Cubic critical, wide range

Percentage Accurate: 18.0% → 99.2%

Time: 15.7s

Alternatives: 7

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 18.0% 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: 99.2% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

4. Applied egg-rr 19.9%

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

   1. div-sub N/A

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

   2. frac-2neg N/A

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

   3. associate-/l/ N/A

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

   4. frac-2neg N/A

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

   5. associate-/l/ N/A

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

6. Applied egg-rr 20.2%

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

7. Taylor expanded in c around 0

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

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

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

   2. *-lowering-*.f64 99.2

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.375, \frac{c \cdot -0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma (/ (* a (* c c)) (* b (* b b))) -0.375 (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	return fma(((a * (c * c)) / (b * (b * b))), -0.375, ((c * -0.5) / b));
}

Julia

function code(a, b, c)
	return fma(Float64(Float64(a * Float64(c * c)) / Float64(b * Float64(b * b))), -0.375, Float64(Float64(c * -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}} + \frac{-1}{2} \cdot \frac{c}{b} \]

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a \cdot {c}^{2}}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

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

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{a \cdot {c}^{2}}}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

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

      \[\leadsto \mathsf{fma}\left(\frac{a \cdot \color{blue}{\left(c \cdot c\right)}}{{b}^{3}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   8. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{{b}^{2}}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{\color{blue}{b \cdot {b}^{2}}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

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

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-3}{8}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   13. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{-3}{8}, \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}}\right) \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{-3}{8}, \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, \frac{-3}{8}, \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b}\right) \]

   16. *-lowering-*.f64 94.4

       \[\leadsto \mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.375, \frac{\color{blue}{c \cdot -0.5}}{b}\right) \]

5. Simplified 94.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot b\right)}, -0.375, \frac{c \cdot -0.5}{b}\right)} \]
6. Add Preprocessing

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, -0.375 \cdot \frac{c \cdot c}{b \cdot b}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (* -0.375 (/ (* c c) (* b b))) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (-0.375 * ((c * c) / (b * b))), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(-0.375 * Float64(Float64(c * c) / Float64(b * b))), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in b around inf

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

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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Simplified 94.4%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, -0.375 \cdot \frac{c \cdot c}{b \cdot b}, c \cdot -0.5\right)}{b}} \]
6. Add Preprocessing

Alternative 4: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.7%

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

4. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{\frac{\left(b - \sqrt{\mathsf{fma}\left(a, -3 \cdot c, \mathsf{fma}\left(b, b, 0\right)\right)}\right) \cdot \frac{1}{a}}{-3}} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{-3}{\left(b - \sqrt{a \cdot \left(-3 \cdot c\right) + \left(b \cdot b + 0\right)}\right) \cdot \frac{1}{a}}}} \]

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

      \[\leadsto \color{blue}{\frac{1}{\frac{-3}{\left(b - \sqrt{a \cdot \left(-3 \cdot c\right) + \left(b \cdot b + 0\right)}\right) \cdot \frac{1}{a}}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{1}{\frac{-3}{\color{blue}{\frac{b - \sqrt{a \cdot \left(-3 \cdot c\right) + \left(b \cdot b + 0\right)}}{a}}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{-3}{b - \sqrt{a \cdot \left(-3 \cdot c\right) + \left(b \cdot b + 0\right)}} \cdot a}} \]

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

      \[\leadsto \frac{1}{\color{blue}{\frac{-3}{b - \sqrt{a \cdot \left(-3 \cdot c\right) + \left(b \cdot b + 0\right)}} \cdot a}} \]

6. Applied egg-rr 19.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{-3}{b - \sqrt{\mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(b, b, 0\right)\right)}} \cdot a}} \]

7. Taylor expanded in a around 0

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

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

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

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

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

   3. associate-*r/ N/A

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

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

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

   5. *-lowering-*.f64 94.3

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

9. Simplified 94.3%

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

10. Final simplification 94.3%

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

Alternative 5: 95.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (fma -0.375 (/ (* a c) (* b b)) -0.5) b)))

C

double code(double a, double b, double c) {
	return c * (fma(-0.375, ((a * c) / (b * b)), -0.5) / b);
}

Julia

function code(a, b, c)
	return Float64(c * Float64(fma(-0.375, Float64(Float64(a * c) / Float64(b * b)), -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-0.375 * N[(N[(a * c), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

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

3. Taylor expanded in c around 0

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

   1. sub-neg N/A

      \[\leadsto c \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   2. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot \left(a \cdot c\right)}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto c \cdot \left(\frac{\color{blue}{\left(\frac{-3}{8} \cdot a\right) \cdot c}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}} \cdot c} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   5. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right)} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

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

      \[\leadsto \color{blue}{c \cdot \left(\left(\frac{-3}{8} \cdot \frac{a}{{b}^{3}}\right) \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right)} \]

   7. associate-*r/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\frac{-3}{8} \cdot a}{{b}^{3}}} \cdot c + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   8. associate-*l/ N/A

      \[\leadsto c \cdot \left(\color{blue}{\frac{\left(\frac{-3}{8} \cdot a\right) \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto c \cdot \left(\frac{\color{blue}{\frac{-3}{8} \cdot \left(a \cdot c\right)}}{{b}^{3}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{3}}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto c \cdot \left(\color{blue}{\frac{a \cdot c}{{b}^{3}} \cdot \frac{-3}{8}} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   12. associate-/l* N/A

       \[\leadsto c \cdot \left(\color{blue}{\left(a \cdot \frac{c}{{b}^{3}}\right)} \cdot \frac{-3}{8} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto c \cdot \left(\color{blue}{a \cdot \left(\frac{c}{{b}^{3}} \cdot \frac{-3}{8}\right)} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)\right) \]

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

       \[\leadsto c \cdot \color{blue}{\mathsf{fma}\left(a, \frac{c}{{b}^{3}} \cdot \frac{-3}{8}, \mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{b}\right)\right)} \]

5. Simplified 94.1%

   \[\leadsto \color{blue}{c \cdot \mathsf{fma}\left(a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot -0.375, \frac{-0.5}{b}\right)} \]

6. Taylor expanded in b around inf

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

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

      \[\leadsto c \cdot \color{blue}{\frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} - \frac{1}{2}}{b}} \]

   2. sub-neg N/A

      \[\leadsto c \cdot \frac{\color{blue}{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{b} \]

   3. metadata-eval N/A

      \[\leadsto c \cdot \frac{\frac{-3}{8} \cdot \frac{a \cdot c}{{b}^{2}} + \color{blue}{\frac{-1}{2}}}{b} \]

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

      \[\leadsto c \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{{b}^{2}}, \frac{-1}{2}\right)}}{b} \]

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

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{-3}{8}, \color{blue}{\frac{a \cdot c}{{b}^{2}}}, \frac{-1}{2}\right)}{b} \]

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

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{\color{blue}{a \cdot c}}{{b}^{2}}, \frac{-1}{2}\right)}{b} \]

   7. unpow2 N/A

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(\frac{-3}{8}, \frac{a \cdot c}{\color{blue}{b \cdot b}}, \frac{-1}{2}\right)}{b} \]

   8. *-lowering-*.f64 94.1

      \[\leadsto c \cdot \frac{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{\color{blue}{b \cdot b}}, -0.5\right)}{b} \]

8. Simplified 94.1%

   \[\leadsto c \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.375, \frac{a \cdot c}{b \cdot b}, -0.5\right)}{b}} \]
9. Add Preprocessing

Alternative 6: 90.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 = (c * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 89.0

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

5. Simplified 89.0%

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

Alternative 7: 90.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 89.0

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

5. Simplified 89.0%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

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

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

   4. /-lowering-/.f64 88.7

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

7. Applied egg-rr 88.7%

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

8. Final simplification 88.7%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))