quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.8% → 86.5%

Time: 9.7s

Alternatives: 8

Speedup: 15.9×

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: 52.8% 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: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{+153}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 7.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6e+153)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 7.7e-53)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6e+153) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 7.7e-53) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6e+153)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 7.7e-53)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6e+153], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 7.7e-53], 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[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.00000000000000037e153

   1. Initial program 37.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 37.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 37.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 100.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   6. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if -6.00000000000000037e153 < b_2 < 7.69999999999999995e-53

   1. Initial program 80.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 80.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 80.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 7.69999999999999995e-53 < b_2

   1. Initial program 15.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 15.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 9.1%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 7.1%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 7.1%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 7.1%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 7.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 7.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 7.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 7.1%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 9.2%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 9.2%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 9.2%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 9.2%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 9.2%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 9.2%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num 9.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow 9.2%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/3 7.1%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow 15.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. metadata-eval 15.4%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.5}} - b_2}\right)}^{-1} \]

      6. pow1/2 15.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}} - b_2}\right)}^{-1} \]

      7. sub-neg 15.4%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-c \cdot a\right)}} - b_2}\right)}^{-1} \]

      8. distribute-rgt-neg-out 15.4%

         \[\leadsto {\left(\frac{a}{\sqrt{b_2 \cdot b_2 + \color{blue}{c \cdot \left(-a\right)}} - b_2}\right)}^{-1} \]

      9. add-sqr-sqrt 13.3%

         \[\leadsto {\left(\frac{a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{c \cdot \left(-a\right)} \cdot \sqrt{c \cdot \left(-a\right)}}} - b_2}\right)}^{-1} \]

      10. hypot-udef 22.4%

          \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)} - b_2}\right)}^{-1} \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 22.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}}} \]

   11. Simplified 22.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}}} \]

   12. Taylor expanded in b_2 around inf 0.0%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   13. Step-by-step derivation

       1. fma-def 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2 \cdot b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b_2}{c}}\right)} \]

       7. metadata-eval 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{-2} \cdot \frac{b_2}{c}\right)} \]

   14. Simplified 90.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6 \cdot 10^{+153}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 7.7 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)}\\ \end{array} \]

Alternative 2: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -8.2e-28)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 3.4e-56)
     (/ (- (sqrt (* a (- c))) b_2) a)
     (/ 1.0 (fma 0.5 (/ a b_2) (* -2.0 (/ b_2 c)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -8.2e-28) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 3.4e-56) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = 1.0 / fma(0.5, (a / b_2), (-2.0 * (b_2 / c)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -8.2e-28)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 3.4e-56)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(1.0 / fma(0.5, Float64(a / b_2), Float64(-2.0 * Float64(b_2 / c))));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -8.2e-28], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.4e-56], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(1.0 / N[(0.5 * N[(a / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -8.2000000000000005e-28

   1. Initial program 64.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 92.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -8.2000000000000005e-28 < b_2 < 3.39999999999999982e-56

   1. Initial program 76.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 76.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 67.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 67.8%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out 67.8%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified 67.8%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   if 3.39999999999999982e-56 < b_2

   1. Initial program 15.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 15.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 9.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 9.1%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 7.1%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 7.1%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 7.1%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 7.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 7.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 7.1%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 7.1%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 9.2%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 9.2%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 9.2%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 9.2%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 9.2%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 9.2%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num 9.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow 9.2%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/3 7.1%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow 15.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. metadata-eval 15.4%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.5}} - b_2}\right)}^{-1} \]

      6. pow1/2 15.4%

         \[\leadsto {\left(\frac{a}{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}} - b_2}\right)}^{-1} \]

      7. sub-neg 15.4%

         \[\leadsto {\left(\frac{a}{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(-c \cdot a\right)}} - b_2}\right)}^{-1} \]

      8. distribute-rgt-neg-out 15.4%

         \[\leadsto {\left(\frac{a}{\sqrt{b_2 \cdot b_2 + \color{blue}{c \cdot \left(-a\right)}} - b_2}\right)}^{-1} \]

      9. add-sqr-sqrt 13.3%

         \[\leadsto {\left(\frac{a}{\sqrt{b_2 \cdot b_2 + \color{blue}{\sqrt{c \cdot \left(-a\right)} \cdot \sqrt{c \cdot \left(-a\right)}}} - b_2}\right)}^{-1} \]

      10. hypot-udef 22.4%

          \[\leadsto {\left(\frac{a}{\color{blue}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right)} - b_2}\right)}^{-1} \]

   9. Applied egg-rr 22.4%

      \[\leadsto \color{blue}{{\left(\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-1 22.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}}} \]

   11. Simplified 22.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\mathsf{hypot}\left(b_2, \sqrt{c \cdot \left(-a\right)}\right) - b_2}}} \]

   12. Taylor expanded in b_2 around inf 0.0%

       \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{a}{b_2} + 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}} \]
   13. Step-by-step derivation

       1. fma-def 0.0%

          \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, 2 \cdot \frac{b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}\right)}} \]

       2. associate-*r/ 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2 \cdot b_2}{c \cdot {\left(\sqrt{-1}\right)}^{2}}}\right)} \]

       3. *-commutative 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot c}}\right)} \]

       4. unpow2 0.0%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot c}\right)} \]

       5. rem-square-sqrt 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \frac{2 \cdot b_2}{\color{blue}{-1} \cdot c}\right)} \]

       6. times-frac 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{\frac{2}{-1} \cdot \frac{b_2}{c}}\right)} \]

       7. metadata-eval 90.3%

          \[\leadsto \frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, \color{blue}{-2} \cdot \frac{b_2}{c}\right)} \]

   14. Simplified 90.3%

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -8.2 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(0.5, \frac{a}{b_2}, -2 \cdot \frac{b_2}{c}\right)}\\ \end{array} \]

Alternative 3: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -6.2 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -6.2e-28)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 4.6e-55) (/ (- (sqrt (* a (- c))) b_2) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.2e-28) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.6e-55) {
		tmp = (sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-6.2d-28)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 4.6d-55) then
        tmp = (sqrt((a * -c)) - b_2) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -6.2e-28) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 4.6e-55) {
		tmp = (Math.sqrt((a * -c)) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -6.2e-28:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 4.6e-55:
		tmp = (math.sqrt((a * -c)) - b_2) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -6.2e-28)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 4.6e-55)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(-c))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -6.2e-28)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 4.6e-55)
		tmp = (sqrt((a * -c)) - b_2) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -6.2e-28], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 4.6e-55], N[(N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -6.19999999999999984e-28

   1. Initial program 64.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 64.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 92.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -6.19999999999999984e-28 < b_2 < 4.60000000000000023e-55

   1. Initial program 76.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 76.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 76.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 67.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg 67.8%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out 67.8%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified 67.8%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   if 4.60000000000000023e-55 < b_2

   1. Initial program 15.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 15.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 89.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.2 \cdot 10^{-28}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 4.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.2e-75)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (if (<= b_2 3.25e-57) (/ (sqrt (* a (- c))) a) (* (/ c b_2) -0.5))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-75) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 3.25e-57) {
		tmp = sqrt((a * -c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-7.2d-75)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else if (b_2 <= 3.25d-57) then
        tmp = sqrt((a * -c)) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.2e-75) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else if (b_2 <= 3.25e-57) {
		tmp = Math.sqrt((a * -c)) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -7.2e-75:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	elif b_2 <= 3.25e-57:
		tmp = math.sqrt((a * -c)) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.2e-75)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	elseif (b_2 <= 3.25e-57)
		tmp = Float64(sqrt(Float64(a * Float64(-c))) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -7.2e-75)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	elseif (b_2 <= 3.25e-57)
		tmp = sqrt((a * -c)) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.2e-75], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.25e-57], N[(N[Sqrt[N[(a * (-c)), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -7.2000000000000001e-75

   1. Initial program 67.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 67.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 67.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 67.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 89.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -7.2000000000000001e-75 < b_2 < 3.24999999999999996e-57

   1. Initial program 73.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 73.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 73.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 73.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. prod-diff 73.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)}} - b_2}{a} \]

      2. *-commutative 73.4%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]

      3. fma-def 73.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(b_2 \cdot b_2 + \left(-a \cdot c\right)\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)} - b_2}{a} \]

      4. associate-+l+ 73.4%

         \[\leadsto \frac{\sqrt{\color{blue}{b_2 \cdot b_2 + \left(\left(-a \cdot c\right) + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}} - b_2}{a} \]

      5. distribute-rgt-neg-in 73.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \left(\color{blue}{a \cdot \left(-c\right)} + \mathsf{fma}\left(-c, a, c \cdot a\right)\right)} - b_2}{a} \]

      6. fma-def 73.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \color{blue}{\mathsf{fma}\left(a, -c, \mathsf{fma}\left(-c, a, c \cdot a\right)\right)}} - b_2}{a} \]

      7. *-commutative 73.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \mathsf{fma}\left(-c, a, \color{blue}{a \cdot c}\right)\right)} - b_2}{a} \]

      8. fma-udef 73.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{\left(-c\right) \cdot a + a \cdot c}\right)} - b_2}{a} \]

      9. distribute-lft-neg-in 73.4%

         \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{\left(-c \cdot a\right)} + a \cdot c\right)} - b_2}{a} \]

      10. *-commutative 73.4%

          \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \left(-\color{blue}{a \cdot c}\right) + a \cdot c\right)} - b_2}{a} \]

      11. distribute-rgt-neg-in 73.4%

          \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{a \cdot \left(-c\right)} + a \cdot c\right)} - b_2}{a} \]

      12. fma-def 73.4%

          \[\leadsto \frac{\sqrt{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \color{blue}{\mathsf{fma}\left(a, -c, a \cdot c\right)}\right)} - b_2}{a} \]

   5. Applied egg-rr 73.4%

      \[\leadsto \frac{\sqrt{\color{blue}{b_2 \cdot b_2 + \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, a \cdot c\right)\right)}} - b_2}{a} \]

   6. Taylor expanded in b_2 around 0 67.0%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \sqrt{c \cdot a + -2 \cdot \left(c \cdot a\right)}} \]
   7. Step-by-step derivation

      1. associate-*l/ 67.1%

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

      2. *-lft-identity 67.1%

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

      3. distribute-rgt1-in 67.5%

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

      4. metadata-eval 67.5%

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

      5. neg-mul-1 67.5%

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

      6. *-commutative 67.5%

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

   8. Simplified 67.5%

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

   if 3.24999999999999996e-57 < b_2

   1. Initial program 15.4%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 15.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 15.4%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 89.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.2 \cdot 10^{-75}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 5: 68.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (+ (* -2.0 (/ b_2 a)) (* 0.5 (/ c b_2)))
   (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= (-5d-310)) then
        tmp = ((-2.0d0) * (b_2 / a)) + (0.5d0 * (c / b_2))
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2))
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(Float64(-2.0 * Float64(b_2 / a)) + Float64(0.5 * Float64(c / b_2)));
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = (-2.0 * (b_2 / a)) + (0.5 * (c / b_2));
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 70.7%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 70.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 69.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 37.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 37.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 37.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 37.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 64.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 6: 47.9% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.4e-235) (/ (- b_2) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-235) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= 1.4d-235) then
        tmp = -b_2 / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-235) {
		tmp = -b_2 / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.4e-235:
		tmp = -b_2 / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.4e-235)
		tmp = Float64(Float64(-b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.4e-235)
		tmp = -b_2 / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.4e-235], N[((-b$95$2) / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.39999999999999998e-235

   1. Initial program 71.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 71.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 71.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube 58.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow3 58.6%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/3 55.8%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow2 55.8%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg 56.0%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative 56.0%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in 56.0%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval 56.0%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr 56.0%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/3 58.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out 58.7%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative 58.7%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg 58.6%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative 58.6%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]

   8. Taylor expanded in c around inf 14.3%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{e^{1.5 \cdot \left(\log \left(-a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}}} - b_2}{a} \]
   9. Step-by-step derivation

      1. *-commutative 14.3%

         \[\leadsto \frac{\sqrt[3]{e^{\color{blue}{\left(\log \left(-a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right) \cdot 1.5}}} - b_2}{a} \]

      2. exp-prod 14.3%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(e^{\log \left(-a\right) + -1 \cdot \log \left(\frac{1}{c}\right)}\right)}^{1.5}}} - b_2}{a} \]

      3. +-commutative 14.3%

         \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{-1 \cdot \log \left(\frac{1}{c}\right) + \log \left(-a\right)}}\right)}^{1.5}} - b_2}{a} \]

      4. mul-1-neg 14.3%

         \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\left(-\log \left(\frac{1}{c}\right)\right)} + \log \left(-a\right)}\right)}^{1.5}} - b_2}{a} \]

      5. log-rec 14.3%

         \[\leadsto \frac{\sqrt[3]{{\left(e^{\left(-\color{blue}{\left(-\log c\right)}\right) + \log \left(-a\right)}\right)}^{1.5}} - b_2}{a} \]

      6. remove-double-neg 14.3%

         \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\log c} + \log \left(-a\right)}\right)}^{1.5}} - b_2}{a} \]

      7. log-prod 34.9%

         \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\log \left(c \cdot \left(-a\right)\right)}}\right)}^{1.5}} - b_2}{a} \]

      8. rem-exp-log 36.3%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{1.5}} - b_2}{a} \]

      9. distribute-rgt-neg-out 36.3%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(-c \cdot a\right)}}^{1.5}} - b_2}{a} \]

      10. *-commutative 36.3%

          \[\leadsto \frac{\sqrt[3]{{\left(-\color{blue}{a \cdot c}\right)}^{1.5}} - b_2}{a} \]

   10. Simplified 36.3%

       \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(-a \cdot c\right)}^{1.5}}} - b_2}{a} \]

   11. Taylor expanded in a around 0 24.7%

       \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   12. Step-by-step derivation

       1. associate-*r/ 24.7%

          \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

       2. neg-mul-1 24.7%

          \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   13. Simplified 24.7%

       \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

   if 1.39999999999999998e-235 < b_2

   1. Initial program 32.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 32.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 32.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 70.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 7: 68.6% accurate, 15.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b_2 \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 1.4e-235) (/ (* b_2 -2.0) a) (* (/ c b_2) -0.5)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-235) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= 1.4d-235) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c / b_2) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 1.4e-235) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 1.4e-235:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c / b_2) * -0.5
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 1.4e-235)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 1.4e-235)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c / b_2) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 1.4e-235], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 1.39999999999999998e-235

   1. Initial program 71.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 71.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 71.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 64.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative 64.2%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   6. Simplified 64.2%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if 1.39999999999999998e-235 < b_2

   1. Initial program 32.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 32.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg 32.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified 32.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 70.9%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 1.4 \cdot 10^{-235}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Alternative 8: 15.8% accurate, 28.0× 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(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 55.9%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 55.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg 55.9%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified 55.9%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
4. Step-by-step derivation

   1. add-cbrt-cube 45.1%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

   2. pow3 45.1%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

   3. pow1/3 42.5%

      \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

   4. sqrt-pow2 42.5%

      \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

   5. fma-neg 42.5%

      \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

   6. *-commutative 42.5%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

   7. distribute-rgt-neg-in 42.5%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

   8. metadata-eval 42.5%

      \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

5. Applied egg-rr 42.5%

   \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
6. Step-by-step derivation

   1. unpow1/3 45.2%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

   2. distribute-rgt-neg-out 45.2%

      \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

   3. *-commutative 45.2%

      \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

   4. fma-neg 45.1%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

   5. *-commutative 45.1%

      \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

7. Simplified 45.1%

   \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]

8. Taylor expanded in c around inf 12.5%

   \[\leadsto \frac{\sqrt[3]{\color{blue}{e^{1.5 \cdot \left(\log \left(-a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right)}}} - b_2}{a} \]
9. Step-by-step derivation

   1. *-commutative 12.5%

      \[\leadsto \frac{\sqrt[3]{e^{\color{blue}{\left(\log \left(-a\right) + -1 \cdot \log \left(\frac{1}{c}\right)\right) \cdot 1.5}}} - b_2}{a} \]

   2. exp-prod 12.5%

      \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(e^{\log \left(-a\right) + -1 \cdot \log \left(\frac{1}{c}\right)}\right)}^{1.5}}} - b_2}{a} \]

   3. +-commutative 12.5%

      \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{-1 \cdot \log \left(\frac{1}{c}\right) + \log \left(-a\right)}}\right)}^{1.5}} - b_2}{a} \]

   4. mul-1-neg 12.5%

      \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\left(-\log \left(\frac{1}{c}\right)\right)} + \log \left(-a\right)}\right)}^{1.5}} - b_2}{a} \]

   5. log-rec 12.5%

      \[\leadsto \frac{\sqrt[3]{{\left(e^{\left(-\color{blue}{\left(-\log c\right)}\right) + \log \left(-a\right)}\right)}^{1.5}} - b_2}{a} \]

   6. remove-double-neg 12.5%

      \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\log c} + \log \left(-a\right)}\right)}^{1.5}} - b_2}{a} \]

   7. log-prod 29.0%

      \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\log \left(c \cdot \left(-a\right)\right)}}\right)}^{1.5}} - b_2}{a} \]

   8. rem-exp-log 30.2%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(c \cdot \left(-a\right)\right)}}^{1.5}} - b_2}{a} \]

   9. distribute-rgt-neg-out 30.2%

      \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(-c \cdot a\right)}}^{1.5}} - b_2}{a} \]

   10. *-commutative 30.2%

       \[\leadsto \frac{\sqrt[3]{{\left(-\color{blue}{a \cdot c}\right)}^{1.5}} - b_2}{a} \]

10. Simplified 30.2%

    \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(-a \cdot c\right)}^{1.5}}} - b_2}{a} \]

11. Taylor expanded in a around 0 15.9%

    \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
12. Step-by-step derivation

    1. associate-*r/ 15.9%

       \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

    2. neg-mul-1 15.9%

       \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

13. Simplified 15.9%

    \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

14. Final simplification 15.9%

    \[\leadsto \frac{-b_2}{a} \]

Reproduce

herbie shell --seed 2023213 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))