quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.3% → 85.6%

Time: 12.3s

Alternatives: 9

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: 52.3% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \mathsf{fma}\left(a, c, 0\right)\right)}}{0 - a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-58)
   (/ (* c -0.5) b_2)
   (if (<= b_2 9.8e+113)
     (- (/ (sqrt (fma b_2 b_2 (- 0.0 (fma a c 0.0)))) (- 0.0 a)) (/ b_2 a))
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-58) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 9.8e+113) {
		tmp = (sqrt(fma(b_2, b_2, (0.0 - fma(a, c, 0.0)))) / (0.0 - a)) - (b_2 / a);
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-58)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 9.8e+113)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(0.0 - fma(a, c, 0.0)))) / Float64(0.0 - a)) - Float64(b_2 / a));
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-58], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 9.8e+113], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(0.0 - N[(a * c + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - a), $MachinePrecision]), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.99999999999999977e-58

   1. Initial program 14.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 91.7

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

   5. Simplified 91.7%

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

   if -4.99999999999999977e-58 < b_2 < 9.80000000000000043e113

   1. Initial program 87.6%

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

      1. sub-neg N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

      6. exp-lowering-exp.f64 N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, \color{blue}{0 - a \cdot c}\right)}}{a} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, \color{blue}{0 - a \cdot c}\right)}}{a} \]

      12. *-lowering-*.f64 66.7

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, 0 - \color{blue}{a \cdot c}\right)}}{a} \]

   4. Applied egg-rr 66.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, 0 - a \cdot c\right)}}}{a} \]
   5. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a} - \frac{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a}} \]

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

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(b\_2\right)}{a}} - \frac{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a} \]

      4. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{0 - b\_2}}{a} - \frac{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a} \]

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

         \[\leadsto \frac{\color{blue}{0 - b\_2}}{a} - \frac{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a} \]

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

         \[\leadsto \frac{0 - b\_2}{a} - \color{blue}{\frac{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a}} \]

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

         \[\leadsto \frac{0 - b\_2}{a} - \frac{\color{blue}{\sqrt{e^{\log b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}}{a} \]

      8. rem-exp-log N/A

         \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{\color{blue}{b\_2} \cdot e^{\log b\_2} + \left(0 - a \cdot c\right)}}{a} \]

      9. rem-exp-log N/A

         \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{b\_2 \cdot \color{blue}{b\_2} + \left(0 - a \cdot c\right)}}{a} \]

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

          \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, 0 - a \cdot c\right)}}}{a} \]

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

          \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{0 - a \cdot c}\right)}}{a} \]

      12. +-lft-identity N/A

          \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \color{blue}{\left(0 + a \cdot c\right)}\right)}}{a} \]

      13. +-commutative N/A

          \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \color{blue}{\left(a \cdot c + 0\right)}\right)}}{a} \]

      14. accelerator-lowering-fma.f64 87.6

          \[\leadsto \frac{0 - b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \color{blue}{\mathsf{fma}\left(a, c, 0\right)}\right)}}{a} \]

   6. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{0 - b\_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \mathsf{fma}\left(a, c, 0\right)\right)}}{a}} \]
   7. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(b\_2\right)}}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \mathsf{fma}\left(a, c, 0\right)\right)}}{a} \]

      2. neg-lowering-neg.f64 87.6

         \[\leadsto \frac{\color{blue}{-b\_2}}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \mathsf{fma}\left(a, c, 0\right)\right)}}{a} \]

   8. Applied egg-rr 87.6%

      \[\leadsto \frac{\color{blue}{-b\_2}}{a} - \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \mathsf{fma}\left(a, c, 0\right)\right)}}{a} \]

   if 9.80000000000000043e113 < b_2

   1. Initial program 58.7%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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. *-lowering-*.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. /-lowering-/.f64 96.2

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

   5. Simplified 96.2%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

      8. /-lowering-/.f64 96.4

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-58}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 9.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, 0 - \mathsf{fma}\left(a, c, 0\right)\right)}}{0 - a} - \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.95 \cdot 10^{-53}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.95e-53)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.5e+115)
     (/ (- (- 0.0 b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (* (/ b_2 a) -2.0))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.95e-53) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.5e+115) {
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	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 <= (-1.95d-53)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.5d+115) then
        tmp = ((0.0d0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.95e-53) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.5e+115) {
		tmp = ((0.0 - b_2) - Math.sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.95e-53:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.5e+115:
		tmp = ((0.0 - b_2) - math.sqrt(((b_2 * b_2) - (c * a)))) / a
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.95e-53)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.5e+115)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.95e-53)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.5e+115)
		tmp = ((0.0 - b_2) - sqrt(((b_2 * b_2) - (c * a)))) / a;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.95e-53], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.5e+115], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - 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 / a), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.9500000000000001e-53

   1. Initial program 14.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 91.7

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

   5. Simplified 91.7%

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

   if -1.9500000000000001e-53 < b_2 < 1.5e115

   1. Initial program 87.6%

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

   if 1.5e115 < b_2

   1. Initial program 58.7%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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. *-lowering-*.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. /-lowering-/.f64 96.2

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

   5. Simplified 96.2%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

      8. /-lowering-/.f64 96.4

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.95 \cdot 10^{-53}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.3e-55)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.1e-13)
     (/ (- (- 0.0 b_2) (sqrt (- 0.0 (* c a)))) a)
     (fma 0.5 (/ c b_2) (/ (* b_2 -2.0) a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.3e-55) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.1e-13) {
		tmp = ((0.0 - b_2) - sqrt((0.0 - (c * a)))) / a;
	} else {
		tmp = fma(0.5, (c / b_2), ((b_2 * -2.0) / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.3e-55)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.1e-13)
		tmp = Float64(Float64(Float64(0.0 - b_2) - sqrt(Float64(0.0 - Float64(c * a)))) / a);
	else
		tmp = fma(0.5, Float64(c / 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, -1.3e-55], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.1e-13], N[(N[(N[(0.0 - b$95$2), $MachinePrecision] - N[Sqrt[N[(0.0 - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(0.5 * N[(c / 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 < -1.2999999999999999e-55

   1. Initial program 14.0%

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

   3. Taylor expanded in b_2 around -inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 91.7

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

   5. Simplified 91.7%

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

   if -1.2999999999999999e-55 < b_2 < 2.09999999999999989e-13

   1. Initial program 85.3%

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

   3. Taylor expanded in b_2 around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{0 - a \cdot c}}}{a} \]

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

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{0 - a \cdot c}}}{a} \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{0 - \color{blue}{c \cdot a}}}{a} \]

      5. *-lowering-*.f64 81.3

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{0 - \color{blue}{c \cdot a}}}{a} \]

   5. Simplified 81.3%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{0 - c \cdot a}}}{a} \]

   if 2.09999999999999989e-13 < b_2

   1. Initial program 71.9%

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

      1. sub-neg N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

      6. exp-lowering-exp.f64 N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, \color{blue}{0 - a \cdot c}\right)}}{a} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, \color{blue}{0 - a \cdot c}\right)}}{a} \]

      12. *-lowering-*.f64 69.7

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, 0 - \color{blue}{a \cdot c}\right)}}{a} \]

   4. Applied egg-rr 69.7%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, 0 - a \cdot c\right)}}}{a} \]

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.8

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

   7. Simplified 87.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(0 - b\_2\right) - \sqrt{0 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, \frac{b\_2 \cdot -2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (/ (* c -0.5) b_2)
   (fma 0.5 (/ c b_2) (/ (* b_2 -2.0) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = fma(0.5, (c / b_2), ((b_2 * -2.0) / a));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = fma(0.5, Float64(c / 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, -2e-310], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 30.4%

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 78.0%

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

      1. sub-neg N/A

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

      2. pow2 N/A

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

      3. pow-to-exp N/A

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

      4. exp-lft-sqr N/A

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

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

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

      6. exp-lowering-exp.f64 N/A

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

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

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

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

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

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

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

      10. neg-sub0 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, \color{blue}{0 - a \cdot c}\right)}}{a} \]

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

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, \color{blue}{0 - a \cdot c}\right)}}{a} \]

      12. *-lowering-*.f64 76.6

          \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, 0 - \color{blue}{a \cdot c}\right)}}{a} \]

   4. Applied egg-rr 76.6%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{\mathsf{fma}\left(e^{\log b\_2}, e^{\log b\_2}, 0 - a \cdot c\right)}}}{a} \]

   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. accelerator-lowering-fma.f64 N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 57.3

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

   7. Simplified 57.3%

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

Alternative 5: 67.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.3 \cdot 10^{-308}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.3e-308) (/ (* c -0.5) b_2) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.3e-308) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-4.3d-308)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.3e-308) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -4.3e-308:
		tmp = (c * -0.5) / b_2
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.3e-308)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -4.3e-308)
		tmp = (c * -0.5) / b_2;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.3e-308], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.3000000000000002e-308

   1. Initial program 30.4%

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

   if -4.3000000000000002e-308 < b_2

   1. Initial program 78.0%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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. *-lowering-*.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. /-lowering-/.f64 57.0

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

   5. Simplified 57.0%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

      8. /-lowering-/.f64 57.2

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

   7. Applied egg-rr 57.2%

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

Alternative 6: 42.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.35e-29) (/ (* c 0.5) b_2) (* (/ b_2 a) -2.0)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.35e-29) {
		tmp = (c * 0.5) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	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 <= (-2.35d-29)) then
        tmp = (c * 0.5d0) / b_2
    else
        tmp = (b_2 / a) * (-2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.35e-29) {
		tmp = (c * 0.5) / b_2;
	} else {
		tmp = (b_2 / a) * -2.0;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.35e-29:
		tmp = (c * 0.5) / b_2
	else:
		tmp = (b_2 / a) * -2.0
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.35e-29)
		tmp = Float64(Float64(c * 0.5) / b_2);
	else
		tmp = Float64(Float64(b_2 / a) * -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.35e-29)
		tmp = (c * 0.5) / b_2;
	else
		tmp = (b_2 / a) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.35e-29], N[(N[(c * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -2.3499999999999999e-29

   1. Initial program 13.1%

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

   3. Taylor expanded in b_2 around inf

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

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

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

      2. sub-neg N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 2.3

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

   5. Simplified 2.3%

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

   6. Taylor expanded in b_2 around 0

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

      1. associate-*r/ N/A

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

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

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

      3. *-lowering-*.f64 31.3

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

   8. Simplified 31.3%

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

   if -2.3499999999999999e-29 < b_2

   1. Initial program 77.7%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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. *-lowering-*.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. /-lowering-/.f64 47.0

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

   5. Simplified 47.0%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

      8. /-lowering-/.f64 47.1

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

   7. Applied egg-rr 47.1%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{c \cdot 0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2}{a} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

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) * (-2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in b_2 around inf

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
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. *-lowering-*.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. /-lowering-/.f64 32.2

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

5. Simplified 32.2%

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

   1. clear-num N/A

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

   2. associate-*r/ N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

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

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

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

   8. /-lowering-/.f64 32.2

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

7. Applied egg-rr 32.2%

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

Alternative 8: 34.4% 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 56.3%

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

3. Taylor expanded in b_2 around inf

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
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. *-lowering-*.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. /-lowering-/.f64 32.2

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

5. Simplified 32.2%

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

Alternative 9: 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 56.3%

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

4. Applied egg-rr 30.6%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(b\_2 - \sqrt{0 - \mathsf{fma}\left(a, c, \mathsf{fma}\left(b\_2, b\_2, 0\right)\right)}\right)} \]

5. Taylor expanded in a around inf

   \[\leadsto \frac{1}{a} \cdot \left(b\_2 - \sqrt{0 - \color{blue}{a \cdot c}}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{a} \cdot \left(b\_2 - \sqrt{0 - \color{blue}{c \cdot a}}\right) \]

   2. *-lowering-*.f64 31.4

      \[\leadsto \frac{1}{a} \cdot \left(b\_2 - \sqrt{0 - \color{blue}{c \cdot a}}\right) \]

7. Simplified 31.4%

   \[\leadsto \frac{1}{a} \cdot \left(b\_2 - \sqrt{0 - \color{blue}{c \cdot a}}\right) \]

8. Taylor expanded in b_2 around inf

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

   1. /-lowering-/.f64 2.5

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

10. Simplified 2.5%

    \[\leadsto \color{blue}{\frac{b\_2}{a}} \]
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 2024194 
(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))