Quadratic roots, medium range

Percentage Accurate: 31.0% → 95.7%

Time: 14.4s

Alternatives: 10

Speedup: 29.0×

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

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (pow(c, 4.0) * ((-5.0 * (pow(a, 3.0) / pow(b, 7.0))) - (((2.0 * (pow(a, 2.0) / pow(b, 5.0))) + (a / (c * pow(b, 3.0)))) / c))) - (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 = ((c ** 4.0d0) * (((-5.0d0) * ((a ** 3.0d0) / (b ** 7.0d0))) - (((2.0d0 * ((a ** 2.0d0) / (b ** 5.0d0))) + (a / (c * (b ** 3.0d0)))) / c))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 94.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around -inf 94.5%

   \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 94.5%

      \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

   2. distribute-frac-neg 94.5%

      \[\leadsto \color{blue}{\frac{-c}{b}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

8. Applied egg-rr 94.5%

   \[\leadsto \color{blue}{\frac{-c}{b}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

9. Final simplification 94.5%

   \[\leadsto {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} - \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{c \cdot {b}^{3}}}{c}\right) - \frac{c}{b} \]
10. Add Preprocessing

Alternative 2: 94.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (pow(c, 2.0) * (((-2.0 * (c * pow(a, 2.0))) / pow(b, 5.0)) - (a / pow(b, 3.0)))) - (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 = ((c ** 2.0d0) * ((((-2.0d0) * (c * (a ** 2.0d0))) / (b ** 5.0d0)) - (a / (b ** 3.0d0)))) - (c / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 94.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + a \cdot \left(-1 \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(-2 \cdot \frac{{c}^{3}}{{b}^{5}} + -0.25 \cdot \frac{a \cdot \left(4 \cdot \frac{{c}^{4}}{{b}^{6}} + 16 \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

6. Taylor expanded in c around -inf 94.5%

   \[\leadsto -1 \cdot \frac{c}{b} + \color{blue}{{c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 94.5%

      \[\leadsto \color{blue}{\left(-\frac{c}{b}\right)} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

   2. distribute-frac-neg 94.5%

      \[\leadsto \color{blue}{\frac{-c}{b}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

8. Applied egg-rr 94.5%

   \[\leadsto \color{blue}{\frac{-c}{b}} + {c}^{4} \cdot \left(-5 \cdot \frac{{a}^{3}}{{b}^{7}} + -1 \cdot \frac{2 \cdot \frac{{a}^{2}}{{b}^{5}} + \frac{a}{{b}^{3} \cdot c}}{c}\right) \]

9. Taylor expanded in c around 0 93.1%

   \[\leadsto \frac{-c}{b} + \color{blue}{{c}^{2} \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right)} \]
10. Step-by-step derivation

    1. mul-1-neg 93.1%

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

    2. unsub-neg 93.1%

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

    3. associate-*r/ 93.1%

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

    4. *-commutative 93.1%

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

11. Simplified 93.1%

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

12. Final simplification 93.1%

    \[\leadsto {c}^{2} \cdot \left(\frac{-2 \cdot \left(c \cdot {a}^{2}\right)}{{b}^{5}} - \frac{a}{{b}^{3}}\right) - \frac{c}{b} \]
13. Add Preprocessing

Alternative 3: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -350:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -350.0) t_0 (- (/ c (- b)) (* a (/ (pow c 2.0) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -350.0) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (pow(c, 2.0) / pow(b, 3.0)));
	}
	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 * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-350.0d0)) then
        tmp = t_0
    else
        tmp = (c / -b) - (a * ((c ** 2.0d0) / (b ** 3.0d0)))
    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 * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -350.0) {
		tmp = t_0;
	} else {
		tmp = (c / -b) - (a * (Math.pow(c, 2.0) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -350.0:
		tmp = t_0
	else:
		tmp = (c / -b) - (a * (math.pow(c, 2.0) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -350.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(c / Float64(-b)) - Float64(a * Float64((c ^ 2.0) / (b ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -350.0)
		tmp = t_0;
	else
		tmp = (c / -b) - (a * ((c ^ 2.0) / (b ^ 3.0)));
	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[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -350.0], t$95$0, N[(N[(c / (-b)), $MachinePrecision] - N[(a * N[(N[Power[c, 2.0], $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -350

   1. Initial program 82.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -350 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 25.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 25.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 25.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 93.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   6. Step-by-step derivation

      1. mul-1-neg 93.0%

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

      2. unsub-neg 93.0%

         \[\leadsto \color{blue}{-1 \cdot \frac{c}{b} - \frac{a \cdot {c}^{2}}{{b}^{3}}} \]

      3. mul-1-neg 93.0%

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

      4. distribute-neg-frac2 93.0%

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

      5. associate-/l* 93.0%

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

   7. Simplified 93.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -350:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b} - a \cdot \frac{{c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -350:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -350.0) t_0 (/ (fma a (pow (/ c b) 2.0) c) (- b)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -350.0) {
		tmp = t_0;
	} else {
		tmp = fma(a, pow((c / b), 2.0), c) / -b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -350.0)
		tmp = t_0;
	else
		tmp = Float64(fma(a, (Float64(c / b) ^ 2.0), c) / Float64(-b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -350.0], t$95$0, N[(N[(a * N[Power[N[(c / b), $MachinePrecision], 2.0], $MachinePrecision] + c), $MachinePrecision] / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -350

   1. Initial program 82.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -350 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 25.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 25.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 25.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 92.7%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 92.7%

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

      2. neg-mul-1 92.7%

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

      3. distribute-lft-neg-in 92.7%

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

   7. Simplified 92.7%

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

   8. Taylor expanded in b around inf 93.0%

      \[\leadsto \color{blue}{\frac{-1 \cdot c + -1 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
   9. Step-by-step derivation

      1. distribute-lft-out 93.0%

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

      2. mul-1-neg 93.0%

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

      3. distribute-neg-frac 93.0%

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

      4. distribute-neg-frac2 93.0%

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

      5. +-commutative 93.0%

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

      6. associate-/l* 93.0%

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

      7. fma-define 93.0%

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

      8. unpow2 93.0%

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

      9. unpow2 93.0%

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

      10. times-frac 93.0%

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

      11. unpow2 93.0%

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

   10. Simplified 93.0%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -350:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, {\left(\frac{c}{b}\right)}^{2}, c\right)}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{{\left(c \cdot a\right)}^{2} \cdot \frac{-2}{{b}^{2}} - c \cdot a}{{b}^{3}} + \frac{-1}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (*
  c
  (+
   (/ (- (* (pow (* c a) 2.0) (/ -2.0 (pow b 2.0))) (* c a)) (pow b 3.0))
   (/ -1.0 b))))

C

double code(double a, double b, double c) {
	return c * ((((pow((c * a), 2.0) * (-2.0 / pow(b, 2.0))) - (c * a)) / pow(b, 3.0)) + (-1.0 / 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 = c * ((((((c * a) ** 2.0d0) * ((-2.0d0) / (b ** 2.0d0))) - (c * a)) / (b ** 3.0d0)) + ((-1.0d0) / b))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((((Math.pow((c * a), 2.0) * (-2.0 / Math.pow(b, 2.0))) - (c * a)) / Math.pow(b, 3.0)) + (-1.0 / b));
}

Python

def code(a, b, c):
	return c * ((((math.pow((c * a), 2.0) * (-2.0 / math.pow(b, 2.0))) - (c * a)) / math.pow(b, 3.0)) + (-1.0 / b))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(Float64(Float64((Float64(c * a) ^ 2.0) * Float64(-2.0 / (b ^ 2.0))) - Float64(c * a)) / (b ^ 3.0)) + Float64(-1.0 / b)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((((((c * a) ^ 2.0) * (-2.0 / (b ^ 2.0))) - (c * a)) / (b ^ 3.0)) + (-1.0 / b));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(N[(N[(N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] * N[(-2.0 / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 92.8%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-2 \cdot \frac{{a}^{2} \cdot c}{{b}^{5}} + -1 \cdot \frac{a}{{b}^{3}}\right) - \frac{1}{b}\right)} \]

6. Taylor expanded in b around inf 92.8%

   \[\leadsto c \cdot \left(\color{blue}{\frac{-2 \cdot \frac{{a}^{2} \cdot {c}^{2}}{{b}^{2}} + -1 \cdot \left(a \cdot c\right)}{{b}^{3}}} - \frac{1}{b}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 92.8%

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

   2. *-commutative 92.8%

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

   3. unsub-neg 92.8%

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

   4. associate-*r/ 92.8%

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

   5. *-commutative 92.8%

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

   6. associate-/l* 92.8%

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

   7. *-commutative 92.8%

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

   8. unpow2 92.8%

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

   9. unpow2 92.8%

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

   10. swap-sqr 92.8%

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

   11. unpow1 92.8%

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

   12. pow-plus 92.8%

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

   13. metadata-eval 92.8%

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

8. Simplified 92.8%

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

9. Final simplification 92.8%

   \[\leadsto c \cdot \left(\frac{{\left(c \cdot a\right)}^{2} \cdot \frac{-2}{{b}^{2}} - c \cdot a}{{b}^{3}} + \frac{-1}{b}\right) \]
10. Add Preprocessing

Alternative 6: 91.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{if}\;t\_0 \leq -350:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))))
   (if (<= t_0 -350.0) t_0 (* c (- (/ -1.0 b) (* (* c a) (pow b -3.0)))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -350.0) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) * pow(b, -3.0)));
	}
	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 * (4.0d0 * a)))) - b) / (a * 2.0d0)
    if (t_0 <= (-350.0d0)) then
        tmp = t_0
    else
        tmp = c * (((-1.0d0) / b) - ((c * a) * (b ** (-3.0d0))))
    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 * (4.0 * a)))) - b) / (a * 2.0);
	double tmp;
	if (t_0 <= -350.0) {
		tmp = t_0;
	} else {
		tmp = c * ((-1.0 / b) - ((c * a) * Math.pow(b, -3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	tmp = 0
	if t_0 <= -350.0:
		tmp = t_0
	else:
		tmp = c * ((-1.0 / b) - ((c * a) * math.pow(b, -3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0))
	tmp = 0.0
	if (t_0 <= -350.0)
		tmp = t_0;
	else
		tmp = Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) * (b ^ -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	tmp = 0.0;
	if (t_0 <= -350.0)
		tmp = t_0;
	else
		tmp = c * ((-1.0 / b) - ((c * a) * (b ^ -3.0)));
	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[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -350.0], t$95$0, N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a)) < -350

   1. Initial program 82.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if -350 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 4 binary64) a) c)))) (*.f64 #s(literal 2 binary64) a))

   1. Initial program 25.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 25.8%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 25.8%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in c around 0 92.7%

      \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 92.7%

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

      2. neg-mul-1 92.7%

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

      3. distribute-lft-neg-in 92.7%

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

   7. Simplified 92.7%

      \[\leadsto \color{blue}{c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
   8. Step-by-step derivation

      1. pow1 92.7%

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

      2. associate-/l* 92.7%

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

      3. fmm-def 92.7%

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

      4. div-inv 92.7%

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

      5. pow-flip 92.7%

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

      6. metadata-eval 92.7%

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

   9. Applied egg-rr 92.7%

      \[\leadsto \color{blue}{{\left(c \cdot \mathsf{fma}\left(-a, c \cdot {b}^{-3}, -\frac{1}{b}\right)\right)}^{1}} \]
   10. Step-by-step derivation

       1. unpow1 92.7%

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

       2. fma-define 92.7%

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

       3. +-commutative 92.7%

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

       4. neg-mul-1 92.7%

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

       5. fma-define 92.7%

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

       6. associate-*r* 92.7%

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

       7. distribute-lft-neg-in 92.7%

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

       8. distribute-lft-neg-out 92.7%

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

       9. fmm-undef 92.7%

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

       10. associate-*r/ 92.7%

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

       11. metadata-eval 92.7%

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

   11. Simplified 92.7%

       \[\leadsto \color{blue}{c \cdot \left(\frac{-1}{b} - \left(a \cdot c\right) \cdot {b}^{-3}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2} \leq -350:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \frac{-1 - a \cdot \frac{c}{{b}^{2}}}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (/ (- -1.0 (* a (/ c (pow b 2.0)))) b)))

C

double code(double a, double b, double c) {
	return c * ((-1.0 - (a * (c / pow(b, 2.0)))) / 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 = c * (((-1.0d0) - (a * (c / (b ** 2.0d0)))) / b)
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 - (a * (c / Math.pow(b, 2.0)))) / b);
}

Python

def code(a, b, c):
	return c * ((-1.0 - (a * (c / math.pow(b, 2.0)))) / b)

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 - Float64(a * Float64(c / (b ^ 2.0)))) / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 - (a * (c / (b ^ 2.0)))) / b);
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 - N[(a * N[(c / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 89.6%

   \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 89.6%

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

   2. neg-mul-1 89.6%

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

   3. distribute-lft-neg-in 89.6%

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

7. Simplified 89.6%

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

8. Taylor expanded in b around inf 89.6%

   \[\leadsto c \cdot \color{blue}{\frac{-1 \cdot \frac{a \cdot c}{{b}^{2}} - 1}{b}} \]
9. Step-by-step derivation

   1. mul-1-neg 89.6%

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

   2. associate-/l* 89.6%

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

10. Applied egg-rr 89.6%

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

11. Final simplification 89.6%

    \[\leadsto c \cdot \frac{-1 - a \cdot \frac{c}{{b}^{2}}}{b} \]
12. Add Preprocessing

Alternative 8: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* c (- (/ -1.0 b) (* (* c a) (pow b -3.0)))))

C

double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) * 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 = c * (((-1.0d0) / b) - ((c * a) * (b ** (-3.0d0))))
end function

Java

public static double code(double a, double b, double c) {
	return c * ((-1.0 / b) - ((c * a) * Math.pow(b, -3.0)));
}

Python

def code(a, b, c):
	return c * ((-1.0 / b) - ((c * a) * math.pow(b, -3.0)))

Julia

function code(a, b, c)
	return Float64(c * Float64(Float64(-1.0 / b) - Float64(Float64(c * a) * (b ^ -3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c * ((-1.0 / b) - ((c * a) * (b ^ -3.0)));
end

Wolfram

code[a_, b_, c_] := N[(c * N[(N[(-1.0 / b), $MachinePrecision] - N[(N[(c * a), $MachinePrecision] * N[Power[b, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 89.6%

   \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 89.6%

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

   2. neg-mul-1 89.6%

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

   3. distribute-lft-neg-in 89.6%

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

7. Simplified 89.6%

   \[\leadsto \color{blue}{c \cdot \left(\frac{\left(-a\right) \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
8. Step-by-step derivation

   1. pow1 89.6%

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

   2. associate-/l* 89.6%

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

   3. fmm-def 89.6%

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

   4. div-inv 89.6%

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

   5. pow-flip 89.6%

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

   6. metadata-eval 89.6%

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

9. Applied egg-rr 89.6%

   \[\leadsto \color{blue}{{\left(c \cdot \mathsf{fma}\left(-a, c \cdot {b}^{-3}, -\frac{1}{b}\right)\right)}^{1}} \]
10. Step-by-step derivation

    1. unpow1 89.6%

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

    2. fma-define 89.6%

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

    3. +-commutative 89.6%

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

    4. neg-mul-1 89.6%

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

    5. fma-define 89.6%

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

    6. associate-*r* 89.6%

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

    7. distribute-lft-neg-in 89.6%

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

    8. distribute-lft-neg-out 89.6%

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

    9. fmm-undef 89.6%

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

    10. associate-*r/ 89.6%

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

    11. metadata-eval 89.6%

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

11. Simplified 89.6%

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

12. Final simplification 89.6%

    \[\leadsto c \cdot \left(\frac{-1}{b} - \left(c \cdot a\right) \cdot {b}^{-3}\right) \]
13. Add Preprocessing

Alternative 9: 81.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ \frac{c}{-b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c (- b)))

C

double code(double a, double b, double c) {
	return 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 = c / -b
end function

Java

public static double code(double a, double b, double c) {
	return c / -b;
}

Python

def code(a, b, c):
	return c / -b

Julia

function code(a, b, c)
	return Float64(c / Float64(-b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / -b;
end

Wolfram

code[a_, b_, c_] := N[(c / (-b)), $MachinePrecision]

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 81.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
6. Step-by-step derivation

   1. associate-*r/ 81.1%

      \[\leadsto \color{blue}{\frac{-1 \cdot c}{b}} \]

   2. mul-1-neg 81.1%

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

7. Simplified 81.1%

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

8. Final simplification 81.1%

   \[\leadsto \frac{c}{-b} \]
9. Add Preprocessing

Alternative 10: 3.2% accurate, 116.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 31.1%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 31.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in c around 0 89.6%

   \[\leadsto \color{blue}{c \cdot \left(-1 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 89.6%

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

   2. neg-mul-1 89.6%

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

   3. distribute-lft-neg-in 89.6%

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

7. Simplified 89.6%

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

8. Taylor expanded in a around 0 80.9%

   \[\leadsto c \cdot \color{blue}{\frac{-1}{b}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 72.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)\right)} \]

   2. expm1-undefine 32.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} - 1} \]

10. Applied egg-rr 32.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} - 1} \]
11. Step-by-step derivation

    1. sub-neg 32.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(c \cdot \frac{-1}{b}\right)} + \left(-1\right)} \]

    2. log1p-undefine 32.4%

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

    3. associate-*r/ 32.4%

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

    4. *-commutative 32.4%

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

    5. neg-mul-1 32.4%

       \[\leadsto e^{\log \left(1 + \frac{\color{blue}{-c}}{b}\right)} + \left(-1\right) \]

    6. distribute-neg-frac 32.4%

       \[\leadsto e^{\log \left(1 + \color{blue}{\left(-\frac{c}{b}\right)}\right)} + \left(-1\right) \]

    7. rem-exp-log 40.7%

       \[\leadsto \color{blue}{\left(1 + \left(-\frac{c}{b}\right)\right)} + \left(-1\right) \]

    8. unsub-neg 40.7%

       \[\leadsto \color{blue}{\left(1 - \frac{c}{b}\right)} + \left(-1\right) \]

    9. metadata-eval 40.7%

       \[\leadsto \left(1 - \frac{c}{b}\right) + \color{blue}{-1} \]

12. Simplified 40.7%

    \[\leadsto \color{blue}{\left(1 - \frac{c}{b}\right) + -1} \]

13. Taylor expanded in c around 0 3.2%

    \[\leadsto \color{blue}{1} + -1 \]

14. Final simplification 3.2%

    \[\leadsto 0 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))