Cubic critical, medium range

Percentage Accurate: 31.1% → 95.3%

Time: 11.0s

Alternatives: 6

Speedup: 23.2×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 31.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.0546875 \cdot \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (* -1.0546875 (/ (pow (* a c) 4.0) (* a (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-1.0546875 * (pow((a * c), 4.0) / (a * pow(b, 7.0))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-1.0546875d0) * (((a * c) ** 4.0d0) / (a * (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-1.0546875 * (Math.pow((a * c), 4.0) / (a * Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-1.0546875 * (math.pow((a * c), 4.0) / (a * math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-1.0546875 * Float64((Float64(a * c) ^ 4.0) / Float64(a * (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-1.0546875 * (((a * c) ^ 4.0) / (a * (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0546875 * N[(N[Power[N[(a * c), $MachinePrecision], 4.0], $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

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

2. Taylor expanded in b around inf 96.1%

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

3. Taylor expanded in c around 0 96.1%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]

5. Simplified 96.1%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}}\right)\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 95.5%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}\right)\right)}\right)\right) \]

   2. expm1-udef 95.4%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}\right)} - 1\right)}\right)\right) \]

   3. associate-*r/ 95.4%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{\frac{a \cdot {b}^{7}}{6.328125}}}\right)} - 1\right)\right)\right) \]

   4. div-inv 95.4%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(\frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{\left(a \cdot {b}^{7}\right) \cdot \frac{1}{6.328125}}}\right)} - 1\right)\right)\right) \]

   5. times-frac 95.4%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{1}{6.328125}}}\right)} - 1\right)\right)\right) \]

   6. metadata-eval 95.4%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \left(e^{\mathsf{log1p}\left(\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\color{blue}{0.1580246913580247}}\right)} - 1\right)\right)\right) \]

7. Applied egg-rr 95.4%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{0.1580246913580247}\right)} - 1\right)}\right)\right) \]
8. Step-by-step derivation

   1. expm1-def 95.5%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{0.1580246913580247}\right)\right)}\right)\right) \]

   2. expm1-log1p 96.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{a \cdot {b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{0.1580246913580247}}\right)\right) \]

   3. times-frac 96.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{\left(a \cdot {b}^{7}\right) \cdot 0.1580246913580247}}\right)\right) \]

   4. *-commutative 96.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{\color{blue}{0.1580246913580247 \cdot \left(a \cdot {b}^{7}\right)}}\right)\right) \]

   5. times-frac 96.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{\frac{-0.16666666666666666}{0.1580246913580247} \cdot \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}}\right)\right) \]

   6. metadata-eval 96.1%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-1.0546875} \cdot \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right) \]

9. Simplified 96.1%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-1.0546875 \cdot \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}}\right)\right) \]

10. Final simplification 96.1%

    \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.0546875 \cdot \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}}\right)\right) \]

Alternative 2: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

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

2. Taylor expanded in b around inf 94.8%

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

3. Final simplification 94.8%

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

Alternative 3: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -7e-5) 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 <= -7e-5) {
		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 <= (-7d-5)) 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 <= -7e-5) {
		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 <= -7e-5:
		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 <= -7e-5)
		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 <= -7e-5)
		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, -7e-5], 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)) < -6.9999999999999994e-5

   1. Initial program 66.5%

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

   if -6.9999999999999994e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 18.3%

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

   2. Taylor expanded in b around inf 91.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 4: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

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)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

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

2. Taylor expanded in b around inf 92.0%

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

3. Final simplification 92.0%

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

Alternative 5: 90.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((-1.125d0) * (((a * c) * (a * c)) * (b ** (-3.0d0)))) + ((-1.5d0) * (c * (a / b)))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.3%

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

2. Taylor expanded in b around inf 91.6%

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

   1. fma-def 91.6%

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

   2. associate-/l* 91.5%

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

   3. associate-/l* 91.5%

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

4. Simplified 91.5%

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

   1. associate-/l* 91.6%

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

   2. associate-/l* 91.6%

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

   3. fma-def 91.6%

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

   4. +-commutative 91.6%

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

   5. associate-/l* 91.6%

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

   6. associate-/l* 91.6%

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

   7. div-inv 91.6%

      \[\leadsto \frac{-1.125 \cdot \color{blue}{\left(\left({a}^{2} \cdot {c}^{2}\right) \cdot \frac{1}{{b}^{3}}\right)} + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

   8. pow-prod-down 91.6%

      \[\leadsto \frac{-1.125 \cdot \left(\color{blue}{{\left(a \cdot c\right)}^{2}} \cdot \frac{1}{{b}^{3}}\right) + -1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

   9. pow-flip 91.6%

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

   10. metadata-eval 91.6%

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

   11. associate-/l* 91.6%

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

   12. associate-/r/ 91.6%

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

6. Applied egg-rr 91.6%

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

   1. unpow2 91.6%

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

8. Applied egg-rr 91.6%

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

9. Final simplification 91.6%

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

Alternative 6: 81.4% 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 31.3%

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

2. Taylor expanded in b around inf 82.0%

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

3. Final simplification 82.0%

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

Reproduce

herbie shell --seed 2023334 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))