quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.9% → 85.7%

Time: 12.9s

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: 52.9% 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: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+118)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.35e-55)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+118) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.35e-55) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / 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 <= (-2d+118)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.35d-55) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / 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 <= -2e+118) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.35e-55) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e+118:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.35e-55:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+118)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.35e-55)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e+118)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.35e-55)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / 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, -2e+118], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-55], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.99999999999999993e118

   1. Initial program 48.5%

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

      1. +-commutative 48.5%

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

      2. unsub-neg 48.5%

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

   3. Simplified 48.5%

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

   5. Taylor expanded in b_2 around -inf 96.9%

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

      1. *-commutative 96.9%

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

   7. Simplified 96.9%

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

   if -1.99999999999999993e118 < b_2 < 1.35000000000000002e-55

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. unsub-neg 82.7%

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

   3. Simplified 82.7%

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

   if 1.35000000000000002e-55 < b_2

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      2. unsub-neg 23.6%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in b_2 around inf 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l/ 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -3.8e-126)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 9.8e-55) (/ (- (sqrt (* a (- c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -3.8e-126) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9.8e-55) {
		tmp = (sqrt((a * -c)) - b_2) / 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 <= (-3.8d-126)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 9.8d-55) then
        tmp = (sqrt((a * -c)) - b_2) / 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 <= -3.8e-126) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 9.8e-55) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -3.8e-126:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 9.8e-55:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -3.8e-126)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 9.8e-55)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -3.8e-126)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 9.8e-55)
		tmp = (sqrt((a * -c)) - b_2) / 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, -3.8e-126], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 9.8e-55], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -3.7999999999999999e-126

   1. Initial program 65.1%

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

      1. +-commutative 65.1%

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

      2. unsub-neg 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in b_2 around -inf 88.9%

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

   if -3.7999999999999999e-126 < b_2 < 9.80000000000000071e-55

   1. Initial program 79.1%

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

      1. +-commutative 79.1%

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

      2. unsub-neg 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in b_2 around 0 75.2%

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

      1. associate-*r* 75.2%

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

      2. neg-mul-1 75.2%

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

   7. Simplified 75.2%

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

   if 9.80000000000000071e-55 < b_2

   1. Initial program 23.6%

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

      1. +-commutative 23.6%

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

      2. unsub-neg 23.6%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in b_2 around inf 84.0%

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

      1. *-commutative 84.0%

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

      2. associate-*l/ 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -3.8 \cdot 10^{-126}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a} + 0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. unsub-neg 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in b_2 around -inf 76.3%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. unsub-neg 41.7%

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

   3. Simplified 41.7%

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

   5. Taylor expanded in b_2 around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l/ 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 68.8%

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

Alternative 4: 22.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.000135:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 0.000135) (/ (- b_2) a) (* c (/ 0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 0.000135) {
		tmp = -b_2 / 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 <= 0.000135d0) then
        tmp = -b_2 / 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 <= 0.000135) {
		tmp = -b_2 / a;
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 0.000135:
		tmp = -b_2 / a
	else:
		tmp = c * (0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 0.000135)
		tmp = Float64(Float64(-b_2) / 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 <= 0.000135)
		tmp = -b_2 / 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, 0.000135], N[((-b$95$2) / a), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.35000000000000002e-4

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. unsub-neg 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b_2 around 0 42.1%

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

      1. associate-*r* 42.1%

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

      2. neg-mul-1 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in a around 0 23.1%

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

      1. associate-*r/ 23.1%

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

      2. neg-mul-1 23.1%

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

   10. Simplified 23.1%

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

   if 1.35000000000000002e-4 < b_2

   1. Initial program 19.4%

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

      1. +-commutative 19.4%

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

      2. unsub-neg 19.4%

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

   3. Simplified 19.4%

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

      1. flip3-- 3.6%

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

      2. div-inv 3.6%

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

      3. pow2 3.6%

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

      4. pow-pow 3.6%

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

      5. metadata-eval 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{\color{blue}{6}} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      6. pow2 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{2}} \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      7. pow2 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{2} \cdot \color{blue}{{b\_2}^{2}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      8. pow-prod-up 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{\left(2 + 2\right)}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      9. metadata-eval 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{\color{blue}{4}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      10. distribute-rgt-out 3.6%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c + b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

      11. add-sqr-sqrt 0.6%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      12. sqrt-unprod 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\sqrt{c \cdot c}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      13. sqr-neg 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \sqrt{\color{blue}{\left(-c\right) \cdot \left(-c\right)}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      14. sqrt-unprod 1.7%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{-c} \cdot \sqrt{-c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      15. add-sqr-sqrt 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(-c\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      16. fma-udef 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

   6. Applied egg-rr 3.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \mathsf{fma}\left(a, c, {b\_2}^{2}\right)}}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 2.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2 + 0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   8. Step-by-step derivation

      1. +-commutative 2.5%

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

      2. *-commutative 2.5%

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

      3. *-commutative 2.5%

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

      4. associate-*r/ 2.9%

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

      5. associate-*l* 2.9%

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

      6. fma-def 2.9%

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

      7. *-commutative 2.9%

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

   9. Simplified 2.9%

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

   10. Taylor expanded in c around inf 27.9%

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

       1. associate-*r/ 27.9%

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

       2. *-commutative 27.9%

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

       3. *-lft-identity 27.9%

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

       4. rgt-mult-inverse 27.9%

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

       5. associate-*r/ 27.9%

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

       6. *-rgt-identity 27.9%

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

       7. associate-/r/ 28.1%

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

       8. associate-/l* 28.3%

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

       9. associate-*r/ 28.1%

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

       10. associate-*r/ 28.1%

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

       11. associate-*r/ 28.3%

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

       12. associate-/l* 28.1%

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

       13. associate-/r/ 27.9%

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

       14. *-rgt-identity 27.9%

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

       15. associate-*r/ 27.9%

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

       16. rgt-mult-inverse 27.9%

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

       17. *-lft-identity 27.9%

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

   12. Simplified 27.9%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.000135:\\ \;\;\;\;\frac{-b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.9% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.00195:\\ \;\;\;\;\frac{-2}{\frac{a}{b\_2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 0.00195) (/ -2.0 (/ a b_2)) (* c (/ 0.5 b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 0.00195) {
		tmp = -2.0 / (a / b_2);
	} 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 <= 0.00195d0) then
        tmp = (-2.0d0) / (a / b_2)
    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 <= 0.00195) {
		tmp = -2.0 / (a / b_2);
	} else {
		tmp = c * (0.5 / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 0.00195:
		tmp = -2.0 / (a / b_2)
	else:
		tmp = c * (0.5 / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 0.00195)
		tmp = Float64(-2.0 / Float64(a / b_2));
	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 <= 0.00195)
		tmp = -2.0 / (a / b_2);
	else
		tmp = c * (0.5 / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 0.00195], N[(-2.0 / N[(a / b$95$2), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 0.0019499999999999999

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. unsub-neg 69.3%

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

   3. Simplified 69.3%

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

      1. flip3-- 23.8%

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

      2. div-inv 23.8%

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

      3. pow2 23.8%

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

      4. pow-pow 23.8%

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

      5. metadata-eval 23.8%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{\color{blue}{6}} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      6. pow2 23.8%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{2}} \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      7. pow2 23.8%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{2} \cdot \color{blue}{{b\_2}^{2}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      8. pow-prod-up 23.8%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{\left(2 + 2\right)}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      9. metadata-eval 23.8%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{\color{blue}{4}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      10. distribute-rgt-out 23.8%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c + b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

      11. add-sqr-sqrt 16.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      12. sqrt-unprod 14.4%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\sqrt{c \cdot c}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      13. sqr-neg 14.4%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \sqrt{\color{blue}{\left(-c\right) \cdot \left(-c\right)}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      14. sqrt-unprod 2.0%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{-c} \cdot \sqrt{-c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      15. add-sqr-sqrt 7.5%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(-c\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      16. fma-udef 7.5%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

   6. Applied egg-rr 23.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \mathsf{fma}\left(a, c, {b\_2}^{2}\right)}}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 52.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2 + 0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   8. Step-by-step derivation

      1. +-commutative 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. associate-*r/ 54.3%

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

      5. associate-*l* 54.3%

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

      6. fma-def 54.3%

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

      7. *-commutative 54.3%

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

   9. Simplified 54.3%

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

   10. Taylor expanded in c around 0 54.1%

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

       1. associate-*r/ 54.1%

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

       2. associate-/l* 53.9%

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

   12. Simplified 53.9%

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

   if 0.0019499999999999999 < b_2

   1. Initial program 19.4%

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

      1. +-commutative 19.4%

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

      2. unsub-neg 19.4%

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

   3. Simplified 19.4%

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

      1. flip3-- 3.6%

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

      2. div-inv 3.6%

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

      3. pow2 3.6%

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

      4. pow-pow 3.6%

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

      5. metadata-eval 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{\color{blue}{6}} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      6. pow2 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{2}} \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      7. pow2 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{2} \cdot \color{blue}{{b\_2}^{2}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      8. pow-prod-up 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{\left(2 + 2\right)}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      9. metadata-eval 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{\color{blue}{4}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      10. distribute-rgt-out 3.6%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c + b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

      11. add-sqr-sqrt 0.6%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      12. sqrt-unprod 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\sqrt{c \cdot c}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      13. sqr-neg 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \sqrt{\color{blue}{\left(-c\right) \cdot \left(-c\right)}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      14. sqrt-unprod 1.7%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{-c} \cdot \sqrt{-c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      15. add-sqr-sqrt 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(-c\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      16. fma-udef 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

   6. Applied egg-rr 3.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \mathsf{fma}\left(a, c, {b\_2}^{2}\right)}}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 2.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2 + 0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   8. Step-by-step derivation

      1. +-commutative 2.5%

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

      2. *-commutative 2.5%

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

      3. *-commutative 2.5%

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

      4. associate-*r/ 2.9%

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

      5. associate-*l* 2.9%

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

      6. fma-def 2.9%

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

      7. *-commutative 2.9%

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

   9. Simplified 2.9%

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

   10. Taylor expanded in c around inf 27.9%

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

       1. associate-*r/ 27.9%

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

       2. *-commutative 27.9%

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

       3. *-lft-identity 27.9%

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

       4. rgt-mult-inverse 27.9%

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

       5. associate-*r/ 27.9%

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

       6. *-rgt-identity 27.9%

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

       7. associate-/r/ 28.1%

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

       8. associate-/l* 28.3%

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

       9. associate-*r/ 28.1%

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

       10. associate-*r/ 28.1%

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

       11. associate-*r/ 28.3%

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

       12. associate-/l* 28.1%

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

       13. associate-/r/ 27.9%

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

       14. *-rgt-identity 27.9%

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

       15. associate-*r/ 27.9%

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

       16. rgt-mult-inverse 27.9%

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

       17. *-lft-identity 27.9%

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

   12. Simplified 27.9%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.00195:\\ \;\;\;\;\frac{-2}{\frac{a}{b\_2}}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.0015:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 0.0015) (/ (* 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 <= 0.0015) {
		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 <= 0.0015d0) 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 <= 0.0015) {
		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 <= 0.0015:
		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 <= 0.0015)
		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 <= 0.0015)
		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, 0.0015], 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 < 0.0015

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. unsub-neg 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b_2 around -inf 54.1%

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

      1. *-commutative 54.1%

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

   7. Simplified 54.1%

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

   if 0.0015 < b_2

   1. Initial program 19.4%

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

      1. +-commutative 19.4%

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

      2. unsub-neg 19.4%

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

   3. Simplified 19.4%

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

      1. flip3-- 3.6%

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

      2. div-inv 3.6%

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

      3. pow2 3.6%

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

      4. pow-pow 3.6%

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

      5. metadata-eval 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{\color{blue}{6}} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      6. pow2 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{2}} \cdot \left(b\_2 \cdot b\_2\right) + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      7. pow2 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{2} \cdot \color{blue}{{b\_2}^{2}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      8. pow-prod-up 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b\_2}^{\left(2 + 2\right)}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      9. metadata-eval 3.6%

         \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{\color{blue}{4}} + \left(\left(a \cdot c\right) \cdot \left(a \cdot c\right) + \left(b\_2 \cdot b\_2\right) \cdot \left(a \cdot c\right)\right)}} - b\_2}{a} \]

      10. distribute-rgt-out 3.6%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \color{blue}{\left(a \cdot c\right) \cdot \left(a \cdot c + b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

      11. add-sqr-sqrt 0.6%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{c} \cdot \sqrt{c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      12. sqrt-unprod 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\sqrt{c \cdot c}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      13. sqr-neg 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \sqrt{\color{blue}{\left(-c\right) \cdot \left(-c\right)}} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      14. sqrt-unprod 1.7%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(\sqrt{-c} \cdot \sqrt{-c}\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      15. add-sqr-sqrt 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \left(a \cdot \color{blue}{\left(-c\right)} + b\_2 \cdot b\_2\right)}} - b\_2}{a} \]

      16. fma-udef 2.3%

          \[\leadsto \frac{\sqrt{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left(a, -c, b\_2 \cdot b\_2\right)}}} - b\_2}{a} \]

   6. Applied egg-rr 3.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left({b\_2}^{6} - {\left(a \cdot c\right)}^{3}\right) \cdot \frac{1}{{b\_2}^{4} + \left(a \cdot c\right) \cdot \mathsf{fma}\left(a, c, {b\_2}^{2}\right)}}} - b\_2}{a} \]

   7. Taylor expanded in b_2 around -inf 2.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2 + 0.5 \cdot \frac{a \cdot c}{b\_2}}}{a} \]
   8. Step-by-step derivation

      1. +-commutative 2.5%

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

      2. *-commutative 2.5%

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

      3. *-commutative 2.5%

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

      4. associate-*r/ 2.9%

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

      5. associate-*l* 2.9%

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

      6. fma-def 2.9%

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

      7. *-commutative 2.9%

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

   9. Simplified 2.9%

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

   10. Taylor expanded in c around inf 27.9%

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

       1. associate-*r/ 27.9%

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

       2. *-commutative 27.9%

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

       3. *-lft-identity 27.9%

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

       4. rgt-mult-inverse 27.9%

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

       5. associate-*r/ 27.9%

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

       6. *-rgt-identity 27.9%

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

       7. associate-/r/ 28.1%

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

       8. associate-/l* 28.3%

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

       9. associate-*r/ 28.1%

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

       10. associate-*r/ 28.1%

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

       11. associate-*r/ 28.3%

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

       12. associate-/l* 28.1%

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

       13. associate-/r/ 27.9%

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

       14. *-rgt-identity 27.9%

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

       15. associate-*r/ 27.9%

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

       16. rgt-mult-inverse 27.9%

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

       17. *-lft-identity 27.9%

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

   12. Simplified 27.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 0.0015:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 3.9 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 3.9e-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 <= 3.9e-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 <= 3.9d-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 <= 3.9e-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 <= 3.9e-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 <= 3.9e-308)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 3.9e-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, 3.9e-308], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 3.8999999999999999e-308

   1. Initial program 67.4%

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

      1. +-commutative 67.4%

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

      2. unsub-neg 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in b_2 around -inf 75.4%

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

      1. *-commutative 75.4%

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

   7. Simplified 75.4%

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

   if 3.8999999999999999e-308 < b_2

   1. Initial program 41.7%

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

      1. +-commutative 41.7%

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

      2. unsub-neg 41.7%

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

   3. Simplified 41.7%

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

   5. Taylor expanded in b_2 around inf 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l/ 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 3.9 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 15.1% 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(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 54.5%

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

   1. +-commutative 54.5%

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

   2. unsub-neg 54.5%

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

3. Simplified 54.5%

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

5. Taylor expanded in b_2 around 0 32.2%

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

   1. associate-*r* 32.2%

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

   2. neg-mul-1 32.2%

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

7. Simplified 32.2%

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

8. Taylor expanded in a around 0 17.1%

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

   1. associate-*r/ 17.1%

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

   2. neg-mul-1 17.1%

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

10. Simplified 17.1%

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

11. Final simplification 17.1%

    \[\leadsto \frac{-b\_2}{a} \]
12. 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 2024040 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :herbie-target
  (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))