Cubic critical, narrow range

Percentage Accurate: 55.5% → 99.4%

Time: 14.4s

Alternatives: 12

Speedup: 2.9×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.5% 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: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot 3}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (fma c (* a -3.0) 0.0) (* a 3.0))
  (+ b (sqrt (fma c (* a -3.0) (* b b))))))

C

double code(double a, double b, double c) {
	return (fma(c, (a * -3.0), 0.0) / (a * 3.0)) / (b + sqrt(fma(c, (a * -3.0), (b * b))));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(c, Float64(a * -3.0), 0.0) / Float64(a * 3.0)) / Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * -3.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

4. Applied egg-rr 57.6%

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

   1. associate-/r* N/A

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

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

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

6. Applied egg-rr 58.3%

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

   1. associate-/l/ N/A

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

   2. associate-*l* N/A

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

   3. associate-/r* N/A

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

   4. /-lowering-/.f64 N/A

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

8. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot 3}}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}}} \]
9. Add Preprocessing

Alternative 2: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{c \cdot a}{b}, b \cdot 2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.025)
   (* (/ 1.0 a) (* 0.3333333333333333 (- (sqrt (fma b b (* a (* c -3.0)))) b)))
   (/ -1.0 (/ (fma -1.5 (/ (* c a) b) (* b 2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.025) {
		tmp = (1.0 / a) * (0.3333333333333333 * (sqrt(fma(b, b, (a * (c * -3.0)))) - b));
	} else {
		tmp = -1.0 / (fma(-1.5, ((c * a) / b), (b * 2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.025)
		tmp = Float64(Float64(1.0 / a) * Float64(0.3333333333333333 * Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b)));
	else
		tmp = Float64(-1.0 / Float64(fma(-1.5, Float64(Float64(c * a) / b), Float64(b * 2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.025], N[(N[(1.0 / a), $MachinePrecision] * N[(0.3333333333333333 * N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.025000000000000001

   1. Initial program 79.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.4

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

   4. Applied egg-rr 79.4%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. inv-pow N/A

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

      8. clear-num N/A

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

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

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

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

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

      11. div-inv N/A

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

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

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

   6. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot 0.3333333333333333\right)} \]

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.0%

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

   4. Applied egg-rr 49.0%

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

   5. Taylor expanded in c around 0

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

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

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b} + 2 \cdot b}{c}}} \]

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

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

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

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

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

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

      5. *-lowering-*.f64 87.8

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, \color{blue}{2 \cdot b}\right)}{c}} \]

   7. Simplified 87.8%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, 2 \cdot b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{1}{a} \cdot \left(0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{c \cdot a}{b}, b \cdot 2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{c \cdot a}{b}, b \cdot 2\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.025)
   (/ (- (sqrt (fma (* c a) -3.0 (* b b))) b) (* a 3.0))
   (/ -1.0 (/ (fma -1.5 (/ (* c a) b) (* b 2.0)) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.025) {
		tmp = (sqrt(fma((c * a), -3.0, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -1.0 / (fma(-1.5, ((c * a) / b), (b * 2.0)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.025)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-1.0 / Float64(fma(-1.5, Float64(Float64(c * a) / b), Float64(b * 2.0)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.025], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(-1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.025000000000000001

   1. Initial program 79.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 79.4

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

   4. Applied egg-rr 79.4%

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

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.0%

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

   4. Applied egg-rr 49.0%

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

   5. Taylor expanded in c around 0

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

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

         \[\leadsto \frac{-1}{\color{blue}{\frac{\frac{-3}{2} \cdot \frac{a \cdot c}{b} + 2 \cdot b}{c}}} \]

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

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

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

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

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

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

      5. *-lowering-*.f64 87.8

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, \color{blue}{2 \cdot b}\right)}{c}} \]

   7. Simplified 87.8%

      \[\leadsto \frac{-1}{\color{blue}{\frac{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, 2 \cdot b\right)}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-1.5, \frac{c \cdot a}{b}, b \cdot 2\right)}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.025)
   (/ (- (sqrt (fma (* c a) -3.0 (* b b))) b) (* a 3.0))
   (/ -1.0 (fma -1.5 (/ a b) (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.025) {
		tmp = (sqrt(fma((c * a), -3.0, (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -1.0 / fma(-1.5, (a / b), (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.025)
		tmp = Float64(Float64(sqrt(fma(Float64(c * a), -3.0, Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-1.0 / fma(-1.5, Float64(a / b), Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.025], N[(N[(N[Sqrt[N[(N[(c * a), $MachinePrecision] * -3.0 + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.5 * N[(a / b), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.025000000000000001

   1. Initial program 79.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 79.4

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

   4. Applied egg-rr 79.4%

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

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.0%

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

   4. Applied egg-rr 49.0%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 87.8

         \[\leadsto \frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

   7. Simplified 87.8%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c \cdot a, -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.025)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ -1.0 (fma -1.5 (/ a b) (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.025) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -1.0 / fma(-1.5, (a / b), (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-1.0 / fma(-1.5, Float64(a / b), Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.025], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.5 * N[(a / b), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.025000000000000001

   1. Initial program 79.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.4

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

   4. Applied egg-rr 79.4%

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

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

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

      2. +-commutative N/A

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

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

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

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

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

      6. sub-neg N/A

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

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

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

      8. *-commutative N/A

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

      9. distribute-rgt-neg-in N/A

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

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

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-in N/A

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

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

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 79.4

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

   6. Applied egg-rr 79.4%

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

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.0%

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

   4. Applied egg-rr 49.0%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 87.8

         \[\leadsto \frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

   7. Simplified 87.8%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.025)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ 0.3333333333333333 a))
   (/ -1.0 (fma -1.5 (/ a b) (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.025) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -1.0 / fma(-1.5, (a / b), (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.025)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-1.0 / fma(-1.5, Float64(a / b), Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.025], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.5 * N[(a / b), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.025000000000000001

   1. Initial program 79.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.4

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

   4. Applied egg-rr 79.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-/r* N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. unsub-neg N/A

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

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

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

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

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

      11. sub-neg N/A

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

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

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

      13. *-commutative N/A

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

      14. distribute-rgt-neg-in N/A

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

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

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

      16. *-commutative N/A

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

      17. distribute-rgt-neg-in N/A

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

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

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

      19. metadata-eval 79.4

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

   6. Applied egg-rr 79.4%

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

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.0%

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

   4. Applied egg-rr 49.0%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 87.8

         \[\leadsto \frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

   7. Simplified 87.8%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.025)
   (* (/ -0.3333333333333333 a) (- b (sqrt (fma a (* c -3.0) (* b b)))))
   (/ -1.0 (fma -1.5 (/ a b) (* 2.0 (/ b c))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.025) {
		tmp = (-0.3333333333333333 / a) * (b - sqrt(fma(a, (c * -3.0), (b * b))));
	} else {
		tmp = -1.0 / fma(-1.5, (a / b), (2.0 * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.025)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - sqrt(fma(a, Float64(c * -3.0), Float64(b * b)))));
	else
		tmp = Float64(-1.0 / fma(-1.5, Float64(a / b), Float64(2.0 * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.025], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(-1.5 * N[(a / b), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.025000000000000001

   1. Initial program 79.4%

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

   4. Applied egg-rr 79.3%

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

   if -0.025000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.0%

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

   4. Applied egg-rr 49.0%

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

   5. Taylor expanded in a around 0

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

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

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

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

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

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

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

      4. /-lowering-/.f64 87.8

         \[\leadsto \frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

   7. Simplified 87.8%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.025:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\left(a \cdot -3\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma c (* a -3.0) 0.0)
  (* (* a -3.0) (- (- b) (sqrt (fma c (* a -3.0) (* b b)))))))

C

double code(double a, double b, double c) {
	return fma(c, (a * -3.0), 0.0) / ((a * -3.0) * (-b - sqrt(fma(c, (a * -3.0), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(fma(c, Float64(a * -3.0), 0.0) / Float64(Float64(a * -3.0) * Float64(Float64(-b) - sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * -3.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(N[(a * -3.0), $MachinePrecision] * N[((-b) - N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

4. Applied egg-rr 57.6%

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

   1. associate-/r* N/A

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

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

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

6. Applied egg-rr 58.3%

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

   1. associate-/l/ N/A

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

   2. associate-*l* N/A

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

   3. frac-2neg N/A

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

   4. /-lowering-/.f64 N/A

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

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right) \cdot \left(a \cdot -3\right)}} \]

9. Final simplification 99.3%

   \[\leadsto \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\left(a \cdot -3\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \]
10. Add Preprocessing

Alternative 9: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot \left(3 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma c (* a -3.0) 0.0)
  (* a (* 3.0 (+ b (sqrt (fma c (* a -3.0) (* b b))))))))

C

double code(double a, double b, double c) {
	return fma(c, (a * -3.0), 0.0) / (a * (3.0 * (b + sqrt(fma(c, (a * -3.0), (b * b))))));
}

Julia

function code(a, b, c)
	return Float64(fma(c, Float64(a * -3.0), 0.0) / Float64(a * Float64(3.0 * Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b)))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(c * N[(a * -3.0), $MachinePrecision] + 0.0), $MachinePrecision] / N[(a * N[(3.0 * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

4. Applied egg-rr 57.6%

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

   1. associate-/r* N/A

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

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

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

6. Applied egg-rr 58.3%

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

   1. associate-/l/ N/A

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

   2. associate-*l* N/A

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

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

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

   4. associate--l+ N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

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

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

   8. +-inverses N/A

      \[\leadsto \frac{c \cdot \left(\mathsf{neg}\left(3 \cdot a\right)\right) + \color{blue}{0}}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)} \]

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

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, \mathsf{neg}\left(3 \cdot a\right), 0\right)}}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(c, \color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot a}, 0\right)}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \color{blue}{-3} \cdot a, 0\right)}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \color{blue}{a \cdot -3}, 0\right)}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(c, \color{blue}{a \cdot -3}, 0\right)}{\left(a \cdot 3\right) \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)} \]

   14. associate-*l* N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\color{blue}{a \cdot \left(3 \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)\right)}} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{\color{blue}{a \cdot \left(3 \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)\right)}} \]

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

       \[\leadsto \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot \color{blue}{\left(3 \cdot \left(b + \sqrt{c \cdot \left(a \cdot -3\right) + b \cdot b}\right)\right)}} \]

8. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right)}{a \cdot \left(3 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)\right)}} \]
9. Add Preprocessing

Alternative 10: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(c, a \cdot -3, 0\right) \cdot 0.3333333333333333}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (* (fma c (* a -3.0) 0.0) 0.3333333333333333)
  (* a (+ b (sqrt (fma c (* a -3.0) (* b b)))))))

C

double code(double a, double b, double c) {
	return (fma(c, (a * -3.0), 0.0) * 0.3333333333333333) / (a * (b + sqrt(fma(c, (a * -3.0), (b * b)))));
}

Julia

function code(a, b, c)
	return Float64(Float64(fma(c, Float64(a * -3.0), 0.0) * 0.3333333333333333) / Float64(a * Float64(b + sqrt(fma(c, Float64(a * -3.0), Float64(b * b))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(c * N[(a * -3.0), $MachinePrecision] + 0.0), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / N[(a * N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

4. Applied egg-rr 57.6%

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

   1. associate-/r* N/A

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

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

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

6. Applied egg-rr 58.3%

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

   1. associate-/r* N/A

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

   2. associate-/l/ N/A

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

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

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

8. Applied egg-rr 99.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, a \cdot -3, 0\right) \cdot 0.3333333333333333}{a \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)}\right)}} \]
9. Add Preprocessing

Alternative 11: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ -1.0 (fma -1.5 (/ a b) (* 2.0 (/ b c)))))

C

double code(double a, double b, double c) {
	return -1.0 / fma(-1.5, (a / b), (2.0 * (b / c)));
}

Julia

function code(a, b, c)
	return Float64(-1.0 / fma(-1.5, Float64(a / b), Float64(2.0 * Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(-1.0 / N[(-1.5 * N[(a / b), $MachinePrecision] + N[(2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

4. Applied egg-rr 56.8%

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

5. Taylor expanded in a around 0

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

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

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

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

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

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

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

   4. /-lowering-/.f64 81.4

      \[\leadsto \frac{-1}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \color{blue}{\frac{b}{c}}\right)} \]

7. Simplified 81.4%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{b}, 2 \cdot \frac{b}{c}\right)}} \]
8. Add Preprocessing

Alternative 12: 64.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in b around inf

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

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

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

   2. /-lowering-/.f64 63.5

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

5. Simplified 63.5%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))