quad2p (problem 3.2.1, positive)

Percentage Accurate: 52.0% → 84.9%

Time: 13.1s

Alternatives: 8

Speedup: 11.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 + Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 + math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) + sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 + sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) + N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.85 \cdot 10^{+89}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.85e+89)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 6e-117)
     (/ (- (sqrt (fma b_2 b_2 (* c (- a)))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.85e+89) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 6e-117) {
		tmp = (sqrt(fma(b_2, b_2, (c * -a))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.85e+89)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 6e-117)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.85e+89], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 6e-117], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.8499999999999999e89

   1. Initial program 45.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 45.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 45.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 96.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 96.2%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 96.2%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -1.8499999999999999e89 < b_2 < 5.99999999999999982e-117

   1. Initial program 85.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 85.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 85.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-neg 85.2%

         \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, -a \cdot c\right)}} - b\_2}{a} \]

      2. *-commutative 85.2%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, -\color{blue}{c \cdot a}\right)} - b\_2}{a} \]

      3. distribute-rgt-neg-in 85.2%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b\_2, b\_2, \color{blue}{c \cdot \left(-a\right)}\right)} - b\_2}{a} \]

   6. Applied egg-rr 85.2%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b\_2, b\_2, c \cdot \left(-a\right)\right)}} - b\_2}{a} \]

   if 5.99999999999999982e-117 < b_2

   1. Initial program 21.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 21.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 21.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 21.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 84.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 84.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 84.8%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.95 \cdot 10^{-50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_2 \leq -1.1 \cdot 10^{-68}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* c (- a))) b_2) a)))
   (if (<= b_2 -7.6e-10)
     (- (* -0.5 (/ c (- b_2))) (* 2.0 (/ b_2 a)))
     (if (<= b_2 -1.95e-50)
       t_0
       (if (<= b_2 -1.1e-68)
         (/ (* b_2 -2.0) a)
         (if (<= b_2 8.2e-117) t_0 (/ (* c -0.5) b_2)))))))

C

double code(double a, double b_2, double c) {
	double t_0 = (sqrt((c * -a)) - b_2) / a;
	double tmp;
	if (b_2 <= -7.6e-10) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= -1.95e-50) {
		tmp = t_0;
	} else if (b_2 <= -1.1e-68) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8.2e-117) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((c * -a)) - b_2) / a
    if (b_2 <= (-7.6d-10)) then
        tmp = ((-0.5d0) * (c / -b_2)) - (2.0d0 * (b_2 / a))
    else if (b_2 <= (-1.95d-50)) then
        tmp = t_0
    else if (b_2 <= (-1.1d-68)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 8.2d-117) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double t_0 = (Math.sqrt((c * -a)) - b_2) / a;
	double tmp;
	if (b_2 <= -7.6e-10) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else if (b_2 <= -1.95e-50) {
		tmp = t_0;
	} else if (b_2 <= -1.1e-68) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8.2e-117) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	t_0 = (math.sqrt((c * -a)) - b_2) / a
	tmp = 0
	if b_2 <= -7.6e-10:
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a))
	elif b_2 <= -1.95e-50:
		tmp = t_0
	elif b_2 <= -1.1e-68:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 8.2e-117:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	t_0 = Float64(Float64(sqrt(Float64(c * Float64(-a))) - b_2) / a)
	tmp = 0.0
	if (b_2 <= -7.6e-10)
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(-b_2))) - Float64(2.0 * Float64(b_2 / a)));
	elseif (b_2 <= -1.95e-50)
		tmp = t_0;
	elseif (b_2 <= -1.1e-68)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 8.2e-117)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	t_0 = (sqrt((c * -a)) - b_2) / a;
	tmp = 0.0;
	if (b_2 <= -7.6e-10)
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	elseif (b_2 <= -1.95e-50)
		tmp = t_0;
	elseif (b_2 <= -1.1e-68)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 8.2e-117)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[b$95$2, -7.6e-10], N[(N[(-0.5 * N[(c / (-b$95$2)), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1.95e-50], t$95$0, If[LessEqual[b$95$2, -1.1e-68], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8.2e-117], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b_2 < -7.5999999999999996e-10

   1. Initial program 56.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 56.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 56.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 56.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 91.0%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(-0.5 \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 91.3%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b\_2} + 2 \cdot \frac{b\_2}{a}\right)} \]

   if -7.5999999999999996e-10 < b_2 < -1.9500000000000001e-50 or -1.10000000000000001e-68 < b_2 < 8.20000000000000063e-117

   1. Initial program 82.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 82.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 82.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 82.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around 0 75.9%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}} - b\_2}{a} \]
   6. Step-by-step derivation

      1. associate-*r* 75.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}} - b\_2}{a} \]

      2. neg-mul-1 75.9%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-a\right)} \cdot c} - b\_2}{a} \]

      3. *-commutative 75.9%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   7. Simplified 75.9%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b\_2}{a} \]

   if -1.9500000000000001e-50 < b_2 < -1.10000000000000001e-68

   1. Initial program 100.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 100.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 100.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 100.0%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if 8.20000000000000063e-117 < b_2

   1. Initial program 21.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 21.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 21.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 21.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 84.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 84.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 84.8%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.6 \cdot 10^{-10}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.95 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{elif}\;b\_2 \leq -1.1 \cdot 10^{-68}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(-a\right)} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1 \cdot 10^{+89}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{elif}\;b\_2 \leq 8 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1e+89)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 8e-118)
     (/ (- (sqrt (- (* b_2 b_2) (* a c))) b_2) a)
     (/ (* c -0.5) b_2))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+89) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8e-118) {
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1d+89)) then
        tmp = (b_2 * (-2.0d0)) / a
    else if (b_2 <= 8d-118) then
        tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1e+89) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 8e-118) {
		tmp = (Math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1e+89:
		tmp = (b_2 * -2.0) / a
	elif b_2 <= 8e-118:
		tmp = (math.sqrt(((b_2 * b_2) - (a * c))) - b_2) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1e+89)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 8e-118)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))) - b_2) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1e+89)
		tmp = (b_2 * -2.0) / a;
	elseif (b_2 <= 8e-118)
		tmp = (sqrt(((b_2 * b_2) - (a * c))) - b_2) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1e+89], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 8e-118], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -9.99999999999999995e88

   1. Initial program 45.0%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 45.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 45.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 45.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 96.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 96.2%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 96.2%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if -9.99999999999999995e88 < b_2 < 7.99999999999999988e-118

   1. Initial program 85.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 85.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 85.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 85.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   if 7.99999999999999988e-118 < b_2

   1. Initial program 21.5%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 21.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 21.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 21.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 84.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 84.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 84.8%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 68.0% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -2e-310)
   (- (* -0.5 (/ c (- b_2))) (* 2.0 (/ b_2 a)))
   (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-2d-310)) then
        tmp = ((-0.5d0) * (c / -b_2)) - (2.0d0 * (b_2 / a))
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -2e-310) {
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -2e-310:
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a))
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -2e-310)
		tmp = Float64(Float64(-0.5 * Float64(c / Float64(-b_2))) - Float64(2.0 * Float64(b_2 / a)));
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -2e-310)
		tmp = (-0.5 * (c / -b_2)) - (2.0 * (b_2 / a));
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -2e-310], N[(N[(-0.5 * N[(c / (-b$95$2)), $MachinePrecision]), $MachinePrecision] - N[(2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -1.999999999999994e-310

   1. Initial program 69.7%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 69.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 69.7%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 60.3%

      \[\leadsto \color{blue}{-1 \cdot \left(b\_2 \cdot \left(-0.5 \cdot \frac{c}{{b\_2}^{2}} + 2 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 60.7%

      \[\leadsto -1 \cdot \color{blue}{\left(-0.5 \cdot \frac{c}{b\_2} + 2 \cdot \frac{b\_2}{a}\right)} \]

   if -1.999999999999994e-310 < b_2

   1. Initial program 36.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 36.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 36.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 36.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 66.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 66.3%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{-b\_2} - 2 \cdot \frac{b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.8% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq 2.2 \cdot 10^{-302}:\\ \;\;\;\;\frac{b\_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 2.2e-302) (/ (* b_2 -2.0) a) (/ (* c -0.5) b_2)))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.2e-302) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= 2.2d-302) then
        tmp = (b_2 * (-2.0d0)) / a
    else
        tmp = (c * (-0.5d0)) / b_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= 2.2e-302) {
		tmp = (b_2 * -2.0) / a;
	} else {
		tmp = (c * -0.5) / b_2;
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= 2.2e-302:
		tmp = (b_2 * -2.0) / a
	else:
		tmp = (c * -0.5) / b_2
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= 2.2e-302)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= 2.2e-302)
		tmp = (b_2 * -2.0) / a;
	else
		tmp = (c * -0.5) / b_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, 2.2e-302], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < 2.20000000000000007e-302

   1. Initial program 70.2%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 70.2%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around -inf 59.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
   6. Step-by-step derivation

      1. *-commutative 59.5%

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   7. Simplified 59.5%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   if 2.20000000000000007e-302 < b_2

   1. Initial program 35.6%

      \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative 35.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

      2. unsub-neg 35.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
   4. Add Preprocessing

   5. Taylor expanded in b_2 around inf 67.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. associate-*r/ 67.2%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b\_2}} \]

      2. *-commutative 67.2%

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   7. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 35.2% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 \cdot -2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (* b_2 -2.0) a))

C

double code(double a, double b_2, double c) {
	return (b_2 * -2.0) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (b_2 * (-2.0d0)) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (b_2 * -2.0) / a;
}

Python

def code(a, b_2, c):
	return (b_2 * -2.0) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 * -2.0) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (b_2 * -2.0) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 52.5%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

   2. unsub-neg 52.5%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

3. Simplified 52.5%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing

5. Taylor expanded in b_2 around -inf 30.4%

   \[\leadsto \frac{\color{blue}{-2 \cdot b\_2}}{a} \]
6. Step-by-step derivation

   1. *-commutative 30.4%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

7. Simplified 30.4%

   \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]
8. Add Preprocessing

Alternative 7: 35.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ b\_2 \cdot \frac{-2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* b_2 (/ -2.0 a)))

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = b_2 * ((-2.0d0) / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Python

def code(a, b_2, c):
	return b_2 * (-2.0 / a)

Julia

function code(a, b_2, c)
	return Float64(b_2 * Float64(-2.0 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = b_2 * (-2.0 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 52.5%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

   2. unsub-neg 52.5%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

3. Simplified 52.5%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 52.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}} \]

   2. associate-/r/ 52.4%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2\right)} \]

   3. sub-neg 52.4%

      \[\leadsto \frac{1}{a} \cdot \left(\sqrt{\color{blue}{b\_2 \cdot b\_2 + \left(-a \cdot c\right)}} - b\_2\right) \]

   4. add-sqr-sqrt 44.2%

      \[\leadsto \frac{1}{a} \cdot \left(\sqrt{b\_2 \cdot b\_2 + \color{blue}{\sqrt{-a \cdot c} \cdot \sqrt{-a \cdot c}}} - b\_2\right) \]

   5. hypot-define 54.2%

      \[\leadsto \frac{1}{a} \cdot \left(\color{blue}{\mathsf{hypot}\left(b\_2, \sqrt{-a \cdot c}\right)} - b\_2\right) \]

   6. *-commutative 54.2%

      \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{-\color{blue}{c \cdot a}}\right) - b\_2\right) \]

   7. distribute-rgt-neg-in 54.2%

      \[\leadsto \frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{\color{blue}{c \cdot \left(-a\right)}}\right) - b\_2\right) \]

6. Applied egg-rr 54.2%

   \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\mathsf{hypot}\left(b\_2, \sqrt{c \cdot \left(-a\right)}\right) - b\_2\right)} \]

7. Taylor expanded in b_2 around -inf 30.4%

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 30.4%

      \[\leadsto \color{blue}{\frac{-2 \cdot b\_2}{a}} \]

   2. *-commutative 30.4%

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   3. *-lft-identity 30.4%

      \[\leadsto \frac{b\_2 \cdot -2}{\color{blue}{1 \cdot a}} \]

   4. times-frac 30.3%

      \[\leadsto \color{blue}{\frac{b\_2}{1} \cdot \frac{-2}{a}} \]

   5. /-rgt-identity 30.3%

      \[\leadsto \color{blue}{b\_2} \cdot \frac{-2}{a} \]

9. Simplified 30.3%

   \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]
10. Add Preprocessing

Alternative 8: 15.6% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2}{-a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ b_2 (- a)))

C

double code(double a, double b_2, double c) {
	return b_2 / -a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = b_2 / -a
end function

Java

public static double code(double a, double b_2, double c) {
	return b_2 / -a;
}

Python

def code(a, b_2, c):
	return b_2 / -a

Julia

function code(a, b_2, c)
	return Float64(b_2 / Float64(-a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = b_2 / -a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(b$95$2 / (-a)), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{\left(-b\_2\right) + \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative 52.5%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} + \left(-b\_2\right)}}{a} \]

   2. unsub-neg 52.5%

      \[\leadsto \frac{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}}{a} \]

3. Simplified 52.5%

   \[\leadsto \color{blue}{\frac{\sqrt{b\_2 \cdot b\_2 - a \cdot c} - b\_2}{a}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 51.9%

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}} \cdot \sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}} - b\_2}{a} \]

   2. pow2 51.9%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}\right)}^{2}} - b\_2}{a} \]

   3. pow1/2 51.9%

      \[\leadsto \frac{{\left(\sqrt{\color{blue}{{\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{0.5}}}\right)}^{2} - b\_2}{a} \]

   4. sqrt-pow1 51.9%

      \[\leadsto \frac{{\color{blue}{\left({\left(b\_2 \cdot b\_2 - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} - b\_2}{a} \]

   5. pow2 51.9%

      \[\leadsto \frac{{\left({\left(\color{blue}{{b\_2}^{2}} - a \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} - b\_2}{a} \]

   6. metadata-eval 51.9%

      \[\leadsto \frac{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{\color{blue}{0.25}}\right)}^{2} - b\_2}{a} \]

6. Applied egg-rr 51.9%

   \[\leadsto \frac{\color{blue}{{\left({\left({b\_2}^{2} - a \cdot c\right)}^{0.25}\right)}^{2}} - b\_2}{a} \]

7. Taylor expanded in a around -inf 19.0%

   \[\leadsto \frac{{\color{blue}{\left(e^{0.25 \cdot \left(\log c + -1 \cdot \log \left(\frac{-1}{a}\right)\right)}\right)}}^{2} - b\_2}{a} \]

8. Taylor expanded in b_2 around inf 12.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{b\_2}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 12.2%

      \[\leadsto \color{blue}{\frac{-1 \cdot b\_2}{a}} \]

   2. neg-mul-1 12.2%

      \[\leadsto \frac{\color{blue}{-b\_2}}{a} \]

10. Simplified 12.2%

    \[\leadsto \color{blue}{\frac{-b\_2}{a}} \]

11. Final simplification 12.2%

    \[\leadsto \frac{b\_2}{-a} \]
12. Add Preprocessing

Developer target: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{t\_1 - b\_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b\_2 + t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ (- t_1 b_2) a) (/ (- c) (+ b_2 t_1)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = (t_1 - b_2) / a;
	} else {
		tmp_1 = -c / (b_2 + t_1);
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = (t_1 - b_2) / a
	else:
		tmp_1 = -c / (b_2 + t_1)
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(Float64(t_1 - b_2) / a);
	else
		tmp_1 = Float64(Float64(-c) / Float64(b_2 + t_1));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = (t_1 - b_2) / a;
	else
		tmp_2 = -c / (b_2 + t_1);
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(N[(t$95$1 - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[((-c) / N[(b$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024086 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  :herbie-expected 10

  :alt
  (if (< b_2 0.0) (/ (- (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))) b_2) a) (/ (- c) (+ b_2 (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c))))) (sqrt (+ (fabs b_2) (* (sqrt (fabs a)) (sqrt (fabs c)))))) (hypot b_2 (* (sqrt (fabs a)) (sqrt (fabs c))))))))

  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))