quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.7% → 85.9%

Time: 10.2s

Alternatives: 10

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.7% 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: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{-0.5 \cdot a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.5e-90)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.4e+44)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (fma (/ c b_2) 0.5 (/ b_2 (* -0.5 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9.5e-90) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.4e+44) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = fma((c / b_2), 0.5, (b_2 / (-0.5 * a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9.5e-90)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.4e+44)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(b_2 / Float64(-0.5 * a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9.5e-90], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.4e+44], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(b$95$2 / N[(-0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.5000000000000003e-90

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 88.4

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

   5. Simplified 88.4%

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

   if -9.5000000000000003e-90 < b_2 < 1.4e44

   1. Initial program 78.7%

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

   if 1.4e44 < b_2

   1. Initial program 56.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. associate-*r/ N/A

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

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

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

      23. /-lowering-/.f64 98.2

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

   5. Simplified 98.2%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9.5 \cdot 10^{-90}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.4 \cdot 10^{+44}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{-0.5 \cdot a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-92}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9e-92)
   (/ (* c -0.5) b_2)
   (if (<= b_2 7.8e-59)
     (/ (- (- b_2) (sqrt (- (* c a)))) a)
     (/ (fma c (/ (* a 0.5) b_2) (* b_2 -2.0)) a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -9e-92) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 7.8e-59) {
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = fma(c, ((a * 0.5) / b_2), (b_2 * -2.0)) / a;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -9e-92)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 7.8e-59)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = Float64(fma(c, Float64(Float64(a * 0.5) / b_2), Float64(b_2 * -2.0)) / a);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -9e-92], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 7.8e-59], N[(N[((-b$95$2) - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * N[(N[(a * 0.5), $MachinePrecision] / b$95$2), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.0000000000000001e-92

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 88.4

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

   5. Simplified 88.4%

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

   if -9.0000000000000001e-92 < b_2 < 7.80000000000000038e-59

   1. Initial program 75.1%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 72.0

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

   5. Simplified 72.0%

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

   if 7.80000000000000038e-59 < b_2

   1. Initial program 66.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

   5. Simplified 91.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -9 \cdot 10^{-92}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 7.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{a \cdot 0.5}{b\_2}, b\_2 \cdot -2\right)}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-90}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.22 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{-0.5 \cdot a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.2e-90)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.22e-58)
     (/ (- (- b_2) (sqrt (- (* c a)))) a)
     (fma (/ c b_2) 0.5 (/ b_2 (* -0.5 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.2e-90) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.22e-58) {
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = fma((c / b_2), 0.5, (b_2 / (-0.5 * a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.2e-90)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.22e-58)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = fma(Float64(c / b_2), 0.5, Float64(b_2 / Float64(-0.5 * a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.2e-90], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.22e-58], N[(N[((-b$95$2) - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * 0.5 + N[(b$95$2 / N[(-0.5 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.19999999999999986e-90

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 88.4

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

   5. Simplified 88.4%

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

   if -2.19999999999999986e-90 < b_2 < 1.2199999999999999e-58

   1. Initial program 75.1%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 72.0

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

   5. Simplified 72.0%

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

   if 1.2199999999999999e-58 < b_2

   1. Initial program 66.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. associate-*r/ N/A

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

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

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

      23. /-lowering-/.f64 91.4

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

   5. Simplified 91.4%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

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

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

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

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

      12. div-inv N/A

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

      13. metadata-eval N/A

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

      14. *-lowering-*.f64 91.7

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

   7. Applied egg-rr 91.7%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.2 \cdot 10^{-90}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.22 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{b\_2}, 0.5, \frac{b\_2}{-0.5 \cdot a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, b\_2 \cdot \frac{-2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -4.6e-92)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.35e-58)
     (/ (- (- b_2) (sqrt (- (* c a)))) a)
     (fma c (/ 0.5 b_2) (* b_2 (/ -2.0 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -4.6e-92) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.35e-58) {
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = fma(c, (0.5 / b_2), (b_2 * (-2.0 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -4.6e-92)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.35e-58)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = fma(c, Float64(0.5 / b_2), Float64(b_2 * Float64(-2.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -4.6e-92], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.35e-58], N[(N[((-b$95$2) - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -4.60000000000000032e-92

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 88.4

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

   5. Simplified 88.4%

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

   if -4.60000000000000032e-92 < b_2 < 1.3499999999999999e-58

   1. Initial program 75.1%

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

   3. Taylor expanded in b_2 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 72.0

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

   5. Simplified 72.0%

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

   if 1.3499999999999999e-58 < b_2

   1. Initial program 66.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

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

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. associate-*r/ N/A

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

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

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

      19. associate-*r/ N/A

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

      20. metadata-eval N/A

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

      21. distribute-neg-frac N/A

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

      22. metadata-eval N/A

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

      23. /-lowering-/.f64 91.4

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

   5. Simplified 91.4%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -4.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.35 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, b\_2 \cdot \frac{-2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b\_2 + b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2.8e-91)
   (/ (* c -0.5) b_2)
   (if (<= b_2 2.1e-46)
     (/ (- (- b_2) (sqrt (- (* c a)))) a)
     (- (/ (+ b_2 b_2) a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-91) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.1e-46) {
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	} else {
		tmp = -((b_2 + b_2) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2.8d-91)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 2.1d-46) then
        tmp = (-b_2 - sqrt(-(c * a))) / a
    else
        tmp = -((b_2 + b_2) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2.8e-91) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 2.1e-46) {
		tmp = (-b_2 - Math.sqrt(-(c * a))) / a;
	} else {
		tmp = -((b_2 + b_2) / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2.8e-91:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 2.1e-46:
		tmp = (-b_2 - math.sqrt(-(c * a))) / a
	else:
		tmp = -((b_2 + b_2) / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2.8e-91)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 2.1e-46)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(-Float64(c * a)))) / a);
	else
		tmp = Float64(-Float64(Float64(b_2 + b_2) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2.8e-91)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 2.1e-46)
		tmp = (-b_2 - sqrt(-(c * a))) / a;
	else
		tmp = -((b_2 + b_2) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2.8e-91], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 2.1e-46], N[(N[((-b$95$2) - N[Sqrt[(-N[(c * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], (-N[(N[(b$95$2 + b$95$2), $MachinePrecision] / a), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.8e-91

   1. Initial program 21.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 88.4

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

   5. Simplified 88.4%

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

   if -2.8e-91 < b_2 < 2.09999999999999987e-46

   1. Initial program 74.3%

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

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. neg-lowering-neg.f64 71.2

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

   5. Simplified 71.2%

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

   if 2.09999999999999987e-46 < b_2

   1. Initial program 67.0%

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

   3. Taylor expanded in b_2 around inf

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

   5. Simplified 91.2%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{b\_2}}{a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 2.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{-c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b\_2 + b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b\_2 + b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310) (/ (* c -0.5) b_2) (- (/ (+ b_2 b_2) 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 = -((b_2 + b_2) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-310)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = -((b_2 + b_2) / a)
    end if
    code = tmp
end function

Java

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

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (c * -0.5) / b_2
	else:
		tmp = -((b_2 + b_2) / 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 = Float64(-Float64(Float64(b_2 + b_2) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (c * -0.5) / b_2;
	else
		tmp = -((b_2 + b_2) / a);
	end
	tmp_2 = 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[(N[(b$95$2 + b$95$2), $MachinePrecision] / a), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 36.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 65.1

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

   5. Simplified 65.1%

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

   if -1.999999999999994e-310 < b_2

   1. Initial program 71.8%

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

   3. Taylor expanded in b_2 around inf

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

   5. Simplified 65.9%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b\_2 + b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e-309) (/ (* c -0.5) b_2) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d-309)) then
        tmp = (c * (-0.5d0)) / b_2
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e-309) {
		tmp = (c * -0.5) / b_2;
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e-309:
		tmp = (c * -0.5) / b_2
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e-309)
		tmp = Float64(Float64(c * -0.5) / b_2);
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e-309)
		tmp = (c * -0.5) / b_2;
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e-309], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.000000000000002e-309

   1. Initial program 36.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 65.1

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

   5. Simplified 65.1%

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

   if -1.000000000000002e-309 < b_2

   1. Initial program 71.8%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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 65.7

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

   5. Simplified 65.7%

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

Alternative 8: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.4 \cdot 10^{-308}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.4e-308) (* c (/ -0.5 b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.4e-308) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.4d-308)) then
        tmp = c * ((-0.5d0) / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.4e-308) {
		tmp = c * (-0.5 / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.4e-308:
		tmp = c * (-0.5 / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.4e-308)
		tmp = Float64(c * Float64(-0.5 / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.4e-308)
		tmp = c * (-0.5 / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.4e-308], N[(c * N[(-0.5 / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.4000000000000002e-308

   1. Initial program 36.6%

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

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   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 65.1

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

   5. Simplified 65.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 65.0

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

   7. Applied egg-rr 65.0%

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

   if -1.4000000000000002e-308 < b_2

   1. Initial program 71.8%

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

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   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 65.7

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

   5. Simplified 65.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.4 \cdot 10^{-308}:\\ \;\;\;\;c \cdot \frac{-0.5}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.5% 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 53.4%

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

3. Taylor expanded in b_2 around inf

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
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 32.7

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

5. Simplified 32.7%

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

Alternative 10: 2.5% 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 53.4%

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

4. Applied egg-rr 25.1%

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

5. Taylor expanded in b_2 around inf

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

   1. /-lowering-/.f64 2.6

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

7. Simplified 2.6%

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

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024205 
(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))