quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.1% → 85.1%

Time: 12.5s

Alternatives: 6

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: 52.1% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -8.5 \cdot 10^{+69}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.5e+69)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 1.9e-56)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.5e+69) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 1.9e-56) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= (-8.5d+69)) then
        tmp = (b_2 / a) * (-2.0d0)
    else if (b_2 <= 1.9d-56) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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 <= -8.5e+69) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 1.9e-56) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -8.5e+69:
		tmp = (b_2 / a) * -2.0
	elif b_2 <= 1.9e-56:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - 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 <= -8.5e+69)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 1.9e-56)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -8.5e+69)
		tmp = (b_2 / a) * -2.0;
	elseif (b_2 <= 1.9e-56)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - 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, -8.5e+69], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 1.9e-56], 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[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.5000000000000002e69

   1. Initial program 59.6%

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

      1. +-commutative 59.6%

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

      2. unsub-neg 59.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in b_2 around -inf 95.2%

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

      1. *-commutative 95.2%

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

   7. Simplified 95.2%

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

   if -8.5000000000000002e69 < b_2 < 1.9000000000000001e-56

   1. Initial program 79.9%

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

      1. +-commutative 79.9%

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

      2. unsub-neg 79.9%

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

   3. Simplified 79.9%

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

   if 1.9000000000000001e-56 < b_2

   1. Initial program 16.3%

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

      1. +-commutative 16.3%

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

      2. unsub-neg 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in b_2 around inf 87.4%

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

Alternative 2: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-98}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.75e-98)
   (* (/ b_2 a) -2.0)
   (if (<= b_2 3e-59) (/ (- (sqrt (* a (- c))) b_2) a) (* -0.5 (/ c b_2)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.75e-98) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 3e-59) {
		tmp = (sqrt((a * -c)) - 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 <= (-2.75d-98)) then
        tmp = (b_2 / a) * (-2.0d0)
    else if (b_2 <= 3d-59) then
        tmp = (sqrt((a * -c)) - 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 <= -2.75e-98) {
		tmp = (b_2 / a) * -2.0;
	} else if (b_2 <= 3e-59) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.75e-98:
		tmp = (b_2 / a) * -2.0
	elif b_2 <= 3e-59:
		tmp = (math.sqrt((a * -c)) - 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 <= -2.75e-98)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	elseif (b_2 <= 3e-59)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.75e-98)
		tmp = (b_2 / a) * -2.0;
	elseif (b_2 <= 3e-59)
		tmp = (sqrt((a * -c)) - 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, -2.75e-98], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[b$95$2, 3e-59], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.7499999999999999e-98

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. unsub-neg 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in b_2 around -inf 90.0%

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

      1. *-commutative 90.0%

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

   7. Simplified 90.0%

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

   if -2.7499999999999999e-98 < b_2 < 3.0000000000000001e-59

   1. Initial program 73.7%

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

      1. +-commutative 73.7%

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

      2. unsub-neg 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in b_2 around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

      3. *-commutative 71.2%

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

   7. Simplified 71.2%

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

   if 3.0000000000000001e-59 < b_2

   1. Initial program 17.4%

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

      1. +-commutative 17.4%

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

      2. unsub-neg 17.4%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in b_2 around inf 86.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.75 \cdot 10^{-98}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{elif}\;b\_2 \leq 3 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.7 \cdot 10^{-294}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.7e-294) (* (/ b_2 a) -2.0) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 5.7e-294) {
		tmp = (b_2 / a) * -2.0;
	} 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 <= 5.7d-294) then
        tmp = (b_2 / a) * (-2.0d0)
    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 <= 5.7e-294) {
		tmp = (b_2 / a) * -2.0;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 5.7e-294:
		tmp = (b_2 / a) * -2.0
	else:
		tmp = -0.5 * (c / b_2)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 5.7e-294)
		tmp = Float64(Float64(b_2 / a) * -2.0);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 5.7e-294)
		tmp = (b_2 / a) * -2.0;
	else
		tmp = -0.5 * (c / b_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 5.7e-294], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.70000000000000032e-294

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

      2. unsub-neg 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in b_2 around -inf 63.2%

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

      1. *-commutative 63.2%

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

   7. Simplified 63.2%

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

   if 5.70000000000000032e-294 < b_2

   1. Initial program 37.7%

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

      1. +-commutative 37.7%

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

      2. unsub-neg 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in b_2 around inf 61.2%

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

Alternative 4: 47.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2 \cdot 10^{-293}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2e-293) (/ b_2 (- a)) (* -0.5 (/ c b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2e-293) {
		tmp = 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 <= 2d-293) then
        tmp = 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 <= 2e-293) {
		tmp = b_2 / -a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2e-293:
		tmp = 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 <= 2e-293)
		tmp = Float64(b_2 / Float64(-a));
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2e-293)
		tmp = 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, 2e-293], N[(b$95$2 / (-a)), $MachinePrecision], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.0000000000000001e-293

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

      2. unsub-neg 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in b_2 around 0 47.2%

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

      1. associate-*r* 47.2%

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

      2. neg-mul-1 47.2%

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

      3. *-commutative 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in b_2 around inf 27.4%

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

      1. mul-1-neg 27.4%

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

   10. Simplified 27.4%

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

   if 2.0000000000000001e-293 < b_2

   1. Initial program 37.7%

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

      1. +-commutative 37.7%

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

      2. unsub-neg 37.7%

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

   3. Simplified 37.7%

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

   5. Taylor expanded in b_2 around inf 61.2%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 2 \cdot 10^{-293}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 23.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (if (<= b_2 1.4e-24) (/ b_2 (- a)) 0.0))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-24) {
		tmp = b_2 / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 1.4d-24) then
        tmp = b_2 / -a
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-24) {
		tmp = b_2 / -a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.4e-24:
		tmp = b_2 / -a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.4e-24)
		tmp = Float64(b_2 / Float64(-a));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.4e-24)
		tmp = b_2 / -a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.4e-24], N[(b$95$2 / (-a)), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.4000000000000001e-24

   1. Initial program 70.8%

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

      1. +-commutative 70.8%

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

      2. unsub-neg 70.8%

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

   3. Simplified 70.8%

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

   5. Taylor expanded in b_2 around 0 50.7%

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

      1. associate-*r* 50.7%

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

      2. neg-mul-1 50.7%

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

      3. *-commutative 50.7%

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

   7. Simplified 50.7%

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

   8. Taylor expanded in b_2 around inf 20.0%

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

      1. mul-1-neg 20.0%

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

   10. Simplified 20.0%

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

   if 1.4000000000000001e-24 < b_2

   1. Initial program 15.9%

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

      1. +-commutative 15.9%

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

      2. unsub-neg 15.9%

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

   3. Simplified 15.9%

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

      1. div-sub 15.1%

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

      2. add-sqr-sqrt 9.4%

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

      3. associate-/l* 11.0%

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

      4. fma-neg 9.5%

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

      5. pow1/2 9.5%

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

      6. sqrt-pow1 9.6%

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

      7. pow2 9.6%

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

      8. metadata-eval 9.6%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}, \frac{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a}, -\frac{b\_2}{a}\right) \]

      9. pow1/2 9.6%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}}{a}, -\frac{b\_2}{a}\right) \]

      10. sqrt-pow1 9.5%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}}{a}, -\frac{b\_2}{a}\right) \]

      11. pow2 9.5%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}{a}, -\frac{b\_2}{a}\right) \]

      12. metadata-eval 9.5%

          \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}}{a}, -\frac{b\_2}{a}\right) \]

   6. Applied egg-rr 9.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 9.5%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - \color{blue}{c \cdot a}\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right) \]

      2. *-commutative 9.5%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - c \cdot a\right)}^{0.25}, \frac{{\left({b\_2}^{2} - \color{blue}{c \cdot a}\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right) \]

      3. distribute-neg-frac 9.5%

         \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - c \cdot a\right)}^{0.25}, \frac{{\left({b\_2}^{2} - c \cdot a\right)}^{0.25}}{a}, \color{blue}{\frac{-b\_2}{a}}\right) \]

   8. Simplified 9.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - c \cdot a\right)}^{0.25}, \frac{{\left({b\_2}^{2} - c \cdot a\right)}^{0.25}}{a}, \frac{-b\_2}{a}\right)} \]

   9. Taylor expanded in c around 0 15.6%

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

       1. distribute-lft1-in 15.6%

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

       2. metadata-eval 15.6%

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

       3. mul0-lft 25.3%

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

   11. Simplified 25.3%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 11.3% accurate, 112.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. +-commutative 56.0%

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

   2. unsub-neg 56.0%

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

3. Simplified 56.0%

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

   1. div-sub 55.8%

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

   2. add-sqr-sqrt 54.1%

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

   3. associate-/l* 54.5%

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

   4. fma-neg 54.3%

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

   5. pow1/2 54.3%

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

   6. sqrt-pow1 54.3%

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

   7. pow2 54.3%

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

   8. metadata-eval 54.3%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}, \frac{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a}, -\frac{b\_2}{a}\right) \]

   9. pow1/2 54.3%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}}{a}, -\frac{b\_2}{a}\right) \]

   10. sqrt-pow1 54.3%

       \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}}{a}, -\frac{b\_2}{a}\right) \]

   11. pow2 54.3%

       \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}}{a}, -\frac{b\_2}{a}\right) \]

   12. metadata-eval 54.3%

       \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}}{a}, -\frac{b\_2}{a}\right) \]

6. Applied egg-rr 54.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right)} \]
7. Step-by-step derivation

   1. *-commutative 54.3%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - \color{blue}{c \cdot a}\right)}^{0.25}, \frac{{\left({b\_2}^{2} - a \cdot c\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right) \]

   2. *-commutative 54.3%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - c \cdot a\right)}^{0.25}, \frac{{\left({b\_2}^{2} - \color{blue}{c \cdot a}\right)}^{0.25}}{a}, -\frac{b\_2}{a}\right) \]

   3. distribute-neg-frac 54.3%

      \[\leadsto \mathsf{fma}\left({\left({b\_2}^{2} - c \cdot a\right)}^{0.25}, \frac{{\left({b\_2}^{2} - c \cdot a\right)}^{0.25}}{a}, \color{blue}{\frac{-b\_2}{a}}\right) \]

8. Simplified 54.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({b\_2}^{2} - c \cdot a\right)}^{0.25}, \frac{{\left({b\_2}^{2} - c \cdot a\right)}^{0.25}}{a}, \frac{-b\_2}{a}\right)} \]

9. Taylor expanded in c around 0 6.3%

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

    1. distribute-lft1-in 6.3%

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

    2. metadata-eval 6.3%

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

    3. mul0-lft 9.0%

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

11. Simplified 9.0%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Developer Target 1: 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 2024141 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

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

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