quad2p (problem 3.2.1, positive)

Percentage Accurate: 51.7% → 85.0%

Time: 10.7s

Alternatives: 8

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.7% 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: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a \cdot -0.125, \frac{c}{b\_2 \cdot b\_2}, -0.5\right)}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e+43)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 7e-54)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c (fma (* a -0.125) (/ c (* b_2 b_2)) -0.5)) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.5e+43) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7e-54) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * fma((a * -0.125), (c / (b_2 * b_2)), -0.5)) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.5e+43)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 7e-54)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * fma(Float64(a * -0.125), Float64(c / Float64(b_2 * b_2)), -0.5)) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.5e+43], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7e-54], 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 * N[(N[(a * -0.125), $MachinePrecision] * N[(c / N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.5e43

   1. Initial program 57.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.0

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

   5. Simplified 97.0%

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

   if -4.5e43 < b_2 < 6.99999999999999964e-54

   1. Initial program 81.2%

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

   if 6.99999999999999964e-54 < b_2

   1. Initial program 12.9%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}}}{b\_2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \frac{a \cdot {c}^{2}}{{b\_2}^{2}} + \frac{-1}{2} \cdot c}}{b\_2} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot {c}^{2}}{{b\_2}^{2}}, \frac{-1}{2} \cdot c\right)}}{b\_2} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\frac{a \cdot {c}^{2}}{{b\_2}^{2}}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{\color{blue}{a \cdot {c}^{2}}}{{b\_2}^{2}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      6. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b\_2}^{2}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b\_2}^{2}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b\_2 \cdot b\_2}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b\_2 \cdot b\_2}}, \frac{-1}{2} \cdot c\right)}{b\_2} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \frac{a \cdot \left(c \cdot c\right)}{b\_2 \cdot b\_2}, \color{blue}{c \cdot \frac{-1}{2}}\right)}{b\_2} \]

      11. *-lowering-*.f64 76.8

          \[\leadsto \frac{\mathsf{fma}\left(-0.125, \frac{a \cdot \left(c \cdot c\right)}{b\_2 \cdot b\_2}, \color{blue}{c \cdot -0.5}\right)}{b\_2} \]

   5. Simplified 76.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \frac{a \cdot \left(c \cdot c\right)}{b\_2 \cdot b\_2}, c \cdot -0.5\right)}{b\_2}} \]

   6. Taylor expanded in c around 0

      \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)}}{b\_2} \]
   7. Step-by-step derivation

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

         \[\leadsto \frac{\color{blue}{c \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} - \frac{1}{2}\right)}}{b\_2} \]

      2. sub-neg N/A

         \[\leadsto \frac{c \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot c}{{b\_2}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}}{b\_2} \]

      3. associate-/l* N/A

         \[\leadsto \frac{c \cdot \left(\frac{-1}{8} \cdot \color{blue}{\left(a \cdot \frac{c}{{b\_2}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b\_2} \]

      4. associate-*r* N/A

         \[\leadsto \frac{c \cdot \left(\color{blue}{\left(\frac{-1}{8} \cdot a\right) \cdot \frac{c}{{b\_2}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}{b\_2} \]

      5. metadata-eval N/A

         \[\leadsto \frac{c \cdot \left(\left(\frac{-1}{8} \cdot a\right) \cdot \frac{c}{{b\_2}^{2}} + \color{blue}{\frac{-1}{2}}\right)}{b\_2} \]

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

         \[\leadsto \frac{c \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{8} \cdot a, \frac{c}{{b\_2}^{2}}, \frac{-1}{2}\right)}}{b\_2} \]

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

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\color{blue}{\frac{-1}{8} \cdot a}, \frac{c}{{b\_2}^{2}}, \frac{-1}{2}\right)}{b\_2} \]

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

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot a, \color{blue}{\frac{c}{{b\_2}^{2}}}, \frac{-1}{2}\right)}{b\_2} \]

      9. unpow2 N/A

         \[\leadsto \frac{c \cdot \mathsf{fma}\left(\frac{-1}{8} \cdot a, \frac{c}{\color{blue}{b\_2 \cdot b\_2}}, \frac{-1}{2}\right)}{b\_2} \]

      10. *-lowering-*.f64 92.3

          \[\leadsto \frac{c \cdot \mathsf{fma}\left(-0.125 \cdot a, \frac{c}{\color{blue}{b\_2 \cdot b\_2}}, -0.5\right)}{b\_2} \]

   8. Simplified 92.3%

      \[\leadsto \frac{\color{blue}{c \cdot \mathsf{fma}\left(-0.125 \cdot a, \frac{c}{b\_2 \cdot b\_2}, -0.5\right)}}{b\_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 7 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \mathsf{fma}\left(a \cdot -0.125, \frac{c}{b\_2 \cdot b\_2}, -0.5\right)}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.5e+43)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 1.6e-48)
     (/ (- (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 <= -4.5e+43) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.6e-48) {
		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 <= (-4.5d+43)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 1.6d-48) 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 <= -4.5e+43) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 1.6e-48) {
		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 <= -4.5e+43:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 1.6e-48:
		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 <= -4.5e+43)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 1.6e-48)
		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 <= -4.5e+43)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 1.6e-48)
		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, -4.5e+43], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 1.6e-48], 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 < -4.5e43

   1. Initial program 57.2%

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

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 97.0

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

   5. Simplified 97.0%

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

   if -4.5e43 < b_2 < 1.5999999999999999e-48

   1. Initial program 81.2%

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

   if 1.5999999999999999e-48 < b_2

   1. Initial program 12.9%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 91.2

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

   5. Simplified 91.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 1.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.85e-62)
   (fma c (/ 0.5 b_2) (/ (* b_2 -2.0) a))
   (if (<= b_2 7.5e-59) (/ (- (sqrt (- (* a c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.85e-62) {
		tmp = fma(c, (0.5 / b_2), ((b_2 * -2.0) / a));
	} else if (b_2 <= 7.5e-59) {
		tmp = (sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.85e-62)
		tmp = fma(c, Float64(0.5 / b_2), Float64(Float64(b_2 * -2.0) / a));
	elseif (b_2 <= 7.5e-59)
		tmp = Float64(Float64(sqrt(Float64(-Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.85e-62], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.5e-59], N[(N[(N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.8499999999999999e-62

   1. Initial program 66.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. mul-1-neg N/A

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

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

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

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

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f64 90.8

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

   5. Simplified 90.8%

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

   6. Taylor expanded in c around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. /-lowering-/.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 91.0

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

   8. Simplified 91.0%

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

   if -1.8499999999999999e-62 < b_2 < 7.50000000000000019e-59

   1. Initial program 78.7%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 71.3

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

   5. Simplified 71.3%

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

   if 7.50000000000000019e-59 < b_2

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 90.2

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

   5. Simplified 90.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.5e-62)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 7.2e-59) (/ (- (sqrt (- (* a c))) b_2) a) (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.5e-62) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.2e-59) {
		tmp = (sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.5d-62)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 7.2d-59) then
        tmp = (sqrt(-(a * c)) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.5e-62) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.2e-59) {
		tmp = (Math.sqrt(-(a * c)) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.5e-62:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 7.2e-59:
		tmp = (math.sqrt(-(a * c)) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.5e-62)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 7.2e-59)
		tmp = Float64(Float64(sqrt(Float64(-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 <= -2.5e-62)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 7.2e-59)
		tmp = (sqrt(-(a * c)) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.5e-62], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.2e-59], N[(N[(N[Sqrt[(-N[(a * c), $MachinePrecision])], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.5000000000000001e-62

   1. Initial program 66.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 91.0

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

   5. Simplified 91.0%

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

   if -2.5000000000000001e-62 < b_2 < 7.20000000000000001e-59

   1. Initial program 78.7%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 71.3

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

   5. Simplified 71.3%

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

   if 7.20000000000000001e-59 < b_2

   1. Initial program 12.7%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 90.2

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

   5. Simplified 90.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\sqrt{-a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 1.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.8e-308) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.8e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 1.8d-308) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.8e-308) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.8e-308:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.8e-308)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.8e-308)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.8e-308], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.7999999999999999e-308

   1. Initial program 71.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 65.1

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

   5. Simplified 65.1%

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

   if 1.7999999999999999e-308 < b_2

   1. Initial program 33.3%

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

   3. Taylor expanded in b_2 around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 66.0

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

   5. Simplified 66.0%

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

Alternative 6: 35.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 \cdot -2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (* b_2 -2.0) a))

C

double code(double a, double b_2, double c) {
	return (b_2 * -2.0) / 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 * (-2.0d0)) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (b_2 * -2.0) / a;
}

Python

def code(a, b_2, c):
	return (b_2 * -2.0) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 * -2.0) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (b_2 * -2.0) / a;
end

Wolfram

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in b_2 around -inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 35.4

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

5. Simplified 35.4%

   \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]
6. Add Preprocessing

Alternative 7: 35.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ b\_2 \cdot \frac{-2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* b_2 (/ -2.0 a)))

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / 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 * ((-2.0d0) / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Python

def code(a, b_2, c):
	return b_2 * (-2.0 / a)

Julia

function code(a, b_2, c)
	return Float64(b_2 * Float64(-2.0 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = b_2 * (-2.0 / a);
end

Wolfram

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in b_2 around -inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 35.4

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

5. Simplified 35.4%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. /-lowering-/.f64 35.3

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

7. Applied egg-rr 35.3%

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

8. Final simplification 35.3%

   \[\leadsto b\_2 \cdot \frac{-2}{a} \]
9. Add Preprocessing

Alternative 8: 14.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in b_2 around inf

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

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

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

   2. unpow2 N/A

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

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

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. associate-/l* N/A

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

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

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

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

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

   9. distribute-neg-frac2 N/A

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

   10. mul-1-neg N/A

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

   11. /-lowering-/.f64 N/A

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

   12. mul-1-neg N/A

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

   13. unpow2 N/A

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

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

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

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

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

   16. neg-lowering-neg.f64 36.0

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

5. Simplified 36.0%

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

   1. rem-square-sqrt N/A

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

   2. rem-sqrt-square N/A

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

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

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

   4. sqrt-prod N/A

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

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

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

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

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

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

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

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

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

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

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

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

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

   12. neg-lowering-neg.f64 53.8

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

7. Applied egg-rr 53.8%

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

8. Taylor expanded in b_2 around inf

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

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} \]

   2. neg-lowering-neg.f64 12.9

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

10. Simplified 12.9%

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

11. Final simplification 12.9%

    \[\leadsto -\frac{b\_2}{a} \]
12. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× 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 2024205 
(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))