quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.4% → 85.2%

Time: 12.5s

Alternatives: 7

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: 52.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.8e+73)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.3e-36)
     (/ (- (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 <= -1.8e+73) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.3e-36) {
		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 <= (-1.8d+73)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.3d-36) 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 <= -1.8e+73) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.3e-36) {
		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 <= -1.8e+73:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.3e-36:
		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 <= -1.8e+73)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.3e-36)
		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 <= -1.8e+73)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.3e-36)
		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, -1.8e+73], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.3e-36], 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 < -1.7999999999999999e73

   1. Initial program 57.9%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 57.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 57.9%

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

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), \color{blue}{a}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      4. *-lowering-*.f64 97.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   7. Simplified 97.0%

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

   if -1.7999999999999999e73 < b_2 < 1.3e-36

   1. Initial program 79.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 79.6%

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

   if 1.3e-36 < b_2

   1. Initial program 18.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 18.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 18.8%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   7. Simplified 91.8%

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

Alternative 2: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(0 - a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.8e-44)
   (+ (/ (* b_2 -2.0) a) (/ (* c 0.5) b_2))
   (if (<= b_2 8.5e-37)
     (/ (- (sqrt (* c (- 0.0 a))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.8e-44) {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	} else if (b_2 <= 8.5e-37) {
		tmp = (sqrt((c * (0.0 - a))) - 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 <= (-4.8d-44)) then
        tmp = ((b_2 * (-2.0d0)) / a) + ((c * 0.5d0) / b_2)
    else if (b_2 <= 8.5d-37) then
        tmp = (sqrt((c * (0.0d0 - a))) - 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 <= -4.8e-44) {
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	} else if (b_2 <= 8.5e-37) {
		tmp = (Math.sqrt((c * (0.0 - a))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.8e-44:
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2)
	elif b_2 <= 8.5e-37:
		tmp = (math.sqrt((c * (0.0 - a))) - 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 <= -4.8e-44)
		tmp = Float64(Float64(Float64(b_2 * -2.0) / a) + Float64(Float64(c * 0.5) / b_2));
	elseif (b_2 <= 8.5e-37)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(0.0 - a))) - 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 <= -4.8e-44)
		tmp = ((b_2 * -2.0) / a) + ((c * 0.5) / b_2);
	elseif (b_2 <= 8.5e-37)
		tmp = (sqrt((c * (0.0 - a))) - 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, -4.8e-44], N[(N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.5e-37], N[(N[(N[Sqrt[N[(c * N[(0.0 - a), $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 < -4.80000000000000017e-44

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 70.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   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 -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}, a\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right), a\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{b\_2 \cdot b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\frac{a \cdot c}{b\_2}}{b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b\_2}\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(0 - b\_2\right)\right), a\right) \]

      14. --lowering--.f64 88.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right), a\right) \]

   7. Simplified 88.2%

      \[\leadsto \frac{\color{blue}{\left(2 + -0.5 \cdot \frac{\frac{c \cdot a}{b\_2}}{b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

   8. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), a\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \mathsf{/.f64}\left(\left(c \cdot \frac{1}{2}\right), b\_2\right)\right) \]

      9. *-lowering-*.f64 89.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\_2\right)\right) \]

   10. Simplified 89.4%

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

   if -4.80000000000000017e-44 < b_2 < 8.5000000000000007e-37

   1. Initial program 73.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 73.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 73.0%

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

   5. Taylor expanded in b_2 around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot c\right)\right)}\right), b\_2\right), a\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(0 - a \cdot c\right)\right), b\_2\right), a\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(c \cdot a\right)\right)\right), b\_2\right), a\right) \]

      5. *-lowering-*.f64 70.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(c, a\right)\right)\right), b\_2\right), a\right) \]

   7. Simplified 70.1%

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

   if 8.5000000000000007e-37 < b_2

   1. Initial program 18.8%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 18.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 18.8%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 91.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   7. Simplified 91.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a} + \frac{c \cdot 0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(0 - a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.7% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (+ (/ (* b_2 -2.0) a) (/ (* c 0.5) 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 = ((b_2 * -2.0) / a) + ((c * 0.5) / 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 = ((b_2 * (-2.0d0)) / a) + ((c * 0.5d0) / 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 = ((b_2 * -2.0) / a) + ((c * 0.5) / 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 = ((b_2 * -2.0) / a) + ((c * 0.5) / 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(Float64(b_2 * -2.0) / a) + Float64(Float64(c * 0.5) / 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 = ((b_2 * -2.0) / a) + ((c * 0.5) / 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[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision] + N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $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 74.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 74.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 74.6%

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

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}, a\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right), a\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{b\_2 \cdot b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\frac{a \cdot c}{b\_2}}{b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b\_2}\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(0 - b\_2\right)\right), a\right) \]

      14. --lowering--.f64 69.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right), a\right) \]

   7. Simplified 69.7%

      \[\leadsto \frac{\color{blue}{\left(2 + -0.5 \cdot \frac{\frac{c \cdot a}{b\_2}}{b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

   8. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(-2 \cdot \frac{b\_2}{a}\right), \color{blue}{\left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-2 \cdot b\_2}{a}\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), a\right), \left(\color{blue}{\frac{1}{2}} \cdot \frac{c}{b\_2}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \left(\frac{1}{2} \cdot \frac{c}{b\_2}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \left(\frac{\frac{1}{2} \cdot c}{\color{blue}{b\_2}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot c\right), \color{blue}{b\_2}\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \mathsf{/.f64}\left(\left(c \cdot \frac{1}{2}\right), b\_2\right)\right) \]

      9. *-lowering-*.f64 70.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{1}{2}\right), b\_2\right)\right) \]

   10. Simplified 70.7%

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

   if -4.999999999999985e-310 < b_2

   1. Initial program 38.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 38.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 38.1%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 64.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   7. Simplified 64.1%

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

Alternative 4: 66.5% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.56 \cdot 10^{-304}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.56e-304) (/ (* 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 <= 1.56e-304) {
		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 <= 1.56d-304) 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 <= 1.56e-304) {
		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 <= 1.56e-304:
		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 <= 1.56e-304)
		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 <= 1.56e-304)
		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, 1.56e-304], 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 < 1.5600000000000001e-304

   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   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

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), \color{blue}{a}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      4. *-lowering-*.f64 70.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   7. Simplified 70.1%

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

   if 1.5600000000000001e-304 < b_2

   1. Initial program 38.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 38.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 38.3%

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

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot c}{\color{blue}{b\_2}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot c\right), \color{blue}{b\_2}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(c \cdot \frac{-1}{2}\right), b\_2\right) \]

      4. *-lowering-*.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, \frac{-1}{2}\right), b\_2\right) \]

   7. Simplified 64.6%

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

Alternative 5: 42.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.5e+72) (/ (* 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 <= 5.5e+72) {
		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 <= 5.5d+72) 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 <= 5.5e+72) {
		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 <= 5.5e+72:
		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 <= 5.5e+72)
		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 <= 5.5e+72)
		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, 5.5e+72], 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 < 5.5e72

   1. Initial program 66.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 66.2%

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

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), \color{blue}{a}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      4. *-lowering-*.f64 45.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   7. Simplified 45.0%

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

   if 5.5e72 < b_2

   1. Initial program 19.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 19.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 19.3%

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

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}, a\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right), a\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{b\_2 \cdot b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\frac{a \cdot c}{b\_2}}{b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b\_2}\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(0 - b\_2\right)\right), a\right) \]

      14. --lowering--.f64 2.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right), a\right) \]

   7. Simplified 2.4%

      \[\leadsto \frac{\color{blue}{\left(2 + -0.5 \cdot \frac{\frac{c \cdot a}{b\_2}}{b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

   8. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \frac{b\_2}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(-2 \cdot \frac{b\_2}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b\_2}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{1}{2} \cdot \frac{1}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a \cdot c}}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \color{blue}{\left(-2 \cdot \frac{b\_2}{a \cdot c}\right)}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{b\_2}\right), \left(\color{blue}{-2} \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{b\_2}\right), \left(-2 \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \left(\color{blue}{-2} \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{b\_2}{a \cdot c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \color{blue}{\left(a \cdot c\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \left(c \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \mathsf{*.f64}\left(c, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 2.4%

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

   11. Taylor expanded in b_2 around 0

       \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{1}{2}}{b\_2}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 34.4%

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b\_2}\right)\right) \]

   13. Simplified 34.4%

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

Alternative 6: 42.0% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 5.5e+72) (* 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 <= 5.5e+72) {
		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 <= 5.5d+72) 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 <= 5.5e+72) {
		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 <= 5.5e+72:
		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 <= 5.5e+72)
		tmp = Float64(b_2 * Float64(-2.0 / a));
	else
		tmp = Float64(c * Float64(0.5 / b_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 5.5e+72)
		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, 5.5e+72], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 5.5e72

   1. Initial program 66.2%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 66.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 66.2%

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

   5. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot b\_2\right), \color{blue}{a}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b\_2 \cdot -2\right), a\right) \]

      4. *-lowering-*.f64 45.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(b\_2, -2\right), a\right) \]

   7. Simplified 45.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-2}{a}\right), \color{blue}{b\_2}\right) \]

      4. /-lowering-/.f64 44.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, a\right), b\_2\right) \]

   9. Applied egg-rr 44.9%

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

   if 5.5e72 < b_2

   1. Initial program 19.3%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

      8. *-lowering-*.f64 19.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

   3. Simplified 19.3%

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

   5. Taylor expanded in b_2 around -inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}, a\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), a\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right), a\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{b\_2 \cdot b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\frac{a \cdot c}{b\_2}}{b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b\_2}\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(0 - b\_2\right)\right), a\right) \]

      14. --lowering--.f64 2.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right), a\right) \]

   7. Simplified 2.4%

      \[\leadsto \frac{\color{blue}{\left(2 + -0.5 \cdot \frac{\frac{c \cdot a}{b\_2}}{b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

   8. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \frac{b\_2}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(-2 \cdot \frac{b\_2}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b\_2}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{1}{2} \cdot \frac{1}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a \cdot c}}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \color{blue}{\left(-2 \cdot \frac{b\_2}{a \cdot c}\right)}\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{b\_2}\right), \left(\color{blue}{-2} \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{b\_2}\right), \left(-2 \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \left(\color{blue}{-2} \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{b\_2}{a \cdot c}\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \color{blue}{\left(a \cdot c\right)}\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \left(c \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 2.4%

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \mathsf{*.f64}\left(c, \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. Simplified 2.4%

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

   11. Taylor expanded in b_2 around 0

       \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{1}{2}}{b\_2}\right)}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 34.4%

          \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b\_2}\right)\right) \]

   13. Simplified 34.4%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq 5.5 \cdot 10^{+72}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 11.1% accurate, 22.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = c * (0.5d0 / b_2)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), \color{blue}{a}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right), a\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\sqrt{b\_2 \cdot b\_2 - a \cdot c}\right), b\_2\right), a\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\left(b\_2 \cdot b\_2 - a \cdot c\right)\right), b\_2\right), a\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(b\_2 \cdot b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \left(a \cdot c\right)\right)\right), b\_2\right), a\right) \]

   8. *-lowering-*.f64 55.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b\_2, b\_2\right), \mathsf{*.f64}\left(a, c\right)\right)\right), b\_2\right), a\right) \]

3. Simplified 55.7%

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

5. Taylor expanded in b_2 around -inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(-1 \cdot \left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right)}, a\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(b\_2 \cdot \left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), a\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot b\_2\right)\right), a\right) \]

   3. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 + \frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{-1}{2} \cdot \frac{a \cdot c}{{b\_2}^{2}}\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{{b\_2}^{2}}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{a \cdot c}{b\_2 \cdot b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   8. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{\frac{a \cdot c}{b\_2}}{b\_2}\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(\frac{a \cdot c}{b\_2}\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(a \cdot c\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(c \cdot a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(\mathsf{neg}\left(b\_2\right)\right)\right), a\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \left(0 - b\_2\right)\right), a\right) \]

   14. --lowering--.f64 34.8%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), b\_2\right), b\_2\right)\right)\right), \mathsf{\_.f64}\left(0, b\_2\right)\right), a\right) \]

7. Simplified 34.8%

   \[\leadsto \frac{\color{blue}{\left(2 + -0.5 \cdot \frac{\frac{c \cdot a}{b\_2}}{b\_2}\right) \cdot \left(0 - b\_2\right)}}{a} \]

8. Taylor expanded in c around inf

   \[\leadsto \color{blue}{c \cdot \left(-2 \cdot \frac{b\_2}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b\_2}\right)} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(-2 \cdot \frac{b\_2}{a \cdot c} + \frac{1}{2} \cdot \frac{1}{b\_2}\right)}\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \left(\frac{1}{2} \cdot \frac{1}{b\_2} + \color{blue}{-2 \cdot \frac{b\_2}{a \cdot c}}\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right), \color{blue}{\left(-2 \cdot \frac{b\_2}{a \cdot c}\right)}\right)\right) \]

   4. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{b\_2}\right), \left(\color{blue}{-2} \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{b\_2}\right), \left(-2 \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \left(\color{blue}{-2} \cdot \frac{b\_2}{a \cdot c}\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \color{blue}{\left(\frac{b\_2}{a \cdot c}\right)}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \color{blue}{\left(a \cdot c\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \left(c \cdot \color{blue}{a}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 28.9%

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, b\_2\right), \mathsf{*.f64}\left(-2, \mathsf{/.f64}\left(b\_2, \mathsf{*.f64}\left(c, \color{blue}{a}\right)\right)\right)\right)\right) \]

10. Simplified 28.9%

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

11. Taylor expanded in b_2 around 0

    \[\leadsto \mathsf{*.f64}\left(c, \color{blue}{\left(\frac{\frac{1}{2}}{b\_2}\right)}\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f64 9.7%

       \[\leadsto \mathsf{*.f64}\left(c, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{b\_2}\right)\right) \]

13. Simplified 9.7%

    \[\leadsto c \cdot \color{blue}{\frac{0.5}{b\_2}} \]
14. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.1× speedup

Math

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

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 2024161 
(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))