Cubic critical, medium range

Percentage Accurate: 31.4% → 95.3%

Time: 16.7s

Alternatives: 5

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.4% 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: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{\frac{-0.16666666666666666}{{b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{a \cdot 0.1580246913580247} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (fma
   -0.5625
   (* (/ a (/ (pow b 5.0) a)) (pow c 3.0))
   (fma -0.5 (/ c b) (/ -0.375 (/ (/ (pow b 3.0) (* c c)) a))))
  (/
   (* (/ -0.16666666666666666 (pow b 7.0)) (pow (* a c) 4.0))
   (* a 0.1580246913580247))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((a / (pow(b, 5.0) / a)) * pow(c, 3.0)), fma(-0.5, (c / b), (-0.375 / ((pow(b, 3.0) / (c * c)) / a)))) + (((-0.16666666666666666 / pow(b, 7.0)) * pow((a * c), 4.0)) / (a * 0.1580246913580247));
}

Julia

function code(a, b, c)
	return Float64(fma(-0.5625, Float64(Float64(a / Float64((b ^ 5.0) / a)) * (c ^ 3.0)), fma(-0.5, Float64(c / b), Float64(-0.375 / Float64(Float64((b ^ 3.0) / Float64(c * c)) / a)))) + Float64(Float64(Float64(-0.16666666666666666 / (b ^ 7.0)) * (Float64(a * c) ^ 4.0)) / Float64(a * 0.1580246913580247)))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(a / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 / N[(N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-0.16666666666666666 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] / N[(a * 0.1580246913580247), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.6%

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

   1. neg-sub0 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-+l- 28.6%

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

   4. sub0-neg 28.6%

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

   5. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

   1. div-inv 28.6%

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

   2. metadata-eval 28.6%

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

   3. *-commutative 28.6%

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

   4. add-sqr-sqrt 28.6%

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

   5. pow2 28.6%

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

5. Applied egg-rr 28.6%

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

6. Taylor expanded in b around inf 95.8%

   \[\leadsto \color{blue}{-1.6875 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5} \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.5 \cdot \frac{c}{b \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3} \cdot {\left(\sqrt{3}\right)}^{2}} + -0.5 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot \left({b}^{7} \cdot {\left(\sqrt{3}\right)}^{2}\right)}\right)\right)} \]

8. Simplified 96.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{-0.5}{3 \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}}} \]
9. Step-by-step derivation

   1. associate-*r/ 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \color{blue}{\frac{\frac{-0.5}{3 \cdot {b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}}} \]

   2. associate-/r* 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{\color{blue}{\frac{\frac{-0.5}{3}}{{b}^{7}}} \cdot {\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}} \]

   3. metadata-eval 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{\frac{\color{blue}{-0.16666666666666666}}{{b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}} \]

   4. div-inv 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{\frac{-0.16666666666666666}{{b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{a \cdot \frac{1}{6.328125}}} \]

   5. metadata-eval 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{\frac{-0.16666666666666666}{{b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{a \cdot \color{blue}{0.1580246913580247}} \]

10. Applied egg-rr 96.8%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \color{blue}{\frac{\frac{-0.16666666666666666}{{b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{a \cdot 0.1580246913580247}} \]

11. Final simplification 96.8%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a}{\frac{{b}^{5}}{a}} \cdot {c}^{3}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \frac{-0.375}{\frac{\frac{{b}^{3}}{c \cdot c}}{a}}\right)\right) + \frac{\frac{-0.16666666666666666}{{b}^{7}} \cdot {\left(a \cdot c\right)}^{4}}{a \cdot 0.1580246913580247} \]

Alternative 2: 93.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{-0.5}{\frac{b}{c}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (* a (/ (pow c 3.0) (/ (pow b 5.0) a)))
  (fma -0.375 (/ a (/ (/ (pow b 3.0) c) c)) (/ -0.5 (/ b c)))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, (a * (pow(c, 3.0) / (pow(b, 5.0) / a))), fma(-0.375, (a / ((pow(b, 3.0) / c) / c)), (-0.5 / (b / c))));
}

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(a * Float64((c ^ 3.0) / Float64((b ^ 5.0) / a))), fma(-0.375, Float64(a / Float64(Float64((b ^ 3.0) / c) / c)), Float64(-0.5 / Float64(b / c))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(a * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.6%

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

   1. neg-sub0 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-+l- 28.6%

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

   4. sub0-neg 28.6%

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

   5. neg-mul-1 28.6%

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

3. Simplified 28.6%

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

   1. clear-num 28.6%

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

   2. inv-pow 28.6%

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

5. Applied egg-rr 28.6%

   \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{{\left(\frac{0.3333333333333333}{a}\right)}^{-1}}} \]
6. Step-by-step derivation

   1. div-sub 28.1%

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

   2. unpow-1 28.1%

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

   3. unpow-1 28.1%

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

7. Applied egg-rr 28.1%

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

8. Taylor expanded in b around inf 95.7%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]
9. Step-by-step derivation

   1. fma-def 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

   2. *-commutative 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{\color{blue}{{c}^{3} \cdot {a}^{2}}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   3. unpow2 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot \color{blue}{\left(a \cdot a\right)}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   4. associate-*r* 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{\color{blue}{\left({c}^{3} \cdot a\right) \cdot a}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   5. *-commutative 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{\color{blue}{\left(a \cdot {c}^{3}\right)} \cdot a}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   6. associate-*l/ 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{a \cdot {c}^{3}}{{b}^{5}} \cdot a}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   7. *-commutative 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{a \cdot \frac{a \cdot {c}^{3}}{{b}^{5}}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   8. *-commutative 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{\color{blue}{{c}^{3} \cdot a}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   9. associate-/l* 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{a}}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   10. +-commutative 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}}\right) \]

   11. fma-def 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)}\right) \]

   12. associate-/l* 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}, -0.5 \cdot \frac{c}{b}\right)\right) \]

   13. unpow2 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}, -0.5 \cdot \frac{c}{b}\right)\right) \]

   14. associate-/r* 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\color{blue}{\frac{\frac{{b}^{3}}{c}}{c}}}, -0.5 \cdot \frac{c}{b}\right)\right) \]

   15. associate-*r/ 95.7%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{-0.5 \cdot c}{b}}\right)\right) \]

   16. associate-/l* 95.3%

       \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{-0.5}{\frac{b}{c}}}\right)\right) \]

10. Simplified 95.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{-0.5}{\frac{b}{c}}\right)\right)} \]

11. Final simplification 95.3%

    \[\leadsto \mathsf{fma}\left(-0.5625, a \cdot \frac{{c}^{3}}{\frac{{b}^{5}}{a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{-0.5}{\frac{b}{c}}\right)\right) \]

Alternative 3: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (/ (* a a) (/ (pow b 5.0) (pow c 3.0)))
  (fma -0.5 (/ c b) (* -0.375 (/ a (/ (pow b 3.0) (* c c)))))))

C

double code(double a, double b, double c) {
	return fma(-0.5625, ((a * a) / (pow(b, 5.0) / pow(c, 3.0))), fma(-0.5, (c / b), (-0.375 * (a / (pow(b, 3.0) / (c * c))))));
}

Julia

function code(a, b, c)
	return fma(-0.5625, Float64(Float64(a * a) / Float64((b ^ 5.0) / (c ^ 3.0))), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(a / Float64((b ^ 3.0) / Float64(c * c))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5625 * N[(N[(a * a), $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.6%

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

   1. sqr-neg 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-*l* 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 95.7%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]
5. Step-by-step derivation

   1. fma-def 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]

   2. associate-/l* 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{a}^{2}}{\frac{{b}^{5}}{{c}^{3}}}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   3. unpow2 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{\color{blue}{a \cdot a}}{\frac{{b}^{5}}{{c}^{3}}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \]

   4. fma-def 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right) \]

   5. associate-/l* 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}\right)\right) \]

   6. unpow2 95.7%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}\right)\right) \]

6. Simplified 95.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right)} \]

7. Final simplification 95.7%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{a}{\frac{{b}^{3}}{c \cdot c}}\right)\right) \]

Alternative 4: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.375, \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma -0.375 (* (* c c) (/ a (pow b 3.0))) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	return fma(-0.375, ((c * c) * (a / pow(b, 3.0))), (-0.5 * (c / b)));
}

Julia

function code(a, b, c)
	return fma(-0.375, Float64(Float64(c * c) * Float64(a / (b ^ 3.0))), Float64(-0.5 * Float64(c / b)))
end

Wolfram

code[a_, b_, c_] := N[(-0.375 * N[(N[(c * c), $MachinePrecision] * N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 28.6%

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

   1. sqr-neg 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-*l* 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 93.0%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
5. Step-by-step derivation

   1. +-commutative 93.0%

      \[\leadsto \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

   2. fma-def 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

   3. associate-/l* 93.0%

      \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}, -0.5 \cdot \frac{c}{b}\right) \]

   4. associate-/r/ 93.0%

      \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{{b}^{3}} \cdot {c}^{2}}, -0.5 \cdot \frac{c}{b}\right) \]

   5. unpow2 93.0%

      \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}} \cdot \color{blue}{\left(c \cdot c\right)}, -0.5 \cdot \frac{c}{b}\right) \]

6. Simplified 93.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}} \cdot \left(c \cdot c\right), -0.5 \cdot \frac{c}{b}\right)} \]

7. Final simplification 93.0%

   \[\leadsto \mathsf{fma}\left(-0.375, \left(c \cdot c\right) \cdot \frac{a}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right) \]

Alternative 5: 81.4% accurate, 23.2× 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 28.6%

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

   1. sqr-neg 28.6%

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

   2. sqr-neg 28.6%

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

   3. associate-*l* 28.6%

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

3. Simplified 28.6%

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

4. Taylor expanded in b around inf 83.8%

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

5. Final simplification 83.8%

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

Reproduce

herbie shell --seed 2023275 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))