Cubic critical

Percentage Accurate: 52.7% → 85.1%

Time: 8.5s

Alternatives: 7

Speedup: 2.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(a \cdot -3, c, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+91)
   (/ (/ b -1.5) a)
   (if (<= b 9e-101)
     (/ (- (sqrt (fma (* a -3.0) c (* b b))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+91) {
		tmp = (b / -1.5) / a;
	} else if (b <= 9e-101) {
		tmp = (sqrt(fma((a * -3.0), c, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+91)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 9e-101)
		tmp = Float64(Float64(sqrt(fma(Float64(a * -3.0), c, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e+91], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9e-101], N[(N[(N[Sqrt[N[(N[(a * -3.0), $MachinePrecision] * c + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.50000000000000001e91

   1. Initial program 52.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 98.0

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

      \[\leadsto b \cdot \color{blue}{\frac{-0.6666666666666666}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.9%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.50000000000000001e91 < b < 8.9999999999999997e-101

   1. Initial program 81.3%

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

   4. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} \cdot c + b \cdot b} - b}{a \cdot 3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 81.3

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

   6. Applied rewrites 81.3%

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

   if 8.9999999999999997e-101 < b

   1. Initial program 17.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

Alternative 2: 85.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.5e+91)
   (/ (/ b -1.5) a)
   (if (<= b 9e-101)
     (/ (- (sqrt (fma b b (* (* c -3.0) a))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.5e+91) {
		tmp = (b / -1.5) / a;
	} else if (b <= 9e-101) {
		tmp = (sqrt(fma(b, b, ((c * -3.0) * a))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.5e+91)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 9e-101)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -3.0) * a))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.5e+91], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9e-101], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.50000000000000001e91

   1. Initial program 52.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 98.0

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

      \[\leadsto b \cdot \color{blue}{\frac{-0.6666666666666666}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.9%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -3.50000000000000001e91 < b < 8.9999999999999997e-101

   1. Initial program 81.3%

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

   4. Applied rewrites 81.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

         \[\leadsto \frac{\sqrt{c \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) + b \cdot b} - b}{a \cdot 3} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} + b \cdot b} - b}{a \cdot 3} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{c \cdot \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) + b \cdot b} - b}{a \cdot 3} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right) \cdot c} + b \cdot b} - b}{a \cdot 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot c + b \cdot b} - b}{a \cdot 3} \]

      11. metadata-eval N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*l* N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 81.2

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 81.2

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

   6. Applied rewrites 81.2%

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

   if 8.9999999999999997e-101 < b

   1. Initial program 17.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

Alternative 3: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7e+64)
   (/ (/ b -1.5) a)
   (if (<= b 9e-101)
     (* (/ (- (sqrt (fma (* -3.0 c) a (* b b))) b) a) 0.3333333333333333)
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7e+64) {
		tmp = (b / -1.5) / a;
	} else if (b <= 9e-101) {
		tmp = ((sqrt(fma((-3.0 * c), a, (b * b))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7e+64)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 9e-101)
		tmp = Float64(Float64(Float64(sqrt(fma(Float64(-3.0 * c), a, Float64(b * b))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7e+64], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 9e-101], N[(N[(N[(N[Sqrt[N[(N[(-3.0 * c), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.9999999999999997e64

   1. Initial program 58.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 98.1

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.0%

      \[\leadsto b \cdot \color{blue}{\frac{-0.6666666666666666}{a}} \]
   8. Step-by-step derivation

   9. Applied rewrites 98.1%

      \[\leadsto \frac{b}{\color{blue}{a \cdot -1.5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.2%

       \[\leadsto \frac{\frac{b}{-1.5}}{\color{blue}{a}} \]

   if -6.9999999999999997e64 < b < 8.9999999999999997e-101

   1. Initial program 80.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 80.0%

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

   if 8.9999999999999997e-101 < b

   1. Initial program 17.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.02 \cdot 10^{-38}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.02e-38)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (if (<= b 5.2e-141)
     (/ (- (sqrt (* -3.0 (* c a))) b) (* a 3.0))
     (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.02e-38) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else if (b <= 5.2e-141) {
		tmp = (sqrt((-3.0 * (c * a))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.02e-38)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	elseif (b <= 5.2e-141)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(c * a))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.02e-38], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-141], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.01999999999999998e-38

   1. Initial program 68.8%

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

   4. Applied rewrites 68.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} \cdot c + b \cdot b} - b}{a \cdot 3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 68.8

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

   6. Applied rewrites 68.8%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot b\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-b\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(-b\right) \cdot \left(\color{blue}{\frac{c}{{b}^{2}} \cdot \frac{-1}{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(-b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, \frac{-1}{2}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, \frac{-1}{2}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \]

      12. lower-/.f64 92.4

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -0.5, \color{blue}{\frac{0.6666666666666666}{a}}\right) \]

   9. Applied rewrites 92.4%

      \[\leadsto \color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -0.5, \frac{0.6666666666666666}{a}\right)} \]

   10. Taylor expanded in a around inf

       \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

   12. Applied rewrites 92.6%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b}}, \frac{b}{a} \cdot -0.6666666666666666\right) \]

   if -1.01999999999999998e-38 < b < 5.20000000000000022e-141

   1. Initial program 77.1%

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if 5.20000000000000022e-141 < b

   1. Initial program 19.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 85.4

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

   5. Applied rewrites 85.4%

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

Alternative 5: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{b}{a} \cdot -0.6666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-311)
   (fma 0.5 (/ c b) (* (/ b a) -0.6666666666666666))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-311) {
		tmp = fma(0.5, (c / b), ((b / a) * -0.6666666666666666));
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-311)
		tmp = fma(0.5, Float64(c / b), Float64(Float64(b / a) * -0.6666666666666666));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-311], N[(0.5 * N[(c / b), $MachinePrecision] + N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -5.00000000000023e-311

   1. Initial program 72.3%

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

   4. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3 \cdot c, a, b \cdot b\right)} - b}{a \cdot 3}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} \cdot c + b \cdot b} - b}{a \cdot 3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c + b \cdot b} - b}{a \cdot 3} \]

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 72.3

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

   6. Applied rewrites 72.3%

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

   7. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot b\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-b\right)} \cdot \left(\frac{-1}{2} \cdot \frac{c}{{b}^{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \]

      5. *-commutative N/A

         \[\leadsto \left(-b\right) \cdot \left(\color{blue}{\frac{c}{{b}^{2}} \cdot \frac{-1}{2}} + \frac{2}{3} \cdot \frac{1}{a}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(-b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{c}{{b}^{2}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right)} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{c}{{b}^{2}}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{\color{blue}{b \cdot b}}, \frac{-1}{2}, \frac{2}{3} \cdot \frac{1}{a}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, \frac{-1}{2}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{a}}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, \frac{-1}{2}, \frac{\color{blue}{\frac{2}{3}}}{a}\right) \]

      12. lower-/.f64 66.5

          \[\leadsto \left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -0.5, \color{blue}{\frac{0.6666666666666666}{a}}\right) \]

   9. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\left(-b\right) \cdot \mathsf{fma}\left(\frac{c}{b \cdot b}, -0.5, \frac{0.6666666666666666}{a}\right)} \]

   10. Taylor expanded in a around inf

       \[\leadsto \frac{-2}{3} \cdot \frac{b}{a} + \color{blue}{\frac{1}{2} \cdot \frac{c}{b}} \]
   11. Step-by-step derivation

   12. Applied rewrites 66.8%

       \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{c}{b}}, \frac{b}{a} \cdot -0.6666666666666666\right) \]

   if -5.00000000000023e-311 < b

   1. Initial program 32.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 68.7

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

   5. Applied rewrites 68.7%

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

Alternative 6: 67.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.1 \cdot 10^{-273}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.1e-273) (* -0.6666666666666666 (/ b a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e-273) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.1d-273) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.1e-273) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.1e-273:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.1e-273)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.1e-273)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.1e-273], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.09999999999999988e-273

   1. Initial program 72.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

      2. lower-/.f64 62.6

         \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

   5. Applied rewrites 62.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]

   if 3.09999999999999988e-273 < b

   1. Initial program 29.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

Alternative 7: 35.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.6666666666666666 \cdot \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.6666666666666666 (/ b a)))

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.6666666666666666d0) * (b / a)
end function

Java

public static double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / a);
}

Python

def code(a, b, c):
	return -0.6666666666666666 * (b / a)

Julia

function code(a, b, c)
	return Float64(-0.6666666666666666 * Float64(b / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.6666666666666666 * (b / a);
end

Wolfram

code[a_, b_, c_] := N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around -inf

   \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{-2}{3} \cdot \frac{b}{a}} \]

   2. lower-/.f64 32.0

      \[\leadsto -0.6666666666666666 \cdot \color{blue}{\frac{b}{a}} \]

5. Applied rewrites 32.0%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))