quad2m (problem 3.2.1, negative)

Percentage Accurate: 51.1% → 85.3%

Time: 7.1s

Alternatives: 6

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: 51.1% 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: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-29)
   (* (/ c b_2) -0.5)
   (if (<= b_2 6.4e+97)
     (/ (+ (sqrt (- (* b_2 b_2) (* a c))) b_2) (- a))
     (fma (/ 0.5 b_2) c (* -2.0 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-29) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 6.4e+97) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) + b_2) / -a;
	} else {
		tmp = fma((0.5 / b_2), c, (-2.0 * (b_2 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-29)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (b_2 <= 6.4e+97)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) + b_2) / Float64(-a));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-29], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b$95$2, 6.4e+97], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.14999999999999996e-29

   1. Initial program 14.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1.14999999999999996e-29 < b_2 < 6.40000000000000032e97

   1. Initial program 85.2%

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

   if 6.40000000000000032e97 < 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. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. 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. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 6.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e-29)
   (* (/ c b_2) -0.5)
   (if (<= b_2 1.3e-40)
     (/ (+ (sqrt (* (- a) c)) b_2) (- a))
     (fma (/ 0.5 b_2) c (* -2.0 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e-29) {
		tmp = (c / b_2) * -0.5;
	} else if (b_2 <= 1.3e-40) {
		tmp = (sqrt((-a * c)) + b_2) / -a;
	} else {
		tmp = fma((0.5 / b_2), c, (-2.0 * (b_2 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e-29)
		tmp = Float64(Float64(c / b_2) * -0.5);
	elseif (b_2 <= 1.3e-40)
		tmp = Float64(Float64(sqrt(Float64(Float64(-a) * c)) + b_2) / Float64(-a));
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e-29], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[b$95$2, 1.3e-40], N[(N[(N[Sqrt[N[((-a) * c), $MachinePrecision]], $MachinePrecision] + b$95$2), $MachinePrecision] / (-a)), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.14999999999999996e-29

   1. Initial program 14.7%

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

   3. Taylor expanded in b_2 around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -1.14999999999999996e-29 < b_2 < 1.3000000000000001e-40

   1. Initial program 81.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if 1.3000000000000001e-40 < b_2

   1. Initial program 69.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. 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. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{elif}\;b\_2 \leq 1.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(-a\right) \cdot c} + b\_2}{-a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-311)
   (* (/ c b_2) -0.5)
   (fma (/ 0.5 b_2) c (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-311) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = fma((0.5 / b_2), c, (-2.0 * (b_2 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-311)
		tmp = Float64(Float64(c / b_2) * -0.5);
	else
		tmp = fma(Float64(0.5 / b_2), c, Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-311], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(0.5 / b$95$2), $MachinePrecision] * c + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.9999999999999e-311

   1. Initial program 35.8%

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

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

      2. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -1.9999999999999e-311 < b_2

   1. Initial program 73.6%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. 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. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 71.3

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

   5. Applied rewrites 71.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{b\_2}, c, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{-297}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.06e-297) (* (/ c b_2) -0.5) (/ (* -2.0 b_2) a)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.06e-297) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = (-2.0 * b_2) / 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 <= (-1.06d-297)) then
        tmp = (c / b_2) * (-0.5d0)
    else
        tmp = ((-2.0d0) * b_2) / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.06e-297) {
		tmp = (c / b_2) * -0.5;
	} else {
		tmp = (-2.0 * b_2) / a;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.06e-297:
		tmp = (c / b_2) * -0.5
	else:
		tmp = (-2.0 * b_2) / a
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.06e-297)
		tmp = Float64(Float64(c / b_2) * -0.5);
	else
		tmp = Float64(Float64(-2.0 * b_2) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.06e-297)
		tmp = (c / b_2) * -0.5;
	else
		tmp = (-2.0 * b_2) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.06e-297], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(-2.0 * b$95$2), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.06e-297

   1. Initial program 34.9%

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

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

      2. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -1.06e-297 < b_2

   1. Initial program 73.5%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.06 \cdot 10^{-297}:\\ \;\;\;\;\frac{c}{b\_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

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

   2. lower-/.f64 31.4

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

5. Applied rewrites 31.4%

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

6. Final simplification 31.4%

   \[\leadsto \frac{c}{b\_2} \cdot -0.5 \]
7. Add Preprocessing

Alternative 6: 10.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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 = (0.5d0 / b_2) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in c around 0

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
4. 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. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 40.1

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

5. Applied rewrites 40.1%

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

6. Taylor expanded in a around inf

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

8. Applied rewrites 8.4%

   \[\leadsto \frac{c}{b\_2} \cdot \color{blue}{0.5} \]
9. Step-by-step derivation

10. Applied rewrites 8.4%

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

Developer Target 1: 99.6% accurate, 0.2× speedup

Math

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

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 2024277 
(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))