quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.6% → 85.1%

Time: 10.5s

Alternatives: 8

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: 51.6% 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: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{+86}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.6e+86)
   (- (* (/ c b_2) (- -0.5)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 2.9e-104)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.6e+86) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 2.9e-104) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = (-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 <= (-2.6d+86)) then
        tmp = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 2.9d-104) then
        tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a
    else
        tmp = ((-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 <= -2.6e+86) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 2.9e-104) {
		tmp = (Math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.6e+86:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a))
	elif b_2 <= 2.9e-104:
		tmp = (math.sqrt(((b_2 * b_2) - (c * a))) - b_2) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.6e+86)
		tmp = Float64(Float64(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 2.9e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.6e+86)
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 2.9e-104)
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.6e+86], N[(N[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 2.9e-104], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.5999999999999998e86

   1. Initial program 47.9%

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

      1. +-commutative 47.9%

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

      2. unsub-neg 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in b_2 around -inf 92.7%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(-0.5 \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 92.9%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b\_2} + 2 \cdot \frac{b\_2}{a}\right)} \]

   if -2.5999999999999998e86 < b_2 < 2.9000000000000001e-104

   1. Initial program 83.6%

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

      1. +-commutative 83.6%

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

      2. unsub-neg 83.6%

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

   3. Simplified 83.6%

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

   if 2.9000000000000001e-104 < b_2

   1. Initial program 16.1%

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

      1. +-commutative 16.1%

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

      2. unsub-neg 16.1%

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

   3. Simplified 16.1%

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

   5. Taylor expanded in b_2 around inf 87.4%

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

      1. associate-*r/ 87.4%

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

      2. *-commutative 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.6 \cdot 10^{+86}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 2.9 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - c \cdot a} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.3e-51)
   (- (* (/ c b_2) (- -0.5)) (* 2.0 (/ b_2 a)))
   (if (<= b_2 7.5e-90) (/ (- (sqrt (* c (- a))) b_2) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.3e-51) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 7.5e-90) {
		tmp = (sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (-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 <= (-2.3d-51)) then
        tmp = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= 7.5d-90) then
        tmp = (sqrt((c * -a)) - b_2) / a
    else
        tmp = ((-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 <= -2.3e-51) {
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else if (b_2 <= 7.5e-90) {
		tmp = (Math.sqrt((c * -a)) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.3e-51:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a))
	elif b_2 <= 7.5e-90:
		tmp = (math.sqrt((c * -a)) - b_2) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.3e-51)
		tmp = Float64(Float64(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 7.5e-90)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.3e-51)
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	elseif (b_2 <= 7.5e-90)
		tmp = (sqrt((c * -a)) - b_2) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.3e-51], N[(N[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-90], N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.30000000000000002e-51

   1. Initial program 65.0%

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

      1. +-commutative 65.0%

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

      2. unsub-neg 65.0%

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

   3. Simplified 65.0%

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

   5. Taylor expanded in b_2 around -inf 86.3%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(-0.5 \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 86.4%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b\_2} + 2 \cdot \frac{b\_2}{a}\right)} \]

   if -2.30000000000000002e-51 < b_2 < 7.4999999999999999e-90

   1. Initial program 78.2%

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

      1. +-commutative 78.2%

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

      2. unsub-neg 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in b_2 around 0 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

      3. *-commutative 67.8%

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

   7. Simplified 67.8%

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

   if 7.4999999999999999e-90 < b_2

   1. Initial program 15.3%

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

      1. +-commutative 15.3%

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

      2. unsub-neg 15.3%

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

   3. Simplified 15.3%

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

   5. Taylor expanded in b_2 around inf 88.1%

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

      1. associate-*r/ 88.1%

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

      2. *-commutative 88.1%

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

   7. Simplified 88.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.3 \cdot 10^{-51}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.7% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (- (* (/ c b_2) (- -0.5)) (* 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 = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else {
		tmp = (-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 = ((c / b_2) * -(-0.5d0)) - (2.0d0 * (b_2 / a))
    else
        tmp = ((-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 = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a))
	else:
		tmp = (-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(Float64(c / b_2) * Float64(-(-0.5))) - Float64(2.0 * Float64(b_2 / a)));
	else
		tmp = Float64(Float64(-0.5 * 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 = ((c / b_2) * -(-0.5)) - (2.0 * (b_2 / a));
	else
		tmp = (-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[(N[(c / b$95$2), $MachinePrecision] * (--0.5)), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 72.1%

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

      1. +-commutative 72.1%

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

      2. unsub-neg 72.1%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in b_2 around -inf 61.6%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(-0.5 \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 62.7%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b\_2} + 2 \cdot \frac{b\_2}{a}\right)} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 29.1%

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

      1. +-commutative 29.1%

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

      2. unsub-neg 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b_2 around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. *-commutative 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b\_2} \cdot \left(--0.5\right) - 2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 43.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{+90}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3e+90) (* b_2 (/ -2.0 a)) (* (/ c b_2) 0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3e+90) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c / b_2) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 3d+90) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = (c / b_2) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 3e+90) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = (c / b_2) * 0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 3e+90:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = (c / b_2) * 0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 3e+90)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(Float64(c / b_2) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 3e+90)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = (c / b_2) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 3e+90], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.99999999999999979e90

   1. Initial program 61.6%

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

      1. +-commutative 61.6%

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

      2. unsub-neg 61.6%

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

   3. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 61.6%

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

      2. add-cube-cbrt 60.6%

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

      3. times-frac 60.6%

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

      4. pow2 60.6%

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

      5. sub-neg 60.6%

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

      6. add-sqr-sqrt 50.3%

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

      7. hypot-define 57.1%

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

      8. *-commutative 57.1%

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

      9. distribute-rgt-neg-in 57.1%

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

   6. Applied egg-rr 57.1%

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

      1. associate-*l/ 57.1%

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

      2. *-lft-identity 57.1%

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

   8. Simplified 57.1%

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

   9. Taylor expanded in b_2 around -inf 40.8%

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

       1. metadata-eval 40.8%

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

       2. distribute-lft-neg-in 40.8%

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

       3. associate-*r/ 40.8%

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

       4. associate-*l/ 40.7%

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

       5. *-commutative 40.7%

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

       6. distribute-rgt-neg-in 40.7%

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

       7. distribute-neg-frac 40.7%

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

       8. metadata-eval 40.7%

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

   11. Simplified 40.7%

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

   if 2.99999999999999979e90 < b_2

   1. Initial program 9.0%

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

      1. +-commutative 9.0%

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

      2. unsub-neg 9.0%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in b_2 around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 75.2%

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

      3. associate-*r* 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-*r/ 75.2%

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

      6. *-commutative 75.2%

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

   7. Simplified 75.2%

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

      1. frac-2neg 75.2%

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

      2. div-inv 75.1%

         \[\leadsto \color{blue}{\left(-a \cdot \frac{c \cdot -0.5}{b\_2}\right) \cdot \frac{1}{-a}} \]

      3. *-commutative 75.1%

         \[\leadsto \left(-a \cdot \frac{\color{blue}{-0.5 \cdot c}}{b\_2}\right) \cdot \frac{1}{-a} \]

      4. *-un-lft-identity 75.1%

         \[\leadsto \left(-a \cdot \frac{-0.5 \cdot c}{\color{blue}{1 \cdot b\_2}}\right) \cdot \frac{1}{-a} \]

      5. times-frac 75.1%

         \[\leadsto \left(-a \cdot \color{blue}{\left(\frac{-0.5}{1} \cdot \frac{c}{b\_2}\right)}\right) \cdot \frac{1}{-a} \]

      6. metadata-eval 75.1%

         \[\leadsto \left(-a \cdot \left(\color{blue}{-0.5} \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{-a} \]

      7. add-sqr-sqrt 37.9%

         \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}} \]

      8. sqrt-unprod 43.6%

         \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}} \]

      9. sqr-neg 43.6%

         \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{a \cdot a}}} \]

      10. sqrt-unprod 23.7%

          \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

      11. add-sqr-sqrt 37.4%

          \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{a}} \]

   9. Applied egg-rr 37.4%

      \[\leadsto \color{blue}{\left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{a}} \]
   10. Step-by-step derivation

       1. distribute-lft-neg-out 37.4%

          \[\leadsto \color{blue}{-\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{a}} \]

       2. associate-*r/ 37.4%

          \[\leadsto -\color{blue}{\frac{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot 1}{a}} \]

       3. *-rgt-identity 37.4%

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

       4. *-commutative 37.4%

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

       5. associate-*r/ 37.1%

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

       6. *-inverses 37.1%

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

       7. *-rgt-identity 37.1%

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

       8. distribute-lft-neg-in 37.1%

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

       9. metadata-eval 37.1%

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

   11. Simplified 37.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 3 \cdot 10^{+90}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b\_2} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.4e-308) (* b_2 (/ -2.0 a)) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.4e-308) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = c * (-0.5 / 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 <= 2.4d-308) then
        tmp = b_2 * ((-2.0d0) / a)
    else
        tmp = c * ((-0.5d0) / 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 <= 2.4e-308) {
		tmp = b_2 * (-2.0 / a);
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.4e-308:
		tmp = b_2 * (-2.0 / a)
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.4e-308)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.4e-308)
		tmp = b_2 * (-2.0 / a);
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.4e-308], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.40000000000000008e-308

   1. Initial program 72.3%

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

      1. +-commutative 72.3%

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

      2. unsub-neg 72.3%

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

   3. Simplified 72.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 72.3%

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

      2. add-cube-cbrt 71.1%

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

      3. times-frac 71.0%

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

      4. pow2 71.0%

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

      5. sub-neg 71.0%

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

      6. add-sqr-sqrt 55.8%

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

      7. hypot-define 66.3%

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

      8. *-commutative 66.3%

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

      9. distribute-rgt-neg-in 66.3%

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

   6. Applied egg-rr 66.3%

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

      1. associate-*l/ 66.4%

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

      2. *-lft-identity 66.4%

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

   8. Simplified 66.4%

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

   9. Taylor expanded in b_2 around -inf 61.6%

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

       1. metadata-eval 61.6%

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

       2. distribute-lft-neg-in 61.6%

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

       3. associate-*r/ 61.6%

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

       4. associate-*l/ 61.5%

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

       5. *-commutative 61.5%

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

       6. distribute-rgt-neg-in 61.5%

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

       7. distribute-neg-frac 61.5%

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

       8. metadata-eval 61.5%

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

   11. Simplified 61.5%

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

   if 2.40000000000000008e-308 < b_2

   1. Initial program 28.5%

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

      1. +-commutative 28.5%

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

      2. unsub-neg 28.5%

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

   3. Simplified 28.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 28.5%

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

      2. add-cube-cbrt 28.1%

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

      3. times-frac 28.1%

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

      4. pow2 28.1%

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

      5. sub-neg 28.1%

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

      6. add-sqr-sqrt 26.4%

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

      7. hypot-define 38.0%

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

      8. *-commutative 38.0%

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

      9. distribute-rgt-neg-in 38.0%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*l/ 38.0%

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

      2. *-lft-identity 38.0%

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

   8. Simplified 38.0%

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

   9. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]
   10. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2} \cdot 0.5} \]

       2. associate-/l* 0.0%

          \[\leadsto \color{blue}{\left(c \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{b\_2}\right)} \cdot 0.5 \]

       3. associate-*r* 0.0%

          \[\leadsto \color{blue}{c \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2}}{b\_2} \cdot 0.5\right)} \]

       4. *-commutative 0.0%

          \[\leadsto c \cdot \color{blue}{\left(0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{b\_2}\right)} \]

       5. associate-*r/ 0.0%

          \[\leadsto c \cdot \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]

       6. unpow2 0.0%

          \[\leadsto c \cdot \frac{0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{b\_2} \]

       7. rem-square-sqrt 69.5%

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

       8. metadata-eval 69.5%

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

   11. Simplified 69.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.4e-308) (/ (* b_2 -2.0) a) (* c (/ -0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.4e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c * (-0.5 / 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 <= 2.4d-308) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = c * ((-0.5d0) / 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 <= 2.4e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = c * (-0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.4e-308:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = c * (-0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.4e-308)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(c * Float64(-0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.4e-308)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = c * (-0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.4e-308], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.40000000000000008e-308

   1. Initial program 72.3%

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

      1. +-commutative 72.3%

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

      2. unsub-neg 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in b_2 around -inf 61.6%

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

      1. *-commutative 61.6%

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

   7. Simplified 61.6%

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

   if 2.40000000000000008e-308 < b_2

   1. Initial program 28.5%

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

      1. +-commutative 28.5%

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

      2. unsub-neg 28.5%

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

   3. Simplified 28.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-un-lft-identity 28.5%

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

      2. add-cube-cbrt 28.1%

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

      3. times-frac 28.1%

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

      4. pow2 28.1%

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

      5. sub-neg 28.1%

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

      6. add-sqr-sqrt 26.4%

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

      7. hypot-define 38.0%

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

      8. *-commutative 38.0%

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

      9. distribute-rgt-neg-in 38.0%

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

   6. Applied egg-rr 38.0%

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

      1. associate-*l/ 38.0%

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

      2. *-lft-identity 38.0%

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

   8. Simplified 38.0%

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

   9. Taylor expanded in b_2 around inf 0.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]
   10. Step-by-step derivation

       1. *-commutative 0.0%

          \[\leadsto \color{blue}{\frac{c \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2} \cdot 0.5} \]

       2. associate-/l* 0.0%

          \[\leadsto \color{blue}{\left(c \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{b\_2}\right)} \cdot 0.5 \]

       3. associate-*r* 0.0%

          \[\leadsto \color{blue}{c \cdot \left(\frac{{\left(\sqrt{-1}\right)}^{2}}{b\_2} \cdot 0.5\right)} \]

       4. *-commutative 0.0%

          \[\leadsto c \cdot \color{blue}{\left(0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{b\_2}\right)} \]

       5. associate-*r/ 0.0%

          \[\leadsto c \cdot \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{b\_2}} \]

       6. unpow2 0.0%

          \[\leadsto c \cdot \frac{0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{b\_2} \]

       7. rem-square-sqrt 69.5%

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

       8. metadata-eval 69.5%

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

   11. Simplified 69.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.4e-308) (/ (* b_2 -2.0) a) (/ (* -0.5 c) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.4e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-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 <= 2.4d-308) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = ((-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 <= 2.4e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.4e-308:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (-0.5 * c) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.4e-308)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.4e-308)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (-0.5 * c) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.4e-308], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.40000000000000008e-308

   1. Initial program 72.3%

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

      1. +-commutative 72.3%

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

      2. unsub-neg 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in b_2 around -inf 61.6%

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

      1. *-commutative 61.6%

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

   7. Simplified 61.6%

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

   if 2.40000000000000008e-308 < b_2

   1. Initial program 28.5%

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

      1. +-commutative 28.5%

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

      2. unsub-neg 28.5%

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

   3. Simplified 28.5%

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

   5. Taylor expanded in b_2 around inf 69.7%

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

      1. associate-*r/ 69.7%

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

      2. *-commutative 69.7%

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

   7. Simplified 69.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.4 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 11.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b\_2} \cdot 0.5 \end{array} \]

FPCore

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

C

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

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 = (c / b_2) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

   1. +-commutative 50.9%

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

   2. unsub-neg 50.9%

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

3. Simplified 50.9%

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

5. Taylor expanded in b_2 around inf 27.8%

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

   1. *-commutative 27.8%

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

   2. associate-/l* 29.3%

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

   3. associate-*r* 29.3%

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

   4. *-commutative 29.3%

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

   5. associate-*r/ 29.3%

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

   6. *-commutative 29.3%

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

7. Simplified 29.3%

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

   1. frac-2neg 29.3%

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

   2. div-inv 29.2%

      \[\leadsto \color{blue}{\left(-a \cdot \frac{c \cdot -0.5}{b\_2}\right) \cdot \frac{1}{-a}} \]

   3. *-commutative 29.2%

      \[\leadsto \left(-a \cdot \frac{\color{blue}{-0.5 \cdot c}}{b\_2}\right) \cdot \frac{1}{-a} \]

   4. *-un-lft-identity 29.2%

      \[\leadsto \left(-a \cdot \frac{-0.5 \cdot c}{\color{blue}{1 \cdot b\_2}}\right) \cdot \frac{1}{-a} \]

   5. times-frac 29.2%

      \[\leadsto \left(-a \cdot \color{blue}{\left(\frac{-0.5}{1} \cdot \frac{c}{b\_2}\right)}\right) \cdot \frac{1}{-a} \]

   6. metadata-eval 29.2%

      \[\leadsto \left(-a \cdot \left(\color{blue}{-0.5} \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{-a} \]

   7. add-sqr-sqrt 15.9%

      \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{-a} \cdot \sqrt{-a}}} \]

   8. sqrt-unprod 14.3%

      \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}} \]

   9. sqr-neg 14.3%

      \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\sqrt{\color{blue}{a \cdot a}}} \]

   10. sqrt-unprod 6.3%

       \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{\sqrt{a} \cdot \sqrt{a}}} \]

   11. add-sqr-sqrt 10.3%

       \[\leadsto \left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{\color{blue}{a}} \]

9. Applied egg-rr 10.3%

   \[\leadsto \color{blue}{\left(-a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{a}} \]
10. Step-by-step derivation

    1. distribute-lft-neg-out 10.3%

       \[\leadsto \color{blue}{-\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot \frac{1}{a}} \]

    2. associate-*r/ 10.3%

       \[\leadsto -\color{blue}{\frac{\left(a \cdot \left(-0.5 \cdot \frac{c}{b\_2}\right)\right) \cdot 1}{a}} \]

    3. *-rgt-identity 10.3%

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

    4. *-commutative 10.3%

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

    5. associate-*r/ 10.3%

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

    6. *-inverses 10.3%

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

    7. *-rgt-identity 10.3%

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

    8. distribute-lft-neg-in 10.3%

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

    9. metadata-eval 10.3%

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

11. Simplified 10.3%

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

12. Final simplification 10.3%

    \[\leadsto \frac{c}{b\_2} \cdot 0.5 \]
13. Add Preprocessing

Developer target: 99.7% accurate, 0.1× speedup

Math

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

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) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	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 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	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(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	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 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	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[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024076 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2) a) (/ (- c) (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))))))

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