Cubic critical, wide range

Percentage Accurate: 17.9% → 97.8%

Time: 22.9s

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: 17.9% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

   1. neg-sub0 19.9%

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

   2. sqr-neg 19.9%

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

   3. associate-+l- 19.9%

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

   4. sub0-neg 19.9%

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

   5. neg-mul-1 19.9%

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

3. Simplified 19.9%

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

   1. div-inv 20.0%

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

   2. metadata-eval 20.0%

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

   3. *-commutative 20.0%

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

   4. add-sqr-sqrt 19.9%

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

   5. pow2 19.9%

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

5. Applied egg-rr 19.9%

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

6. Taylor expanded in b around inf 96.8%

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

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

Alternative 2: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

   1. neg-sub0 19.9%

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

   2. sqr-neg 19.9%

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

   3. associate-+l- 19.9%

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

   4. sub0-neg 19.9%

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

   5. neg-mul-1 19.9%

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

3. Simplified 19.9%

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

   1. div-inv 20.0%

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

   2. metadata-eval 20.0%

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

   3. *-commutative 20.0%

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

   4. add-sqr-sqrt 19.9%

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

   5. pow2 19.9%

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

5. Applied egg-rr 19.9%

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

6. Taylor expanded in b around inf 96.0%

   \[\leadsto \color{blue}{-1.6875 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5} \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.5 \cdot \frac{c}{b \cdot {\left(\sqrt{3}\right)}^{2}} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3} \cdot {\left(\sqrt{3}\right)}^{2}}\right)} \]
7. Step-by-step derivation

   1. +-commutative 96.0%

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

8. Simplified 97.2%

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

9. Final simplification 97.2%

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

Alternative 3: 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 19.9%

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

2. Taylor expanded in b around inf 95.3%

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

   1. fma-def 95.3%

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

   2. *-commutative 95.3%

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

   3. associate-/l* 95.3%

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

   4. associate-*l/ 95.3%

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

   5. unpow2 95.3%

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

4. Simplified 95.3%

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

5. Final simplification 95.3%

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

Alternative 4: 95.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.375d0) * ((c * c) * (a / (b ** 3.0d0)))) + ((-0.5d0) / (b / c))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.9%

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

2. Taylor expanded in b around inf 94.7%

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

3. Taylor expanded in a around 0 95.3%

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

   1. +-commutative 95.3%

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

   2. fma-def 95.3%

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

   3. associate-/l* 95.3%

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

   4. unpow2 95.3%

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

   5. associate-*r/ 95.3%

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

5. Simplified 95.3%

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

   1. fma-udef 95.3%

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

   2. associate-/r/ 95.3%

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

   3. associate-/l* 94.9%

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

7. Applied egg-rr 94.9%

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

8. Final simplification 94.9%

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

Alternative 5: 90.5% 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 19.9%

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

2. Taylor expanded in b around inf 89.0%

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

3. Final simplification 89.0%

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

Reproduce

herbie shell --seed 2023279 
(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)))