Cubic critical, wide range

Percentage Accurate: 18.2% → 97.7%

Time: 8.1s

Alternatives: 6

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: 18.2% 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: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.6%

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

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. *-commutative 97.8%

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

   3. unpow2 97.8%

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

   4. fma-def 97.8%

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

   5. fma-def 97.8%

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

4. Simplified 97.8%

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

5. Taylor expanded in c around 0 97.8%

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

   1. distribute-rgt-out 97.8%

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

   2. associate-*r* 97.8%

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

   3. *-commutative 97.8%

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

   4. times-frac 97.8%

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

7. Simplified 97.8%

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

   1. fma-udef 97.8%

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

   2. associate-/r/ 97.8%

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

   3. *-commutative 97.8%

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

9. Applied egg-rr 97.8%

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

10. Final simplification 97.8%

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

Alternative 2: 96.9% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5625
  (/ (* (pow c 3.0) (* a a)) (pow b 5.0))
  (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, ((pow(c, 3.0) * (a * a)) / pow(b, 5.0)), 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((c ^ 3.0) * Float64(a * a)) / (b ^ 5.0)), 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[(N[Power[c, 3.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $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 19.6%

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

2. Taylor expanded in b around inf 96.5%

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

   1. fma-def 96.5%

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

   2. *-commutative 96.5%

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

   3. unpow2 96.5%

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

   4. fma-def 96.5%

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

   5. associate-/l* 96.5%

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

   6. unpow2 96.5%

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

4. Simplified 96.5%

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

5. Final simplification 96.5%

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

Alternative 3: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -5e-10) t_0 (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -5e-10) {
		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 * (3.0d0 * a)))) - b) / (3.0d0 * a)
    if (t_0 <= (-5d-10)) 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 * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -5e-10) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -5e-10:
		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(3.0 * a)))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -5e-10)
		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 * (3.0 * a)))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -5e-10)
		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[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-10], 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)) < -5.00000000000000031e-10

   1. Initial program 66.3%

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

   if -5.00000000000000031e-10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 8.8%

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

   2. Taylor expanded in b around inf 96.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 95.1% 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 19.6%

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

2. Taylor expanded in b around inf 94.2%

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

   1. fma-def 94.2%

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

   2. associate-*r/ 94.2%

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

   3. associate-*r* 94.2%

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

   4. unpow2 94.2%

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

4. Simplified 94.2%

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

5. Final simplification 94.2%

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

Alternative 5: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1.5 \cdot \frac{c \cdot a}{b} + \frac{-1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (+ (* -1.5 (/ (* c a) b)) (/ (* -1.125 (* (* c a) (* c a))) (pow b 3.0)))
  (* 3.0 a)))

C

double code(double a, double b, double c) {
	return ((-1.5 * ((c * a) / b)) + ((-1.125 * ((c * a) * (c * a))) / pow(b, 3.0))) / (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 = (((-1.5d0) * ((c * a) / b)) + (((-1.125d0) * ((c * a) * (c * a))) / (b ** 3.0d0))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return ((-1.5 * ((c * a) / b)) + ((-1.125 * ((c * a) * (c * a))) / Math.pow(b, 3.0))) / (3.0 * a);
}

Python

def code(a, b, c):
	return ((-1.5 * ((c * a) / b)) + ((-1.125 * ((c * a) * (c * a))) / math.pow(b, 3.0))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-1.5 * Float64(Float64(c * a) / b)) + Float64(Float64(-1.125 * Float64(Float64(c * a) * Float64(c * a))) / (b ^ 3.0))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = ((-1.5 * ((c * a) / b)) + ((-1.125 * ((c * a) * (c * a))) / (b ^ 3.0))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.125 * N[(N[(c * a), $MachinePrecision] * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.6%

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

2. Taylor expanded in b around inf 93.5%

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

   1. associate-*r/ 93.5%

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

   2. pow-prod-down 93.5%

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

4. Applied egg-rr 93.5%

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

   1. unpow2 93.5%

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

6. Applied egg-rr 93.5%

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

7. Final simplification 93.5%

   \[\leadsto \frac{-1.5 \cdot \frac{c \cdot a}{b} + \frac{-1.125 \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot a\right)\right)}{{b}^{3}}}{3 \cdot a} \]

Alternative 6: 90.1% 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 19.6%

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

2. Taylor expanded in b around inf 89.2%

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

3. Final simplification 89.2%

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

Reproduce

herbie shell --seed 2023271 
(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)))