quad2p (problem 3.2.1, positive)

Percentage Accurate: 53.2% → 85.6%

Time: 9.5s

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: 53.2% 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: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e+154)
   (* -2.0 (/ b_2 a))
   (if (<= b_2 7.8e-25)
     (/ (- (sqrt (fma (- c) a (* b_2 b_2))) b_2) a)
     (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e+154) {
		tmp = -2.0 * (b_2 / a);
	} else if (b_2 <= 7.8e-25) {
		tmp = (sqrt(fma(-c, a, (b_2 * b_2))) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e+154)
		tmp = Float64(-2.0 * Float64(b_2 / a));
	elseif (b_2 <= 7.8e-25)
		tmp = Float64(Float64(sqrt(fma(Float64(-c), a, Float64(b_2 * b_2))) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e+154], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 7.8e-25], N[(N[(N[Sqrt[N[((-c) * a + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.00000000000000007e154

   1. Initial program 48.6%

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

   3. Taylor expanded in b_2 around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -2.00000000000000007e154 < b_2 < 7.8e-25

   1. Initial program 80.5%

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

   4. Applied rewrites 45.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 45.4

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

   6. Applied rewrites 80.5%

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

   if 7.8e-25 < b_2

   1. Initial program 14.2%

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

   3. Taylor expanded in b_2 around inf

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

      1. lower-/.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 86.3%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{+154}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-c, a, b\_2 \cdot b\_2\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.05 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.05e-122)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.1e-26) (/ (- (sqrt (* (- a) c)) b_2) a) (/ (* -0.5 c) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.05e-122) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.1e-26) {
		tmp = (sqrt((-a * c)) - b_2) / a;
	} else {
		tmp = (-0.5 * c) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.05e-122)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.1e-26)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) - b_2) / a);
	else
		tmp = Float64(Float64(-0.5 * c) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.05e-122], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.1e-26], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.04999999999999996e-122

   1. Initial program 69.1%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   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. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 81.9%

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

   if -1.04999999999999996e-122 < b_2 < 1.1e-26

   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 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. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.9

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

   5. Applied rewrites 67.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 67.9

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

   7. Applied rewrites 67.9%

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

   if 1.1e-26 < b_2

   1. Initial program 16.5%

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

   3. Taylor expanded in b_2 around inf

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

      1. lower-/.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 88.9%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.05 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.6% 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 2024229 
(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))