Cubic critical, wide range

Percentage Accurate: 17.5% → 97.8%

Time: 22.0s

Alternatives: 9

Speedup: 23.2×

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: 17.5% 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: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{{\left(a \cdot c\right)}^{4}}{a}}{\frac{{b}^{7}}{-1.0546875}}\right)\right)\right) \end{array} \]

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)
   (fma
    -0.375
    (/ a (/ (pow b 3.0) (* c c)))
    (/ (/ (pow (* a c) 4.0) a) (/ (pow b 7.0) -1.0546875))))))

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), fma(-0.375, (a / (pow(b, 3.0) / (c * c))), ((pow((a * c), 4.0) / a) / (pow(b, 7.0) / -1.0546875)))));
}

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), fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(Float64((Float64(a * c) ^ 4.0) / a) / Float64((b ^ 7.0) / -1.0546875)))))
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] + N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] / N[(N[Power[b, 7.0], $MachinePrecision] / -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

2. Taylor expanded in b around inf 97.8%

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

   1. fma-def 97.8%

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

   2. associate-/l* 97.8%

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

   3. unpow2 97.8%

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

   4. fma-def 97.8%

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

   5. fma-def 97.8%

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

4. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.8%

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

7. Simplified 97.8%

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

   1. frac-times 97.8%

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

   2. div-inv 97.8%

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

   3. metadata-eval 97.8%

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

9. Applied egg-rr 97.8%

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

    1. associate-/r* 97.8%

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

    2. times-frac 97.8%

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

    3. metadata-eval 97.8%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \color{blue}{-1.0546875}}{{b}^{7}}\right)\right)\right) \]

    4. associate-/l* 97.8%

       \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{4}}{a}}{\frac{{b}^{7}}{-1.0546875}}}\right)\right)\right) \]

11. Simplified 97.8%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \color{blue}{\frac{\frac{{\left(a \cdot c\right)}^{4}}{a}}{\frac{{b}^{7}}{-1.0546875}}}\right)\right)\right) \]

12. Final simplification 97.8%

    \[\leadsto \mathsf{fma}\left(-0.5625, \frac{a \cdot a}{\frac{{b}^{5}}{{c}^{3}}}, \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, \frac{\frac{{\left(a \cdot c\right)}^{4}}{a}}{\frac{{b}^{7}}{-1.0546875}}\right)\right)\right) \]

Alternative 2: 97.1% 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 18.6%

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

2. Taylor expanded in b around inf 97.1%

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

   1. fma-def 97.1%

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

   2. associate-/l* 97.1%

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

   3. unpow2 97.1%

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

   4. fma-def 97.1%

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

   5. associate-/l* 97.1%

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

   6. unpow2 97.1%

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

4. Simplified 97.1%

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

5. Final simplification 97.1%

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

Alternative 3: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6.5e-8)
   (* (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ 0.3333333333333333 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6.5e-8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) * (0.3333333333333333 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	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)) <= -6.5e-8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	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], -6.5e-8], 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6.49999999999999997e-8

   1. Initial program 69.6%

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

      1. neg-sub0 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-+l- 69.6%

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

      4. sub0-neg 69.6%

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

      5. neg-mul-1 69.6%

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

   3. Simplified 69.8%

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

      1. div-inv 69.8%

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

      2. metadata-eval 69.8%

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

      3. *-commutative 69.8%

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

      4. expm1-log1p-u 69.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]

   5. Applied egg-rr 69.6%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot a\right)\right)}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 69.8%

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

      2. div-sub 68.2%

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

      3. associate-*r* 68.2%

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

      4. *-commutative 68.2%

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

      5. *-commutative 68.2%

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

   7. Applied egg-rr 68.2%

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

      1. div-sub 69.8%

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

      2. *-lft-identity 69.8%

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

      3. associate-*l/ 69.7%

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

      4. *-commutative 69.7%

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

      5. associate-/r* 69.7%

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

      6. metadata-eval 69.7%

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

      7. associate-*l* 69.7%

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

   9. Simplified 69.7%

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

   if -6.49999999999999997e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.9%

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

   2. Taylor expanded in b around inf 96.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6.5e-8)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6.5e-8) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} else {
		tmp = -0.5 * (c / b);
	}
	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)) <= -6.5e-8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	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], -6.5e-8], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6.49999999999999997e-8

   1. Initial program 69.6%

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

      1. neg-sub0 69.6%

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

      2. sqr-neg 69.6%

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

      3. associate-+l- 69.6%

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

      4. sub0-neg 69.6%

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

      5. neg-mul-1 69.6%

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

   3. Simplified 69.8%

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

   if -6.49999999999999997e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.9%

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

   2. Taylor expanded in b around inf 96.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 5: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -6.5e-8) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -6.5e-8) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-6.5d-8)) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -6.5e-8) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -6.5e-8:
		tmp = t_0
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -6.5e-8)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -6.5e-8)
		tmp = t_0;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -6.5e-8], t$95$0, N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6.49999999999999997e-8

   1. Initial program 69.6%

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

   if -6.49999999999999997e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.9%

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

   2. Taylor expanded in b around inf 96.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 6: 91.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \frac{c}{0.3333333333333333}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -6.5e-8)
   (/ (- (sqrt (- (* b b) (* a (/ c 0.3333333333333333)))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6.5e-8) {
		tmp = (sqrt(((b * b) - (a * (c / 0.3333333333333333)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-6.5d-8)) then
        tmp = (sqrt(((b * b) - (a * (c / 0.3333333333333333d0)))) - b) / (a * 3.0d0)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6.5e-8) {
		tmp = (Math.sqrt(((b * b) - (a * (c / 0.3333333333333333)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6.5e-8:
		tmp = (math.sqrt(((b * b) - (a * (c / 0.3333333333333333)))) - b) / (a * 3.0)
	else:
		tmp = -0.5 * (c / b)
	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)) <= -6.5e-8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c / 0.3333333333333333)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -6.5e-8)
		tmp = (sqrt(((b * b) - (a * (c / 0.3333333333333333)))) - b) / (a * 3.0);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = 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], -6.5e-8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -6.49999999999999997e-8

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. metadata-eval 69.6%

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

      3. associate-/r/ 69.6%

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

      4. un-div-inv 69.6%

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

   3. Applied egg-rr 69.6%

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

      1. associate-/r/ 69.6%

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

   5. Simplified 69.6%

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

   if -6.49999999999999997e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.9%

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

   2. Taylor expanded in b around inf 96.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \frac{c}{0.3333333333333333}} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 95.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $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 18.6%

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

2. Taylor expanded in b around inf 95.4%

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

   1. fma-def 95.4%

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

   2. *-commutative 95.4%

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

   3. associate-/l* 95.4%

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

   4. associate-*l/ 95.4%

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

   5. unpow2 95.4%

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

4. Simplified 95.4%

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

5. Final simplification 95.4%

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

Alternative 8: 95.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

2. Taylor expanded in b around inf 94.8%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}}}{3 \cdot a} \]
3. Step-by-step derivation

   1. fma-def 94.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a \cdot c}{b}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}}{3 \cdot a} \]

   2. associate-/l* 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, \color{blue}{\frac{a}{\frac{b}{c}}}, -1.125 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{3}}\right)}{3 \cdot a} \]

   3. associate-/l* 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, -1.125 \cdot \color{blue}{\frac{{a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}\right)}{3 \cdot a} \]

   4. associate-*r/ 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, \color{blue}{\frac{-1.125 \cdot {a}^{2}}{\frac{{b}^{3}}{{c}^{2}}}}\right)}{3 \cdot a} \]

   5. unpow2 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, \frac{-1.125 \cdot \color{blue}{\left(a \cdot a\right)}}{\frac{{b}^{3}}{{c}^{2}}}\right)}{3 \cdot a} \]

   6. unpow2 94.8%

      \[\leadsto \frac{\mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, \frac{-1.125 \cdot \left(a \cdot a\right)}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}\right)}{3 \cdot a} \]

4. Simplified 94.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1.5, \frac{a}{\frac{b}{c}}, \frac{-1.125 \cdot \left(a \cdot a\right)}{\frac{{b}^{3}}{c \cdot c}}\right)}}{3 \cdot a} \]

5. Taylor expanded in a around 0 95.4%

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

   1. fma-def 95.4%

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

   2. associate-*r/ 95.4%

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

   3. unpow2 95.4%

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

   4. associate-*r* 95.4%

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

7. Simplified 95.4%

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

8. Final simplification 95.4%

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

Alternative 9: 90.6% 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 18.6%

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

2. Taylor expanded in b around inf 89.8%

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

3. Final simplification 89.8%

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

Reproduce

herbie shell --seed 2023275 
(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)))