Cubic critical, wide range

Percentage Accurate: 18.0% → 97.8%

Time: 21.9s

Alternatives: 5

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: 18.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.1× speedup

Math

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

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
    -0.16666666666666666
    (* (/ (pow (* a c) 4.0) a) (/ 6.328125 (pow b 7.0)))
    (* (/ a (pow b 3.0)) (* (* c c) -0.375))))))

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(-0.16666666666666666, ((pow((a * c), 4.0) / a) * (6.328125 / pow(b, 7.0))), ((a / pow(b, 3.0)) * ((c * c) * -0.375)))));
}

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(-0.16666666666666666, Float64(Float64((Float64(a * c) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0))), Float64(Float64(a / (b ^ 3.0)) * Float64(Float64(c * c) * -0.375)))))
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[(-0.16666666666666666 * N[(N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2%

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

   1. neg-sub0 18.2%

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

   2. sqr-neg 18.2%

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

   3. associate-+l- 18.2%

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

   4. sub0-neg 18.2%

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

   5. neg-mul-1 18.2%

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

3. Simplified 18.3%

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

   1. div-inv 18.3%

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

   2. metadata-eval 18.3%

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

   3. *-commutative 18.3%

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

   4. expm1-log1p-u 18.2%

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

5. Applied egg-rr 18.2%

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

6. Taylor expanded in b around inf 97.1%

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

8. Simplified 97.1%

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

9. Final simplification 97.1%

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

Alternative 2: 97.0% 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 18.2%

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

   1. sqr-neg 18.2%

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

   2. sqr-neg 18.2%

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

   3. associate-*l* 18.2%

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

3. Simplified 18.2%

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

4. Taylor expanded in b around inf 96.2%

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

   1. fma-def 96.2%

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

   2. associate-/l* 96.2%

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

   3. unpow2 96.2%

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

   4. fma-def 96.2%

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

   5. associate-/l* 96.2%

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

   6. unpow2 96.2%

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

6. Simplified 96.2%

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

7. Final simplification 96.2%

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

Alternative 3: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2.25 \cdot \frac{a \cdot a}{b \cdot b}\\ \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{t_0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{t_0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 2.25 (/ (* a a) (* b b)))))
   (/
    (fma
     -0.5
     (+
      (/ (* c c) (/ b t_0))
      (/
       (pow c 3.0)
       (/ b (fma 1.5 (* (/ a b) (/ t_0 b)) (/ (* -3.0 (* a 0.0)) (* b b))))))
     (/ (* a (* c -1.5)) b))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = 2.25 * ((a * a) / (b * b));
	return fma(-0.5, (((c * c) / (b / t_0)) + (pow(c, 3.0) / (b / fma(1.5, ((a / b) * (t_0 / b)), ((-3.0 * (a * 0.0)) / (b * b)))))), ((a * (c * -1.5)) / b)) / (a * 3.0);
}

Julia

function code(a, b, c)
	t_0 = Float64(2.25 * Float64(Float64(a * a) / Float64(b * b)))
	return Float64(fma(-0.5, Float64(Float64(Float64(c * c) / Float64(b / t_0)) + Float64((c ^ 3.0) / Float64(b / fma(1.5, Float64(Float64(a / b) * Float64(t_0 / b)), Float64(Float64(-3.0 * Float64(a * 0.0)) / Float64(b * b)))))), Float64(Float64(a * Float64(c * -1.5)) / b)) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(2.25 * N[(N[(a * a), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(-0.5 * N[(N[(N[(c * c), $MachinePrecision] / N[(b / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[c, 3.0], $MachinePrecision] / N[(b / N[(1.5 * N[(N[(a / b), $MachinePrecision] * N[(t$95$0 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-3.0 * N[(a * 0.0), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(c * -1.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 18.2%

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

   1. sqr-neg 18.2%

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

   2. sqr-neg 18.2%

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

   3. associate-*l* 18.2%

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

3. Simplified 18.2%

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

   1. flip3-- 18.2%

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

   2. div-inv 18.3%

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

   3. pow2 18.3%

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

   4. pow-pow 18.3%

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

   5. metadata-eval 18.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{\color{blue}{6}} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   6. pow2 18.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   7. pow2 18.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   8. pow-prod-up 18.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   9. metadata-eval 18.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{{b}^{\color{blue}{4}} + \left(\left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right)\right)}}}{3 \cdot a} \]

   10. distribute-rgt-out 18.3%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\left({b}^{6} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \color{blue}{\left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + b \cdot b\right)}}}}{3 \cdot a} \]

5. Applied egg-rr 18.3%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left({b}^{6} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right) \cdot \frac{1}{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right) + b \cdot b\right)}}}}{3 \cdot a} \]

6. Taylor expanded in c around 0 95.5%

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b} + \left(-0.5 \cdot \frac{{c}^{2} \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2}}{{b}^{2}} + {\left(-1.5 \cdot \frac{a}{b}\right)}^{2}\right)\right)}{b} + -0.5 \cdot \frac{{c}^{3} \cdot \left(-3 \cdot \frac{a \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + 9 \cdot \frac{{a}^{2}}{{b}^{2}}\right)}{{b}^{2}} + 1.5 \cdot \frac{a \cdot \left(-9 \cdot \frac{{a}^{2}}{{b}^{2}} + \left(9 \cdot \frac{{a}^{2}}{{b}^{2}} + {\left(-1.5 \cdot \frac{a}{b}\right)}^{2}\right)\right)}{{b}^{2}}\right)}{b}\right)}}{3 \cdot a} \]

8. Simplified 95.6%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}}{3 \cdot a} \]

9. Taylor expanded in a around 0 95.6%

   \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{\color{blue}{2.25 \cdot \frac{{a}^{2}}{{b}^{2}}} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]
10. Step-by-step derivation

    1. unpow2 95.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{\color{blue}{a \cdot a}}{{b}^{2}} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]

    2. unpow2 95.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{a \cdot a}{\color{blue}{b \cdot b}} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]

11. Simplified 95.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{\color{blue}{2.25 \cdot \frac{a \cdot a}{b \cdot b}} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]

12. Taylor expanded in a around 0 95.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{\color{blue}{2.25 \cdot \frac{{a}^{2}}{{b}^{2}}} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{a \cdot a}{b \cdot b} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]
13. Step-by-step derivation

    1. unpow2 95.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{\color{blue}{a \cdot a}}{{b}^{2}} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]

    2. unpow2 95.6%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{a \cdot a}{\color{blue}{b \cdot b}} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]

14. Simplified 95.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{\color{blue}{2.25 \cdot \frac{a \cdot a}{b \cdot b}} + 0}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{a \cdot a}{b \cdot b} + 0}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{3 \cdot a} \]

15. Final simplification 95.6%

    \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{2.25 \cdot \frac{a \cdot a}{b \cdot b}}} + \frac{{c}^{3}}{\frac{b}{\mathsf{fma}\left(1.5, \frac{a}{b} \cdot \frac{2.25 \cdot \frac{a \cdot a}{b \cdot b}}{b}, \frac{-3 \cdot \left(a \cdot 0\right)}{b \cdot b}\right)}}, \frac{a \cdot \left(c \cdot -1.5\right)}{b}\right)}{a \cdot 3} \]

Alternative 4: 95.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (-0.375 * ((a / pow(b, 3.0)) * (c * c))) + (-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 = ((-0.375d0) * ((a / (b ** 3.0d0)) * (c * c))) + ((-0.5d0) * (c / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

   1. sqr-neg 18.2%

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

   2. sqr-neg 18.2%

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

   3. associate-*l* 18.2%

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

3. Simplified 18.2%

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

4. Taylor expanded in b around inf 94.5%

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

   1. +-commutative 94.5%

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

   2. fma-def 94.5%

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

   3. associate-/l* 94.5%

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

   4. associate-/r/ 94.5%

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

   5. unpow2 94.5%

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

6. Simplified 94.5%

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

   1. fma-udef 94.5%

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

   2. *-commutative 94.5%

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

8. Applied egg-rr 94.5%

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

9. Final simplification 94.5%

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

Alternative 5: 90.3% 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 18.2%

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

   1. sqr-neg 18.2%

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

   2. sqr-neg 18.2%

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

   3. associate-*l* 18.2%

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

3. Simplified 18.2%

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

4. Taylor expanded in b around inf 89.8%

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

5. Final simplification 89.8%

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

Reproduce

herbie shell --seed 2023283 
(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)))