quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.5% → 86.5%

Time: 11.1s

Alternatives: 12

Speedup: 11.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.5% 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: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-154}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)}{-a} - \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.2e-53)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1e-154)
     (- (/ (hypot b_2 (sqrt (* c (- a)))) (- a)) (/ b_2 a))
     (if (<= b_2 6e+103)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (/ (* b_2 -2.0) a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-154) {
		tmp = (hypot(b_2, sqrt((c * -a))) / -a) - (b_2 / a);
	} else if (b_2 <= 6e+103) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.2e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-154) {
		tmp = (Math.hypot(b_2, Math.sqrt((c * -a))) / -a) - (b_2 / a);
	} else if (b_2 <= 6e+103) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.2e-53:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1e-154:
		tmp = (math.hypot(b_2, math.sqrt((c * -a))) / -a) - (b_2 / a)
	elif b_2 <= 6e+103:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.2e-53)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1e-154)
		tmp = Float64(Float64(hypot(b_2, sqrt(Float64(c * Float64(-a)))) / Float64(-a)) - Float64(b_2 / a));
	elseif (b_2 <= 6e+103)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.2e-53)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1e-154)
		tmp = (hypot(b_2, sqrt((c * -a))) / -a) - (b_2 / a);
	elseif (b_2 <= 6e+103)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.2e-53], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1e-154], N[(N[(N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision] / (-a)), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 6e+103], 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 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.2000000000000001e-53

   1. Initial program 19.0%

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

   3. Taylor expanded in b_2 around -inf 82.9%

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

      1. associate-*r/ 82.9%

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

   5. Simplified 82.9%

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

   if -3.2000000000000001e-53 < b_2 < 9.9999999999999997e-155

   1. Initial program 74.2%

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

      1. div-sub 74.2%

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

      2. neg-mul-1 74.2%

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

      3. associate-/l* 74.2%

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

      4. add-sqr-sqrt 24.8%

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

      5. sqrt-prod 72.5%

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

      6. sqr-neg 72.5%

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

      7. sqrt-unprod 49.2%

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

      8. add-sqr-sqrt 72.4%

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

      9. fma-neg 72.4%

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

      10. add-sqr-sqrt 49.2%

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

      11. sqrt-unprod 72.5%

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

      12. sqr-neg 72.5%

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

      13. sqrt-prod 24.8%

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

      14. add-sqr-sqrt 74.2%

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

   4. Applied egg-rr 80.9%

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

      1. fma-undefine 80.9%

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

      2. unsub-neg 80.9%

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

      3. mul-1-neg 80.9%

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

      4. distribute-frac-neg2 80.9%

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

      5. distribute-rgt-neg-out 80.9%

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

      6. *-commutative 80.9%

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

      7. distribute-rgt-neg-in 80.9%

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

   6. Simplified 80.9%

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

   if 9.9999999999999997e-155 < b_2 < 6e103

   1. Initial program 92.3%

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

   if 6e103 < b_2

   1. Initial program 51.0%

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

   3. Taylor expanded in b_2 around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. *-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-154}:\\ \;\;\;\;\frac{\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)}{-a} - \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{+103}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-162}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.1e-53)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1e-162)
     (/ (- (- b_2) (hypot b_2 (sqrt (* c (- a))))) a)
     (if (<= b_2 9.2e+104)
       (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
       (/ (* b_2 -2.0) a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.1e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-162) {
		tmp = (-b_2 - hypot(b_2, sqrt((c * -a)))) / a;
	} else if (b_2 <= 9.2e+104) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.1e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1e-162) {
		tmp = (-b_2 - Math.hypot(b_2, Math.sqrt((c * -a)))) / a;
	} else if (b_2 <= 9.2e+104) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.1e-53:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1e-162:
		tmp = (-b_2 - math.hypot(b_2, math.sqrt((c * -a)))) / a
	elif b_2 <= 9.2e+104:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.1e-53)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1e-162)
		tmp = Float64(Float64(Float64(-b_2) - hypot(b_2, sqrt(Float64(c * Float64(-a))))) / a);
	elseif (b_2 <= 9.2e+104)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.1e-53)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1e-162)
		tmp = (-b_2 - hypot(b_2, sqrt((c * -a)))) / a;
	elseif (b_2 <= 9.2e+104)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.1e-53], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1e-162], N[(N[((-b$95$2) - N[Sqrt[b$95$2 ^ 2 + N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 9.2e+104], 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 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -3.10000000000000015e-53

   1. Initial program 19.0%

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

   3. Taylor expanded in b_2 around -inf 82.9%

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

      1. associate-*r/ 82.9%

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

   5. Simplified 82.9%

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

   if -3.10000000000000015e-53 < b_2 < 9.99999999999999954e-163

   1. Initial program 74.2%

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

      1. sub-neg 74.2%

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

      2. distribute-neg-out 74.2%

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

      3. add-sqr-sqrt 24.8%

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

      4. sqrt-prod 72.5%

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

      5. sqr-neg 72.5%

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

      6. sqrt-unprod 49.2%

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

      7. add-sqr-sqrt 72.4%

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

      8. add-sqr-sqrt 49.2%

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

      9. sqrt-unprod 72.5%

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

      10. sqr-neg 72.5%

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

      11. sqrt-prod 24.8%

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

      12. add-sqr-sqrt 74.2%

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

      13. sub-neg 74.2%

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

      14. add-sqr-sqrt 74.2%

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

      15. hypot-define 80.9%

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

      16. distribute-rgt-neg-in 80.9%

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

   4. Applied egg-rr 80.9%

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

      1. +-commutative 80.9%

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

      2. distribute-neg-in 80.9%

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

      3. sub-neg 80.9%

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

      4. distribute-rgt-neg-out 80.9%

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

      5. *-commutative 80.9%

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

      6. distribute-rgt-neg-in 80.9%

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

   6. Simplified 80.9%

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

   if 9.99999999999999954e-163 < b_2 < 9.19999999999999938e104

   1. Initial program 92.3%

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

   if 9.19999999999999938e104 < b_2

   1. Initial program 51.0%

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

   3. Taylor expanded in b_2 around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. *-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.1 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 10^{-162}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right)}{a}\\ \mathbf{elif}\;b\_2 \leq 9.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.4 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-53)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.4e+104)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.4e+104) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-3.8d-53)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 1.4d+104) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.4e+104) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e-53:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 1.4e+104:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e-53)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.4e+104)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e-53)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 1.4e+104)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -3.8e-53], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.4e+104], 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 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.7999999999999998e-53

   1. Initial program 19.0%

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

   3. Taylor expanded in b_2 around -inf 82.9%

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

      1. associate-*r/ 82.9%

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

   5. Simplified 82.9%

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

   if -3.7999999999999998e-53 < b_2 < 1.4e104

   1. Initial program 82.4%

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

   if 1.4e104 < b_2

   1. Initial program 51.0%

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

   3. Taylor expanded in b_2 around inf 98.4%

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

      1. associate-*r/ 98.4%

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

      2. *-commutative 98.4%

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

   5. Simplified 98.4%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.4 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-53)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 9e-87)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9e-87) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-53)) then
        tmp = ((-0.5d0) * c) / b_2
    else if (b_2 <= 9d-87) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-53) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 9e-87) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-53:
		tmp = (-0.5 * c) / b_2
	elif b_2 <= 9e-87:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-53)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 9e-87)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-53)
		tmp = (-0.5 * c) / b_2;
	elseif (b_2 <= 9e-87)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-53], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9e-87], 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[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.79999999999999985e-53

   1. Initial program 19.0%

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

   3. Taylor expanded in b_2 around -inf 82.9%

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

      1. associate-*r/ 82.9%

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

   5. Simplified 82.9%

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

   if -2.79999999999999985e-53 < b_2 < 8.99999999999999915e-87

   1. Initial program 76.8%

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

   3. Taylor expanded in b_2 around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. neg-mul-1 76.7%

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

   5. Simplified 76.7%

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

   if 8.99999999999999915e-87 < b_2

   1. Initial program 68.6%

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

   3. Taylor expanded in c around 0 88.7%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = ((-0.5d0) * c) / b_2
    else
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 37.2%

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

   3. Taylor expanded in b_2 around -inf 63.2%

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

      1. associate-*r/ 63.2%

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

   5. Simplified 63.2%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 69.8%

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

   3. Taylor expanded in c around 0 73.1%

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

Alternative 6: 68.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-305) (/ (* -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 <= -2.25e-305) {
		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 <= (-2.25d-305)) 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 <= -2.25e-305) {
		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 <= -2.25e-305:
		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 <= -2.25e-305)
		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 <= -2.25e-305)
		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, -2.25e-305], 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 < -2.2500000000000001e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

   5. Simplified 63.6%

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

   if -2.2500000000000001e-305 < b_2

   1. Initial program 70.0%

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

   3. Taylor expanded in b_2 around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

   5. Simplified 72.3%

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

Alternative 7: 68.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.15e-305) (/ (* -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 <= -2.15e-305) {
		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 <= (-2.15d-305)) 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 <= -2.15e-305) {
		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 <= -2.15e-305:
		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 <= -2.15e-305)
		tmp = Float64(Float64(-0.5 * 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 <= -2.15e-305)
		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, -2.15e-305], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.1500000000000001e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

   5. Simplified 63.6%

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

   if -2.1500000000000001e-305 < b_2

   1. Initial program 70.0%

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

   3. Taylor expanded in b_2 around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. associate-/l* 72.2%

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

      2. *-commutative 72.2%

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.15 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5}{\frac{b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-305) (/ -0.5 (/ b_2 c)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-305) {
		tmp = -0.5 / (b_2 / c);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.9d-305)) then
        tmp = (-0.5d0) / (b_2 / c)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-305) {
		tmp = -0.5 / (b_2 / c);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.9e-305:
		tmp = -0.5 / (b_2 / c)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.9e-305)
		tmp = Float64(-0.5 / Float64(b_2 / c));
	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 <= -1.9e-305)
		tmp = -0.5 / (b_2 / c);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.9e-305], N[(-0.5 / N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

   5. Simplified 63.6%

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

      1. add-cube-cbrt 62.5%

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

      2. pow3 62.5%

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

      3. associate-/l* 62.5%

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

   7. Applied egg-rr 62.5%

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

      1. rem-cube-cbrt 63.6%

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

      2. clear-num 63.4%

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

      3. un-div-inv 63.4%

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

   9. Applied egg-rr 63.4%

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

   if -1.9e-305 < b_2

   1. Initial program 70.0%

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

   3. Taylor expanded in b_2 around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. associate-/l* 72.2%

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

      2. *-commutative 72.2%

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;\frac{-0.5}{\frac{b\_2}{c}}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-305) (* c (/ -0.5 b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-305) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.9d-305)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-305) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.9e-305:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.9e-305)
		tmp = Float64(c * Float64(-0.5 / 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 <= -1.9e-305)
		tmp = c * (-0.5 / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.9e-305], N[(c * N[(-0.5 / 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 < -1.9e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

   5. Simplified 63.6%

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

   6. Taylor expanded in c around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*r/ 63.4%

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

   8. Simplified 63.4%

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

   if -1.9e-305 < b_2

   1. Initial program 70.0%

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

   3. Taylor expanded in b_2 around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. associate-/l* 72.2%

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

      2. *-commutative 72.2%

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

   7. Applied egg-rr 72.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.9 \cdot 10^{-305}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.9e-305) (* c (/ -0.5 b_2)) (/ b_2 (- a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-305) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.9d-305)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = b_2 / -a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.9e-305) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 / -a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.9e-305:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = b_2 / -a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.9e-305)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 / Float64(-a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.9e-305)
		tmp = c * (-0.5 / b_2);
	else
		tmp = b_2 / -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.9e-305], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 / (-a)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9e-305

   1. Initial program 36.7%

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

   3. Taylor expanded in b_2 around -inf 63.6%

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

      1. associate-*r/ 63.6%

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

   5. Simplified 63.6%

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

   6. Taylor expanded in c around 0 63.6%

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

      1. associate-*r/ 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-*r/ 63.4%

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

   8. Simplified 63.4%

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

   if -1.9e-305 < b_2

   1. Initial program 70.0%

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

   3. Taylor expanded in b_2 around 0 40.4%

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

      1. associate-*r* 40.4%

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

      2. neg-mul-1 40.4%

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

   5. Simplified 40.4%

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

   6. Taylor expanded in b_2 around inf 29.2%

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

      1. neg-mul-1 29.2%

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

      2. distribute-neg-frac2 29.2%

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

   8. Simplified 29.2%

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

Alternative 11: 15.8% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b_2 around 0 36.1%

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

   1. associate-*r* 36.1%

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

   2. neg-mul-1 36.1%

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

5. Simplified 36.1%

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

6. Taylor expanded in b_2 around inf 16.2%

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

   1. neg-mul-1 16.2%

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

   2. distribute-neg-frac2 16.2%

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

8. Simplified 16.2%

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

Alternative 12: 2.5% accurate, 37.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b_2 around 0 36.1%

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

   1. associate-*r* 36.1%

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

   2. neg-mul-1 36.1%

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

5. Simplified 36.1%

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

6. Taylor expanded in b_2 around inf 16.2%

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

   1. neg-mul-1 16.2%

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

   2. distribute-neg-frac2 16.2%

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

8. Simplified 16.2%

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

   1. neg-sub0 16.2%

      \[\leadsto \frac{b\_2}{\color{blue}{0 - a}} \]

   2. sub-neg 16.2%

      \[\leadsto \frac{b\_2}{\color{blue}{0 + \left(-a\right)}} \]

   3. add-sqr-sqrt 7.1%

      \[\leadsto \frac{b\_2}{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}} \]

   4. sqrt-unprod 10.3%

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

   5. sqr-neg 10.3%

      \[\leadsto \frac{b\_2}{0 + \sqrt{\color{blue}{a \cdot a}}} \]

   6. sqrt-unprod 1.4%

      \[\leadsto \frac{b\_2}{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

   7. add-sqr-sqrt 2.7%

      \[\leadsto \frac{b\_2}{0 + \color{blue}{a}} \]

10. Applied egg-rr 2.7%

    \[\leadsto \frac{b\_2}{\color{blue}{0 + a}} \]
11. Step-by-step derivation

    1. +-lft-identity 2.7%

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

12. Simplified 2.7%

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

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

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

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