quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.6% → 84.5%

Time: 10.2s

Alternatives: 10

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: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.7 \cdot 10^{-80}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 3.3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(b\_2, \frac{-1}{a}, \frac{-1}{a} \cdot \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.7e-80)
   (/ (* c -0.5) b_2)
   (if (<= b_2 3.3e-19)
     (fma b_2 (/ -1.0 a) (* (/ -1.0 a) (sqrt (fma c (- a) (* b_2 b_2)))))
     (/ (fma c (/ (* a 0.5) b_2) (* b_2 -2.0)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.7e-80) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 3.3e-19) {
		tmp = fma(b_2, (-1.0 / a), ((-1.0 / a) * sqrt(fma(c, -a, (b_2 * b_2)))));
	} else {
		tmp = fma(c, ((a * 0.5) / b_2), (b_2 * -2.0)) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.7e-80)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 3.3e-19)
		tmp = fma(b_2, Float64(-1.0 / a), Float64(Float64(-1.0 / a) * sqrt(fma(c, Float64(-a), Float64(b_2 * b_2)))));
	else
		tmp = Float64(fma(c, Float64(Float64(a * 0.5) / b_2), Float64(b_2 * -2.0)) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.7e-80], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 3.3e-19], N[(b$95$2 * N[(-1.0 / a), $MachinePrecision] + N[(N[(-1.0 / a), $MachinePrecision] * N[Sqrt[N[(c * (-a) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(N[(a * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.7e-80

   1. Initial program 14.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-/.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. lower-*.f64 89.7

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

   5. Applied rewrites 89.7%

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

   if -1.7e-80 < b_2 < 3.2999999999999998e-19

   1. Initial program 74.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. div-inv N/A

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

      8. *-commutative N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-lft-in N/A

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

      12. associate-/r/ N/A

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

      13. clear-num N/A

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

      14. frac-2neg N/A

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

      15. div-inv N/A

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

      16. lift-neg.f64 N/A

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

      17. remove-double-neg N/A

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

   4. Applied rewrites 74.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. remove-double-div N/A

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

      8. metadata-eval N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-div N/A

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

      11. lift-/.f64 N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-/.f64 N/A

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

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

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

   6. Applied rewrites 74.5%

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

   if 3.2999999999999998e-19 < b_2

   1. Initial program 69.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-101}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-101)
   (/ (* c -0.5) b_2)
   (if (<= b_2 5e+100)
     (/ (- (- b_2) (sqrt (fma b_2 b_2 (* c (- a))))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-101) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 5e+100) {
		tmp = (-b_2 - sqrt(fma(b_2, b_2, (c * -a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e-101)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 5e+100)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(fma(b_2, b_2, Float64(c * Float64(-a))))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.25e-101], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+100], N[(N[((-b$95$2) - N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.2499999999999999e-101

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-/.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. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.2499999999999999e-101 < b_2 < 4.9999999999999999e100

   1. Initial program 81.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 81.2

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

   4. Applied rewrites 81.2%

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

   if 4.9999999999999999e100 < b_2

   1. Initial program 60.3%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-101}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-101}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.25e-101)
   (/ (* c -0.5) b_2)
   (if (<= b_2 5e+100)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-101) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 5e+100) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= (-2.25d-101)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 5d+100) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.25e-101) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 5e+100) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.25e-101:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 5e+100:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.25e-101)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 5e+100)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.25e-101)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 5e+100)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.25e-101], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 5e+100], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.2499999999999999e-101

   1. Initial program 15.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-/.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. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   if -2.2499999999999999e-101 < b_2 < 4.9999999999999999e100

   1. Initial program 81.2%

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

   if 4.9999999999999999e100 < b_2

   1. Initial program 60.3%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.25 \cdot 10^{-101}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-134)
   (/ (* c -0.5) b_2)
   (if (<= b_2 8.2e-42)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (/ (fma c (/ (* a 0.5) b_2) (* b_2 -2.0)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8.2e-42) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = fma(c, ((a * 0.5) / b_2), (b_2 * -2.0)) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-134)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 8.2e-42)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(fma(c, Float64(Float64(a * 0.5) / b_2), Float64(b_2 * -2.0)) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-134], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 8.2e-42], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(N[(a * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.000000000000001e-134

   1. Initial program 18.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-/.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. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -9.000000000000001e-134 < b_2 < 8.2000000000000003e-42

   1. Initial program 75.9%

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

   3. Taylor expanded in b_2 around 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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 8.2000000000000003e-42 < b_2

   1. Initial program 70.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-134}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-134)
   (/ (* c -0.5) b_2)
   (if (<= b_2 8.2e-42)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8.2e-42) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= (-9d-134)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 8.2d-42) then
        tmp = (-b_2 - sqrt((c * -a))) / a
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-134) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 8.2e-42) {
		tmp = (-b_2 - Math.sqrt((c * -a))) / a;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -9e-134:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 8.2e-42:
		tmp = (-b_2 - math.sqrt((c * -a))) / a
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-134)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 8.2e-42)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -9e-134)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 8.2e-42)
		tmp = (-b_2 - sqrt((c * -a))) / a;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-134], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 8.2e-42], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.000000000000001e-134

   1. Initial program 18.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-/.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. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -9.000000000000001e-134 < b_2 < 8.2000000000000003e-42

   1. Initial program 75.9%

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

   3. Taylor expanded in b_2 around 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. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 8.2000000000000003e-42 < b_2

   1. Initial program 70.9%

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

   3. Taylor expanded in b_2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.9

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

   5. Applied rewrites 96.9%

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

Alternative 6: 68.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -6 \cdot 10^{-309}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e-309) (/ (* c -0.5) b_2) (/ (* b_2 -2.0) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	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 <= (-6d-309)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 * (-2.0d0)) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 * -2.0) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6e-309:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 * -2.0) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e-309)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 * -2.0) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6e-309)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 * -2.0) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e-309], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -6.000000000000001e-309

   1. Initial program 26.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-/.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. lower-*.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -6.000000000000001e-309 < b_2

   1. Initial program 75.4%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

Alternative 7: 34.7% 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 51.9%

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

3. Taylor expanded in b_2 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 39.7

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

5. Applied rewrites 39.7%

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

Alternative 8: 34.6% 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 51.9%

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

3. Taylor expanded in b_2 around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. lower-*.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 39.6

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

5. Applied rewrites 39.6%

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

Alternative 9: 2.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double a, double b_2, double c) {
	return (b_2 + 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 + b_2) / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

4. Applied rewrites 31.7%

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

5. Taylor expanded in b_2 around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 2.4

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

7. Applied rewrites 2.4%

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

   1. lift-neg.f64 N/A

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

   2. lift--.f64 N/A

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

   3. clear-num N/A

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

   4. remove-double-div N/A

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

   5. lower-/.f64 2.4

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-neg.f64 N/A

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

   9. remove-double-neg N/A

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

   10. lower-+.f64 2.4

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

9. Applied rewrites 2.4%

   \[\leadsto \color{blue}{\frac{b\_2 + b\_2}{a}} \]
10. Add Preprocessing

Alternative 10: 2.6% accurate, 3.3× 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(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 51.9%

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

4. Applied rewrites 17.0%

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

5. Taylor expanded in b_2 around inf

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

   1. lower-/.f64 2.4

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

7. Applied rewrites 2.4%

   \[\leadsto \color{blue}{\frac{b\_2}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

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

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

Reproduce

herbie shell --seed 2024214 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :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) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

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