Cubic critical, medium range

Percentage Accurate: 31.4% → 95.7%

Time: 14.0s

Alternatives: 6

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: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666, \frac{{c}^{4} \cdot 6.328125}{\frac{{b}^{7}}{{a}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, a \cdot \frac{c}{\frac{{b}^{3}}{c}}, -0.5625 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.16666666666666666
  (/ (* (pow c 4.0) 6.328125) (/ (pow b 7.0) (pow a 3.0)))
  (fma
   -0.5
   (/ c b)
   (fma
    -0.375
    (* a (/ c (/ (pow b 3.0) c)))
    (* -0.5625 (* (/ (pow c 3.0) (pow b 5.0)) (* a a)))))))

C

double code(double a, double b, double c) {
	return fma(-0.16666666666666666, ((pow(c, 4.0) * 6.328125) / (pow(b, 7.0) / pow(a, 3.0))), fma(-0.5, (c / b), fma(-0.375, (a * (c / (pow(b, 3.0) / c))), (-0.5625 * ((pow(c, 3.0) / pow(b, 5.0)) * (a * a))))));
}

Julia

function code(a, b, c)
	return fma(-0.16666666666666666, Float64(Float64((c ^ 4.0) * 6.328125) / Float64((b ^ 7.0) / (a ^ 3.0))), fma(-0.5, Float64(c / b), fma(-0.375, Float64(a * Float64(c / Float64((b ^ 3.0) / c))), Float64(-0.5625 * Float64(Float64((c ^ 3.0) / (b ^ 5.0)) * Float64(a * a))))))
end

Wolfram

code[a_, b_, c_] := N[(-0.16666666666666666 * N[(N[(N[Power[c, 4.0], $MachinePrecision] * 6.328125), $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / N[Power[a, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(a * N[(c / N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5625 * N[(N[(N[Power[c, 3.0], $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.7%

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

2. Taylor expanded in b around 0 30.7%

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

   1. div-inv 30.7%

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

   2. +-commutative 30.7%

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

   3. unpow2 30.7%

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

   4. associate-*r* 30.7%

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

   5. *-commutative 30.7%

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

   6. *-commutative 30.7%

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

   7. fma-udef 30.8%

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

   8. sub-neg 30.8%

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

   9. *-commutative 30.8%

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

4. Applied egg-rr 30.8%

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

5. Taylor expanded in a around 0 95.5%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot \frac{{a}^{3} \cdot \left(5.0625 \cdot \frac{{c}^{4}}{{b}^{6}} + {\left(-1.125 \cdot \frac{{c}^{2}}{{b}^{3}}\right)}^{2}\right)}{b} + \left(-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

7. Simplified 95.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{a}^{3}}{b} \cdot \mathsf{fma}\left(5.0625, \frac{{c}^{4}}{{b}^{6}}, \frac{{c}^{4}}{{b}^{6}} \cdot 1.265625\right), \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, -0.5625 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right)\right)\right)} \]

8. Taylor expanded in b around 0 95.5%

   \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{\left(5.0625 \cdot {c}^{4} + 1.265625 \cdot {c}^{4}\right) \cdot {a}^{3}}{{b}^{7}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, -0.5625 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. associate-/l* 95.5%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\frac{5.0625 \cdot {c}^{4} + 1.265625 \cdot {c}^{4}}{\frac{{b}^{7}}{{a}^{3}}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, -0.5625 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

   2. distribute-rgt-out 95.5%

      \[\leadsto \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{{c}^{4} \cdot \left(5.0625 + 1.265625\right)}}{\frac{{b}^{7}}{{a}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{c}{\frac{{b}^{3}}{c}} \cdot a, -0.5625 \cdot \left(\frac{{c}^{3}}{{b}^{5}} \cdot \left(a \cdot a\right)\right)\right)\right)\right) \]

   3. metadata-eval 95.5%

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

10. Simplified 95.5%

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

11. Final simplification 95.5%

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

Alternative 2: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\\ \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{t_0 \cdot t_0 + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{{b}^{7} \cdot a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* -1.125 (* c (* c (* a a))))))
   (fma
    -0.5625
    (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
    (fma
     -0.16666666666666666
     (/ (+ (* t_0 t_0) (* 5.0625 (pow (* c a) 4.0))) (* (pow b 7.0) a))
     (fma -0.5 (/ c b) (* -0.375 (/ (* c c) (/ (pow b 3.0) a))))))))

C

double code(double a, double b, double c) {
	double t_0 = -1.125 * (c * (c * (a * a)));
	return fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.16666666666666666, (((t_0 * t_0) + (5.0625 * pow((c * a), 4.0))) / (pow(b, 7.0) * a)), fma(-0.5, (c / b), (-0.375 * ((c * c) / (pow(b, 3.0) / a))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(-1.125 * Float64(c * Float64(c * Float64(a * a))))
	return fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.16666666666666666, Float64(Float64(Float64(t_0 * t_0) + Float64(5.0625 * (Float64(c * a) ^ 4.0))) / Float64((b ^ 7.0) * a)), fma(-0.5, Float64(c / b), Float64(-0.375 * Float64(Float64(c * c) / Float64((b ^ 3.0) / a))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(-1.125 * N[(c * N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(5.0625 * N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.375 * N[(N[(c * c), $MachinePrecision] / N[(N[Power[b, 3.0], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.7%

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

2. Taylor expanded in b around inf 95.5%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}} + \left(-0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]
3. Step-by-step derivation

   1. fma-def 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3} \cdot {a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right)} \]

   2. associate-/l* 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \color{blue}{\frac{{c}^{3}}{\frac{{b}^{5}}{{a}^{2}}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   3. unpow2 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{\color{blue}{a \cdot a}}}, -0.16666666666666666 \cdot \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)\right) \]

   4. fma-def 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left({c}^{2} \cdot {a}^{2}\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{{c}^{2} \cdot a}{{b}^{3}}\right)}\right) \]

4. Simplified 95.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left({c}^{4} \cdot {a}^{4}\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({c}^{4} \cdot {a}^{4}\right)\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   2. expm1-udef 94.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({c}^{4} \cdot {a}^{4}\right)} - 1\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   3. pow-prod-down 94.8%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(c \cdot a\right)}^{4}}\right)} - 1\right)}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

6. Applied egg-rr 94.8%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(c \cdot a\right)}^{4}\right)} - 1\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
7. Step-by-step derivation

   1. expm1-def 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(c \cdot a\right)}^{4}\right)\right)}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   2. expm1-log1p 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{{\left(c \cdot a\right)}^{4}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

8. Simplified 95.5%

   \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)}^{2} + 5.0625 \cdot \color{blue}{{\left(c \cdot a\right)}^{4}}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]
9. Step-by-step derivation

   1. unpow2 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right)} + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   2. associate-*l* 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(-1.125 \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}\right) \cdot \left(-1.125 \cdot \left(\left(c \cdot c\right) \cdot \left(a \cdot a\right)\right)\right) + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

   3. associate-*l* 95.5%

      \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(-1.125 \cdot \color{blue}{\left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)}\right) + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

10. Applied egg-rr 95.5%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\color{blue}{\left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right)} + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

11. Final simplification 95.5%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.16666666666666666, \frac{\left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(-1.125 \cdot \left(c \cdot \left(c \cdot \left(a \cdot a\right)\right)\right)\right) + 5.0625 \cdot {\left(c \cdot a\right)}^{4}}{{b}^{7} \cdot a}, \mathsf{fma}\left(-0.5, \frac{c}{b}, -0.375 \cdot \frac{c \cdot c}{\frac{{b}^{3}}{a}}\right)\right)\right) \]

Alternative 3: 94.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

2. Taylor expanded in b around inf 93.8%

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

   1. fma-def 93.8%

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

   2. associate-/l* 93.8%

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

   3. unpow2 93.8%

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

   4. fma-def 93.8%

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

   5. associate-/l* 93.8%

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

   6. associate-*r/ 93.8%

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

   7. unpow2 93.8%

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

4. Simplified 93.8%

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

5. Final simplification 93.8%

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

Alternative 4: 94.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.7%

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

2. Taylor expanded in b around 0 30.7%

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

   1. div-inv 30.7%

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

   2. +-commutative 30.7%

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

   3. unpow2 30.7%

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

   4. associate-*r* 30.7%

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

   5. *-commutative 30.7%

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

   6. *-commutative 30.7%

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

   7. fma-udef 30.8%

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

   8. sub-neg 30.8%

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

   9. *-commutative 30.8%

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

4. Applied egg-rr 30.8%

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

5. Taylor expanded in b around inf 93.8%

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

   1. +-commutative 93.8%

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

   2. associate-+l+ 93.8%

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

   3. fma-def 93.8%

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

   4. fma-def 93.8%

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

   5. associate-/l* 93.8%

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

   6. associate-/r/ 93.8%

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

   7. unpow2 93.8%

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

   8. associate-/l* 93.8%

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

   9. associate-/l* 93.8%

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

   10. associate-/r/ 93.8%

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

   11. unpow2 93.8%

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

7. Simplified 93.8%

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

8. Final simplification 93.8%

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

Alternative 5: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + \frac{a \cdot -0.375}{\frac{{b}^{3}}{c \cdot c}} \end{array} \]

FPCore

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

C

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

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)) + ((a * (-0.375d0)) / ((b ** 3.0d0) / (c * c)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + ((a * -0.375) / (Math.pow(b, 3.0) / (c * c)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + ((a * -0.375) / (math.pow(b, 3.0) / (c * c)))

Julia

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

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + ((a * -0.375) / ((b ^ 3.0) / (c * c)));
end

Wolfram

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

Derivation

1. Initial program 30.7%

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

2. Taylor expanded in b around 0 30.7%

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

3. Taylor expanded in b around inf 90.6%

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

   1. fma-def 90.6%

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

   2. associate-*r/ 90.6%

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

   3. *-commutative 90.6%

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

   4. associate-*r* 90.6%

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

   5. unpow2 90.6%

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

5. Simplified 90.6%

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

   1. fma-udef 90.6%

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

   2. associate-/l* 90.6%

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

   3. *-commutative 90.6%

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

7. Applied egg-rr 90.6%

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

8. Final simplification 90.6%

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

Alternative 6: 81.3% 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 30.7%

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

2. Taylor expanded in b around inf 81.7%

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

3. Final simplification 81.7%

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

Reproduce

herbie shell --seed 2023279 
(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)))