Quadratic roots, full range

Percentage Accurate: 51.6% → 84.6%

Time: 12.6s

Alternatives: 8

Speedup: 19.1×

Specification

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: 51.6% 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: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e+154)
   (/ (- b) a)
   (if (<= b 7e-136)
     (/ (- (sqrt (- (* b b) (* (* a 4.0) c))) b) (* a 2.0))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e+154) {
		tmp = -b / a;
	} else if (b <= 7e-136) {
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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) :: tmp
    if (b <= (-1.4d+154)) then
        tmp = -b / a
    else if (b <= 7d-136) then
        tmp = (sqrt(((b * b) - ((a * 4.0d0) * c))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e+154) {
		tmp = -b / a;
	} else if (b <= 7e-136) {
		tmp = (Math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e+154:
		tmp = -b / a
	elif b <= 7e-136:
		tmp = (math.sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e+154)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 7e-136)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 4.0) * c))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.4e+154)
		tmp = -b / a;
	elseif (b <= 7e-136)
		tmp = (sqrt(((b * b) - ((a * 4.0) * c))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e+154], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 7e-136], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 4.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.4e154

   1. Initial program 44.8%

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

      1. *-commutative 44.8%

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

   3. Simplified 44.8%

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

   4. Taylor expanded in b around -inf 97.8%

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

      1. associate-*r/ 97.8%

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

      2. mul-1-neg 97.8%

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

   6. Simplified 97.8%

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

   if -1.4e154 < b < 7.00000000000000058e-136

   1. Initial program 91.0%

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

   if 7.00000000000000058e-136 < b

   1. Initial program 26.3%

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

      1. *-commutative 26.3%

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

   3. Simplified 26.3%

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

   5. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 30.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   7. Simplified 30.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   8. Taylor expanded in b around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. unpow2 0.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      5. rem-square-sqrt 77.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      6. metadata-eval 77.0%

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

   10. Simplified 77.0%

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

       1. frac-2neg 77.0%

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

       2. metadata-eval 77.0%

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

       3. div-inv 77.0%

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

       4. distribute-neg-in 77.0%

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

       5. add-sqr-sqrt 41.2%

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

       6. sqrt-unprod 40.3%

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

       7. mul-1-neg 40.3%

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

       8. mul-1-neg 40.3%

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

       9. sqr-neg 40.3%

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

       10. sqrt-unprod 12.0%

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

       11. add-sqr-sqrt 23.6%

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

       12. mul-1-neg 23.6%

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

       13. sub-neg 23.6%

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

   12. Applied egg-rr 77.0%

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

       1. associate-*r/ 77.0%

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

       2. metadata-eval 77.0%

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

   14. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]

Alternative 2: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-136}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-146)
   (/ (- b) a)
   (if (<= b 6.6e-136)
     (* (- (sqrt (* c (* a -4.0))) b) (/ 0.5 a))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-146) {
		tmp = -b / a;
	} else if (b <= 6.6e-136) {
		tmp = (sqrt((c * (a * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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) :: tmp
    if (b <= (-3.2d-146)) then
        tmp = -b / a
    else if (b <= 6.6d-136) then
        tmp = (sqrt((c * (a * (-4.0d0)))) - b) * (0.5d0 / a)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-146) {
		tmp = -b / a;
	} else if (b <= 6.6e-136) {
		tmp = (Math.sqrt((c * (a * -4.0))) - b) * (0.5 / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-146:
		tmp = -b / a
	elif b <= 6.6e-136:
		tmp = (math.sqrt((c * (a * -4.0))) - b) * (0.5 / a)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-146)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-136)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -4.0))) - b) * Float64(0.5 / a));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.2e-146)
		tmp = -b / a;
	elseif (b <= 6.6e-136)
		tmp = (sqrt((c * (a * -4.0))) - b) * (0.5 / a);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-146], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-136], N[(N[(N[Sqrt[N[(c * N[(a * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1999999999999999e-146

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in b around -inf 84.7%

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

      1. associate-*r/ 84.7%

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

      2. mul-1-neg 84.7%

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

   6. Simplified 84.7%

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

   if -3.1999999999999999e-146 < b < 6.60000000000000035e-136

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in b around 0 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*r* 83.5%

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

   6. Simplified 83.5%

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

      1. +-commutative 83.5%

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

      2. unsub-neg 83.5%

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

   8. Applied egg-rr 83.5%

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

      1. expm1-log1p-u 58.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\right)\right)} \]

      2. expm1-udef 20.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\right)} - 1} \]

      3. *-un-lft-identity 20.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{1 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}}{a \cdot 2}\right)} - 1 \]

      4. *-commutative 20.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{1 \cdot \left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}{\color{blue}{2 \cdot a}}\right)} - 1 \]

      5. times-frac 20.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{2} \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}}\right)} - 1 \]

      6. metadata-eval 20.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.5} \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}\right)} - 1 \]

   10. Applied egg-rr 20.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 58.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}\right)\right)} \]

       2. expm1-log1p 83.5%

          \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a}} \]

       3. *-commutative 83.5%

          \[\leadsto \color{blue}{\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a} \cdot 0.5} \]

       4. associate-*l/ 83.5%

          \[\leadsto \color{blue}{\frac{\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot 0.5}{a}} \]

       5. *-lft-identity 83.5%

          \[\leadsto \frac{\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right) \cdot 0.5}{\color{blue}{1 \cdot a}} \]

       6. times-frac 83.4%

          \[\leadsto \color{blue}{\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{1} \cdot \frac{0.5}{a}} \]

   12. Simplified 83.4%

       \[\leadsto \color{blue}{\left(\sqrt{c \cdot \left(-4 \cdot a\right)} - b\right) \cdot \frac{0.5}{a}} \]

   if 6.60000000000000035e-136 < b

   1. Initial program 26.3%

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

      1. *-commutative 26.3%

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

   3. Simplified 26.3%

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

   5. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 30.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   7. Simplified 30.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   8. Taylor expanded in b around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. unpow2 0.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      5. rem-square-sqrt 77.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      6. metadata-eval 77.0%

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

   10. Simplified 77.0%

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

       1. frac-2neg 77.0%

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

       2. metadata-eval 77.0%

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

       3. div-inv 77.0%

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

       4. distribute-neg-in 77.0%

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

       5. add-sqr-sqrt 41.2%

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

       6. sqrt-unprod 40.3%

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

       7. mul-1-neg 40.3%

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

       8. mul-1-neg 40.3%

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

       9. sqr-neg 40.3%

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

       10. sqrt-unprod 12.0%

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

       11. add-sqr-sqrt 23.6%

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

       12. mul-1-neg 23.6%

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

       13. sub-neg 23.6%

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

   12. Applied egg-rr 77.0%

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

       1. associate-*r/ 77.0%

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

       2. metadata-eval 77.0%

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

   14. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-136}:\\ \;\;\;\;\left(\sqrt{c \cdot \left(a \cdot -4\right)} - b\right) \cdot \frac{0.5}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]

Alternative 3: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e-146)
   (/ (- b) a)
   (if (<= b 6.6e-136)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ -1.0 (- (/ b c) (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-146) {
		tmp = -b / a;
	} else if (b <= 6.6e-136) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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) :: tmp
    if (b <= (-3.2d-146)) then
        tmp = -b / a
    else if (b <= 6.6d-136) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e-146) {
		tmp = -b / a;
	} else if (b <= 6.6e-136) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e-146:
		tmp = -b / a
	elif b <= 6.6e-136:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e-146)
		tmp = Float64(Float64(-b) / a);
	elseif (b <= 6.6e-136)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.2e-146)
		tmp = -b / a;
	elseif (b <= 6.6e-136)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e-146], N[((-b) / a), $MachinePrecision], If[LessEqual[b, 6.6e-136], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.1999999999999999e-146

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in b around -inf 84.7%

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

      1. associate-*r/ 84.7%

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

      2. mul-1-neg 84.7%

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

   6. Simplified 84.7%

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

   if -3.1999999999999999e-146 < b < 6.60000000000000035e-136

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in b around 0 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*r* 83.5%

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

   6. Simplified 83.5%

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

      1. +-commutative 83.5%

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

      2. unsub-neg 83.5%

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

   8. Applied egg-rr 83.5%

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

   if 6.60000000000000035e-136 < b

   1. Initial program 26.3%

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

      1. *-commutative 26.3%

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

   3. Simplified 26.3%

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

   5. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 30.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   7. Simplified 30.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   8. Taylor expanded in b around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. unpow2 0.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      5. rem-square-sqrt 77.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      6. metadata-eval 77.0%

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

   10. Simplified 77.0%

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

       1. frac-2neg 77.0%

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

       2. metadata-eval 77.0%

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

       3. div-inv 77.0%

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

       4. distribute-neg-in 77.0%

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

       5. add-sqr-sqrt 41.2%

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

       6. sqrt-unprod 40.3%

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

       7. mul-1-neg 40.3%

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

       8. mul-1-neg 40.3%

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

       9. sqr-neg 40.3%

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

       10. sqrt-unprod 12.0%

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

       11. add-sqr-sqrt 23.6%

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

       12. mul-1-neg 23.6%

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

       13. sub-neg 23.6%

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

   12. Applied egg-rr 77.0%

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

       1. associate-*r/ 77.0%

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

       2. metadata-eval 77.0%

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

   14. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]

Alternative 4: 68.3% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (- (/ c b) (/ b a)) (/ -1.0 (- (/ b c) (/ a b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / 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) :: tmp
    if (b <= (-2d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = (-1.0d0) / ((b / c) - (a / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = -1.0 / ((b / c) - (a / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = -1.0 / ((b / c) - (a / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(-1.0 / Float64(Float64(b / c) - Float64(a / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = -1.0 / ((b / c) - (a / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 78.6%

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

      1. *-commutative 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in b around -inf 71.9%

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

      1. +-commutative 71.9%

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

      2. mul-1-neg 71.9%

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

      3. unsub-neg 71.9%

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

   6. Simplified 71.9%

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

   if -1.999999999999994e-310 < b

   1. Initial program 36.4%

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

      1. *-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Applied egg-rr 39.3%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 39.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   7. Simplified 39.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

   8. Taylor expanded in b around inf 0.0%

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

      1. associate-*r/ 0.0%

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

      2. *-commutative 0.0%

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

      3. times-frac 0.0%

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

      4. unpow2 0.0%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      5. rem-square-sqrt 64.1%

         \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

      6. metadata-eval 64.1%

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

   10. Simplified 64.1%

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

       1. frac-2neg 64.1%

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

       2. metadata-eval 64.1%

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

       3. div-inv 64.1%

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

       4. distribute-neg-in 64.1%

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

       5. add-sqr-sqrt 34.5%

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

       6. sqrt-unprod 34.0%

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

       7. mul-1-neg 34.0%

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

       8. mul-1-neg 34.0%

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

       9. sqr-neg 34.0%

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

       10. sqrt-unprod 10.1%

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

       11. add-sqr-sqrt 19.8%

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

       12. mul-1-neg 19.8%

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

       13. sub-neg 19.8%

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

   12. Applied egg-rr 64.1%

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

       1. associate-*r/ 64.1%

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

       2. metadata-eval 64.1%

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

   14. Simplified 64.1%

       \[\leadsto \color{blue}{\frac{-1}{\frac{b}{c} - \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{b}{c} - \frac{a}{b}}\\ \end{array} \]

Alternative 5: 43.4% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 4.6e+15) (/ (- b) a) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6e+15) {
		tmp = -b / a;
	} else {
		tmp = 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) :: tmp
    if (b <= 4.6d+15) then
        tmp = -b / a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.6e+15) {
		tmp = -b / a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 4.6e+15:
		tmp = -b / a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.6e+15)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 4.6e+15)
		tmp = -b / a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.6e+15], N[((-b) / a), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.6e15

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in b around -inf 50.2%

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

      1. associate-*r/ 50.2%

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

      2. mul-1-neg 50.2%

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

   6. Simplified 50.2%

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

   if 4.6e15 < b

   1. Initial program 20.6%

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

      1. *-commutative 20.6%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in b around inf 69.2%

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

      1. associate-/l* 74.8%

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

   6. Simplified 74.8%

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

      1. add-log-exp 32.1%

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

      2. *-commutative 32.1%

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

      3. times-frac 32.1%

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

      4. metadata-eval 32.1%

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

      5. div-inv 32.1%

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

      6. clear-num 32.1%

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

   8. Applied egg-rr 32.1%

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

      1. rem-log-exp 75.4%

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

      2. add-sqr-sqrt 55.2%

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

      3. sqrt-unprod 48.3%

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

      4. mul-1-neg 48.3%

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

      5. mul-1-neg 48.3%

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

      6. sqr-neg 48.3%

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

      7. sqrt-unprod 30.1%

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

      8. add-sqr-sqrt 30.8%

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

      9. div-inv 30.8%

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

      10. clear-num 30.8%

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

      11. un-div-inv 30.8%

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

   10. Applied egg-rr 30.8%

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

   11. Taylor expanded in a around 0 30.5%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]

Alternative 6: 68.4% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 5e-304) (/ (- b) a) (/ (- c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e-304) {
		tmp = -b / a;
	} else {
		tmp = -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) :: tmp
    if (b <= 5d-304) then
        tmp = -b / a
    else
        tmp = -c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 5e-304) {
		tmp = -b / a;
	} else {
		tmp = -c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5e-304:
		tmp = -b / a
	else:
		tmp = -c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5e-304)
		tmp = Float64(Float64(-b) / a);
	else
		tmp = Float64(Float64(-c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 5e-304)
		tmp = -b / a;
	else
		tmp = -c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5e-304], N[((-b) / a), $MachinePrecision], N[((-c) / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.99999999999999965e-304

   1. Initial program 78.0%

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

      1. *-commutative 78.0%

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

   3. Simplified 78.0%

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

   4. Taylor expanded in b around -inf 71.3%

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

      1. associate-*r/ 71.3%

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

      2. mul-1-neg 71.3%

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

   6. Simplified 71.3%

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

   if 4.99999999999999965e-304 < b

   1. Initial program 36.7%

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

      1. *-commutative 36.7%

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

   3. Simplified 36.7%

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

   4. Taylor expanded in b around inf 64.4%

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

      1. mul-1-neg 64.4%

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

      2. distribute-neg-frac 64.4%

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

   6. Simplified 64.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-304}:\\ \;\;\;\;\frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array} \]

Alternative 7: 2.5% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. *-commutative 56.8%

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

3. Simplified 56.8%

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

5. Applied egg-rr 53.4%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 53.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

7. Simplified 53.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot -2}{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -4\right)}\right)}}} \]

8. Taylor expanded in b around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. times-frac 0.0%

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

   4. unpow2 0.0%

      \[\leadsto \frac{1}{\frac{4}{\color{blue}{\sqrt{-4} \cdot \sqrt{-4}}} \cdot \frac{b}{c} + \frac{a}{b}} \]

   5. rem-square-sqrt 34.4%

      \[\leadsto \frac{1}{\frac{4}{\color{blue}{-4}} \cdot \frac{b}{c} + \frac{a}{b}} \]

   6. metadata-eval 34.4%

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

10. Simplified 34.4%

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

11. Taylor expanded in b around 0 2.7%

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

12. Final simplification 2.7%

    \[\leadsto \frac{b}{a} \]

Alternative 8: 11.3% accurate, 38.7× 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 / 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 56.8%

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

   1. *-commutative 56.8%

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

3. Simplified 56.8%

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

4. Taylor expanded in b around inf 26.3%

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

   1. associate-/l* 28.5%

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

6. Simplified 28.5%

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

   1. add-log-exp 13.1%

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

   2. *-commutative 13.1%

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

   3. times-frac 13.1%

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

   4. metadata-eval 13.1%

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

   5. div-inv 13.1%

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

   6. clear-num 13.1%

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

8. Applied egg-rr 13.1%

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

   1. rem-log-exp 28.7%

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

   2. add-sqr-sqrt 20.0%

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

   3. sqrt-unprod 17.9%

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

   4. mul-1-neg 17.9%

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

   5. mul-1-neg 17.9%

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

   6. sqr-neg 17.9%

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

   7. sqrt-unprod 9.8%

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

   8. add-sqr-sqrt 11.1%

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

   9. div-inv 11.1%

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

   10. clear-num 11.1%

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

   11. un-div-inv 11.1%

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

10. Applied egg-rr 11.1%

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

11. Taylor expanded in a around 0 11.0%

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

12. Final simplification 11.0%

    \[\leadsto \frac{c}{b} \]

Reproduce

herbie shell --seed 2023325 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))