} \end{array} \]

FPCore

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

C

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

Fortran

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 = 0.5d0 * (c / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.8%

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

2. Taylor expanded in b_2 around inf 31.8%

   \[\leadsto \frac{\color{blue}{-2 \cdot b_2 + 0.5 \cdot \frac{c \cdot a}{b_2}}}{a} \]

3. Taylor expanded in b_2 around 0 13.4%

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

4. Final simplification 13.4%

   \[\leadsto 0.5 \cdot \frac{c}{b_2} \]

Reproduce

herbie shell --seed 2023257 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))