quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.6% → 85.7%

Time: 10.4s

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: 52.6% 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: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b\_2, \frac{-1}{a}, \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e-123)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.1e+119)
     (fma b_2 (/ -1.0 a) (/ (sqrt (- (* b_2 b_2) (* a c))) (- a)))
     (fma 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 <= -2.1e-123) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.1e+119) {
		tmp = fma(b_2, (-1.0 / a), (sqrt(((b_2 * b_2) - (a * c))) / -a));
	} else {
		tmp = fma(c, (0.5 / b_2), ((b_2 * -2.0) / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.1e-123)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.1e+119)
		tmp = fma(b_2, Float64(-1.0 / a), Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) / Float64(-a)));
	else
		tmp = fma(c, Float64(0.5 / b_2), Float64(Float64(b_2 * -2.0) / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.1e-123], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.1e+119], N[(b$95$2 * N[(-1.0 / a), $MachinePrecision] + N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-a)), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.0999999999999999e-123

   1. Initial program 22.1%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

   if -2.0999999999999999e-123 < b_2 < 1.1000000000000001e119

   1. Initial program 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. frac-2neg N/A

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

      6. div-inv N/A

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

      7. lift-neg.f64 N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-/.f64 85.1

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

   4. Applied rewrites 85.1%

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

   if 1.1000000000000001e119 < b_2

   1. Initial program 54.9%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 54.9

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

   4. Applied rewrites 54.9%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 90.7

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

   7. Applied rewrites 90.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(b\_2, \frac{-1}{a}, \frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}{-a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.1 \cdot 10^{+119}:\\ \;\;\;\;\frac{b\_2}{-a} - \frac{\sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.1e-123)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.1e+119)
     (- (/ b_2 (- a)) (/ (sqrt (fma c (- a) (* b_2 b_2))) a))
     (fma 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 <= -2.1e-123) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.1e+119) {
		tmp = (b_2 / -a) - (sqrt(fma(c, -a, (b_2 * b_2))) / a);
	} else {
		tmp = fma(c, (0.5 / b_2), ((b_2 * -2.0) / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.1e-123)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.1e+119)
		tmp = Float64(Float64(b_2 / Float64(-a)) - Float64(sqrt(fma(c, Float64(-a), Float64(b_2 * b_2))) / a));
	else
		tmp = fma(c, Float64(0.5 / b_2), Float64(Float64(b_2 * -2.0) / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.1e-123], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.1e+119], N[(N[(b$95$2 / (-a)), $MachinePrecision] - N[(N[Sqrt[N[(c * (-a) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.0999999999999999e-123

   1. Initial program 20.4%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 81.5

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

   5. Applied rewrites 81.5%

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

   if -2.0999999999999999e-123 < b_2 < 1.1000000000000001e119

   1. Initial program 84.5%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 83.9

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

   4. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. remove-double-div N/A

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

      9. sub-div N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower--.f64 N/A

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

   6. Applied rewrites 84.5%

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

   if 1.1000000000000001e119 < b_2

   1. Initial program 51.8%

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

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 97.0

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

   7. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{0.5}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)} \]
3. Recombined 3 regimes into one program.
4. 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 2024229 
(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))