sqrt sqr

Percentage Accurate: 50.6% → 100.0%

Time: 8.8s

Alternatives: 6

Speedup: 111.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))

C

double code(double x) {
	return (x / x) - ((1.0 / x) * sqrt((x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / x) - ((1.0d0 / x) * sqrt((x * x)))
end function

Java

public static double code(double x) {
	return (x / x) - ((1.0 / x) * Math.sqrt((x * x)));
}

Python

def code(x):
	return (x / x) - ((1.0 / x) * math.sqrt((x * x)))

Julia

function code(x)
	return Float64(Float64(x / x) - Float64(Float64(1.0 / x) * sqrt(Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = (x / x) - ((1.0 / x) * sqrt((x * x)));
end

Wolfram

code[x_] := N[(N[(x / x), $MachinePrecision] - N[(N[(1.0 / x), $MachinePrecision] * N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 50.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))

C

double code(double x) {
	return (x / x) - ((1.0 / x) * sqrt((x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / x) - ((1.0d0 / x) * sqrt((x * x)))
end function

Java

public static double code(double x) {
	return (x / x) - ((1.0 / x) * Math.sqrt((x * x)));
}

Python

def code(x):
	return (x / x) - ((1.0 / x) * math.sqrt((x * x)))

Julia

function code(x)
	return Float64(Float64(x / x) - Float64(Float64(1.0 / x) * sqrt(Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = (x / x) - ((1.0 / x) * sqrt((x * x)));
end

Wolfram

code[x_] := N[(N[(x / x), $MachinePrecision] - N[(N[(1.0 / x), $MachinePrecision] * N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{\left|x\right|}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (/ (fabs x) x)))

C

double code(double x) {
	return 1.0 - (fabs(x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (abs(x) / x)
end function

Java

public static double code(double x) {
	return 1.0 - (Math.abs(x) / x);
}

Python

def code(x):
	return 1.0 - (math.fabs(x) / x)

Julia

function code(x)
	return Float64(1.0 - Float64(abs(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (abs(x) / x);
end

Wolfram

code[x_] := N[(1.0 - N[(N[Abs[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

   \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 49.7%

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

   2. *-inverses 49.7%

      \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

   3. distribute-frac-neg2 49.7%

      \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

   4. sqr-neg 49.7%

      \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

   5. *-inverses 49.7%

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

   6. cancel-sign-sub 49.7%

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

   7. *-inverses 49.7%

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

   8. *-inverses 49.7%

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

   9. distribute-neg-frac 49.7%

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

   10. *-inverses 49.7%

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

   11. metadata-eval 49.7%

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

   12. associate-*l/ 51.2%

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

   13. neg-mul-1 51.2%

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

   14. distribute-neg-frac 51.2%

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

   15. distribute-neg-frac2 51.2%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 58.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{\frac{1 + \left(-1 - x \cdot -2\right)}{2 - x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310)
   (- 1.0 (/ (/ (+ 1.0 (- -1.0 (* x -2.0))) (- 2.0 x)) x))
   0.0))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 1.0 - (((1.0 + (-1.0 - (x * -2.0))) / (2.0 - x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-310)) then
        tmp = 1.0d0 - (((1.0d0 + ((-1.0d0) - (x * (-2.0d0)))) / (2.0d0 - x)) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 1.0 - (((1.0 + (-1.0 - (x * -2.0))) / (2.0 - x)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = 1.0 - (((1.0 + (-1.0 - (x * -2.0))) / (2.0 - x)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(1.0 - Float64(Float64(Float64(1.0 + Float64(-1.0 - Float64(x * -2.0))) / Float64(2.0 - x)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = 1.0 - (((1.0 + (-1.0 - (x * -2.0))) / (2.0 - x)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(1.0 - N[(N[(N[(1.0 + N[(-1.0 - N[(x * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 - x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 50.7%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 50.7%

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

      2. *-inverses 50.7%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 50.7%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 50.7%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 50.7%

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

      6. cancel-sign-sub 50.7%

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

      7. *-inverses 50.7%

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

      8. *-inverses 50.7%

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

      9. distribute-neg-frac 50.7%

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

      10. *-inverses 50.7%

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

      11. metadata-eval 50.7%

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

      12. associate-*l/ 50.7%

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

      13. neg-mul-1 50.7%

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

      14. distribute-neg-frac 50.7%

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

      15. distribute-neg-frac2 50.7%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 96.4%

         \[\leadsto 1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)}}{x} \]

      2. expm1-undefine 52.4%

         \[\leadsto 1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right|\right)} - 1}}{x} \]

      3. log1p-undefine 52.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 10.1%

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

      7. rem-exp-log 11.5%

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

   6. Applied egg-rr 11.5%

      \[\leadsto 1 - \frac{\color{blue}{\left(1 + x\right) - 1}}{x} \]
   7. Step-by-step derivation

      1. associate--l+ 11.6%

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

      2. flip-+ 11.1%

         \[\leadsto 1 - \frac{\color{blue}{\frac{1 \cdot 1 - \left(x - 1\right) \cdot \left(x - 1\right)}{1 - \left(x - 1\right)}}}{x} \]

      3. metadata-eval 11.1%

         \[\leadsto 1 - \frac{\frac{\color{blue}{1} - \left(x - 1\right) \cdot \left(x - 1\right)}{1 - \left(x - 1\right)}}{x} \]

      4. sub-neg 11.1%

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

      5. metadata-eval 11.1%

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

      6. sub-neg 11.1%

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

      7. metadata-eval 11.1%

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

      8. sub-neg 11.1%

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

      9. metadata-eval 11.1%

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

   8. Applied egg-rr 11.1%

      \[\leadsto 1 - \frac{\color{blue}{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{1 - \left(x + -1\right)}}}{x} \]
   9. Step-by-step derivation

      1. metadata-eval 11.1%

         \[\leadsto 1 - \frac{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{\color{blue}{\left(2 - 1\right)} - \left(x + -1\right)}}{x} \]

      2. associate--r+ 11.1%

         \[\leadsto 1 - \frac{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{\color{blue}{2 - \left(1 + \left(x + -1\right)\right)}}}{x} \]

      3. +-commutative 11.1%

         \[\leadsto 1 - \frac{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{2 - \color{blue}{\left(\left(x + -1\right) + 1\right)}}}{x} \]

      4. associate-+l+ 11.1%

         \[\leadsto 1 - \frac{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{2 - \color{blue}{\left(x + \left(-1 + 1\right)\right)}}}{x} \]

      5. metadata-eval 11.1%

         \[\leadsto 1 - \frac{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{2 - \left(x + \color{blue}{0}\right)}}{x} \]

      6. +-rgt-identity 11.1%

         \[\leadsto 1 - \frac{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{2 - \color{blue}{x}}}{x} \]

   10. Simplified 11.1%

       \[\leadsto 1 - \frac{\color{blue}{\frac{1 - \left(x + -1\right) \cdot \left(x + -1\right)}{2 - x}}}{x} \]

   11. Taylor expanded in x around 0 18.6%

       \[\leadsto 1 - \frac{\frac{1 - \color{blue}{\left(1 + -2 \cdot x\right)}}{2 - x}}{x} \]
   12. Step-by-step derivation

       1. *-commutative 18.6%

          \[\leadsto 1 - \frac{\frac{1 - \left(1 + \color{blue}{x \cdot -2}\right)}{2 - x}}{x} \]

   13. Simplified 18.6%

       \[\leadsto 1 - \frac{\frac{1 - \color{blue}{\left(1 + x \cdot -2\right)}}{2 - x}}{x} \]

   if -3.999999999999988e-310 < x

   1. Initial program 48.7%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 48.7%

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

      2. *-inverses 48.7%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 48.7%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 48.7%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. *-inverses 48.7%

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

      8. *-inverses 48.7%

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

      9. distribute-neg-frac 48.7%

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

      10. *-inverses 48.7%

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

      11. metadata-eval 48.7%

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

      12. associate-*l/ 51.6%

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

      13. neg-mul-1 51.6%

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

      14. distribute-neg-frac 51.6%

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

      15. distribute-neg-frac2 51.6%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
   6. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

      2. rem-square-sqrt 49.1%

         \[\leadsto \frac{x}{x} - \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} \]

      3. fabs-sqr 49.1%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} \]

      4. rem-square-sqrt 100.0%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{x}}{x} \]

      5. *-inverses 100.0%

         \[\leadsto \color{blue}{1} - \frac{x}{x} \]

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{\frac{1 + \left(-1 - x \cdot -2\right)}{2 - x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+16}:\\ \;\;\;\;1 - x \cdot \left(\left(x + 2\right) \cdot \frac{1}{x \cdot \left(x + 2\right)}\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{1 + \left(x + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e+16)
   (- 1.0 (* x (* (+ x 2.0) (/ 1.0 (* x (+ x 2.0))))))
   (if (<= x -4e-310) (- 1.0 (/ (+ 1.0 (+ x -1.0)) x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -5e+16) {
		tmp = 1.0 - (x * ((x + 2.0) * (1.0 / (x * (x + 2.0)))));
	} else if (x <= -4e-310) {
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5d+16)) then
        tmp = 1.0d0 - (x * ((x + 2.0d0) * (1.0d0 / (x * (x + 2.0d0)))))
    else if (x <= (-4d-310)) then
        tmp = 1.0d0 - ((1.0d0 + (x + (-1.0d0))) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5e+16) {
		tmp = 1.0 - (x * ((x + 2.0) * (1.0 / (x * (x + 2.0)))));
	} else if (x <= -4e-310) {
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5e+16:
		tmp = 1.0 - (x * ((x + 2.0) * (1.0 / (x * (x + 2.0)))))
	elif x <= -4e-310:
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e+16)
		tmp = Float64(1.0 - Float64(x * Float64(Float64(x + 2.0) * Float64(1.0 / Float64(x * Float64(x + 2.0))))));
	elseif (x <= -4e-310)
		tmp = Float64(1.0 - Float64(Float64(1.0 + Float64(x + -1.0)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5e+16)
		tmp = 1.0 - (x * ((x + 2.0) * (1.0 / (x * (x + 2.0)))));
	elseif (x <= -4e-310)
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5e+16], N[(1.0 - N[(x * N[(N[(x + 2.0), $MachinePrecision] * N[(1.0 / N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-310], N[(1.0 - N[(N[(1.0 + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -5e16

   1. Initial program 48.9%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 48.9%

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

      2. *-inverses 48.9%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 48.9%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 48.9%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 48.9%

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

      6. cancel-sign-sub 48.9%

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

      7. *-inverses 48.9%

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

      8. *-inverses 48.9%

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

      9. distribute-neg-frac 48.9%

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

      10. *-inverses 48.9%

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

      11. metadata-eval 48.9%

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

      12. associate-*l/ 48.9%

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

      13. neg-mul-1 48.9%

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

      14. distribute-neg-frac 48.9%

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

      15. distribute-neg-frac2 48.9%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 92.0%

         \[\leadsto 1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)}}{x} \]

      2. expm1-undefine 92.0%

         \[\leadsto 1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right|\right)} - 1}}{x} \]

      3. log1p-undefine 92.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. rem-exp-log 3.1%

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

   6. Applied egg-rr 3.1%

      \[\leadsto 1 - \frac{\color{blue}{\left(1 + x\right) - 1}}{x} \]
   7. Step-by-step derivation

      1. clear-num 3.1%

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

      2. associate-/r/ 4.1%

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

      3. add-exp-log 0.0%

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

      4. expm1-define 0.0%

         \[\leadsto 1 - \frac{1}{x} \cdot \color{blue}{\mathsf{expm1}\left(\log \left(1 + x\right)\right)} \]

      5. log1p-define 0.0%

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

      6. expm1-log1p-u 4.1%

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

   8. Applied egg-rr 4.1%

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

      1. *-inverses 4.1%

         \[\leadsto 1 - \frac{\color{blue}{\frac{x + 2}{x + 2}}}{x} \cdot x \]

      2. associate-/r* 12.1%

         \[\leadsto 1 - \color{blue}{\frac{x + 2}{\left(x + 2\right) \cdot x}} \cdot x \]

      3. clear-num 12.2%

         \[\leadsto 1 - \color{blue}{\frac{1}{\frac{\left(x + 2\right) \cdot x}{x + 2}}} \cdot x \]

      4. associate-/r/ 12.1%

         \[\leadsto 1 - \color{blue}{\left(\frac{1}{\left(x + 2\right) \cdot x} \cdot \left(x + 2\right)\right)} \cdot x \]

      5. *-commutative 12.1%

         \[\leadsto 1 - \left(\frac{1}{\color{blue}{x \cdot \left(x + 2\right)}} \cdot \left(x + 2\right)\right) \cdot x \]

   10. Applied egg-rr 12.1%

       \[\leadsto 1 - \color{blue}{\left(\frac{1}{x \cdot \left(x + 2\right)} \cdot \left(x + 2\right)\right)} \cdot x \]

   if -5e16 < x < -3.999999999999988e-310

   1. Initial program 52.2%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 52.2%

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

      2. *-inverses 52.2%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 52.2%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 52.2%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 52.2%

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

      6. cancel-sign-sub 52.2%

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

      7. *-inverses 52.2%

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

      8. *-inverses 52.2%

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

      9. distribute-neg-frac 52.2%

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

      10. *-inverses 52.2%

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

      11. metadata-eval 52.2%

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

      12. associate-*l/ 52.2%

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

      13. neg-mul-1 52.2%

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

      14. distribute-neg-frac 52.2%

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

      15. distribute-neg-frac2 52.2%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 99.9%

         \[\leadsto 1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)}}{x} \]

      2. expm1-undefine 20.3%

         \[\leadsto 1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right|\right)} - 1}}{x} \]

      3. log1p-undefine 20.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 18.2%

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

      7. rem-exp-log 18.3%

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

   6. Applied egg-rr 18.3%

      \[\leadsto 1 - \frac{\color{blue}{\left(1 + x\right) - 1}}{x} \]
   7. Step-by-step derivation

      1. associate--l+ 18.4%

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

      2. +-commutative 18.4%

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

      3. sub-neg 18.4%

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

      4. metadata-eval 18.4%

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

   8. Applied egg-rr 18.4%

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

   if -3.999999999999988e-310 < x

   1. Initial program 48.7%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 48.7%

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

      2. *-inverses 48.7%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 48.7%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 48.7%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. *-inverses 48.7%

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

      8. *-inverses 48.7%

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

      9. distribute-neg-frac 48.7%

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

      10. *-inverses 48.7%

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

      11. metadata-eval 48.7%

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

      12. associate-*l/ 51.6%

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

      13. neg-mul-1 51.6%

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

      14. distribute-neg-frac 51.6%

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

      15. distribute-neg-frac2 51.6%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
   6. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

      2. rem-square-sqrt 49.1%

         \[\leadsto \frac{x}{x} - \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} \]

      3. fabs-sqr 49.1%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} \]

      4. rem-square-sqrt 100.0%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{x}}{x} \]

      5. *-inverses 100.0%

         \[\leadsto \color{blue}{1} - \frac{x}{x} \]

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+16}:\\ \;\;\;\;1 - x \cdot \left(\left(x + 2\right) \cdot \frac{1}{x \cdot \left(x + 2\right)}\right)\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{1 + \left(x + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+20}:\\ \;\;\;\;1 - \left(x + 2\right) \cdot \frac{x}{x \cdot \left(x + 2\right)}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{1 + \left(x + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e+20)
   (- 1.0 (* (+ x 2.0) (/ x (* x (+ x 2.0)))))
   (if (<= x -4e-310) (- 1.0 (/ (+ 1.0 (+ x -1.0)) x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1e+20) {
		tmp = 1.0 - ((x + 2.0) * (x / (x * (x + 2.0))));
	} else if (x <= -4e-310) {
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1d+20)) then
        tmp = 1.0d0 - ((x + 2.0d0) * (x / (x * (x + 2.0d0))))
    else if (x <= (-4d-310)) then
        tmp = 1.0d0 - ((1.0d0 + (x + (-1.0d0))) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e+20) {
		tmp = 1.0 - ((x + 2.0) * (x / (x * (x + 2.0))));
	} else if (x <= -4e-310) {
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e+20:
		tmp = 1.0 - ((x + 2.0) * (x / (x * (x + 2.0))))
	elif x <= -4e-310:
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e+20)
		tmp = Float64(1.0 - Float64(Float64(x + 2.0) * Float64(x / Float64(x * Float64(x + 2.0)))));
	elseif (x <= -4e-310)
		tmp = Float64(1.0 - Float64(Float64(1.0 + Float64(x + -1.0)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e+20)
		tmp = 1.0 - ((x + 2.0) * (x / (x * (x + 2.0))));
	elseif (x <= -4e-310)
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e+20], N[(1.0 - N[(N[(x + 2.0), $MachinePrecision] * N[(x / N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e-310], N[(1.0 - N[(N[(1.0 + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1e20

   1. Initial program 48.9%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 48.9%

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

      2. *-inverses 48.9%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 48.9%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 48.9%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 48.9%

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

      6. cancel-sign-sub 48.9%

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

      7. *-inverses 48.9%

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

      8. *-inverses 48.9%

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

      9. distribute-neg-frac 48.9%

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

      10. *-inverses 48.9%

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

      11. metadata-eval 48.9%

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

      12. associate-*l/ 48.9%

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

      13. neg-mul-1 48.9%

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

      14. distribute-neg-frac 48.9%

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

      15. distribute-neg-frac2 48.9%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 92.0%

         \[\leadsto 1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)}}{x} \]

      2. expm1-undefine 92.0%

         \[\leadsto 1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right|\right)} - 1}}{x} \]

      3. log1p-undefine 92.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. rem-exp-log 3.1%

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

   6. Applied egg-rr 3.1%

      \[\leadsto 1 - \frac{\color{blue}{\left(1 + x\right) - 1}}{x} \]
   7. Step-by-step derivation

      1. add-exp-log 3.1%

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

      2. log-div 0.0%

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

      3. associate--l+ 0.0%

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

      4. log1p-define 0.0%

         \[\leadsto 1 - e^{\color{blue}{\mathsf{log1p}\left(x - 1\right)} - \log x} \]

      5. add-exp-log 0.0%

         \[\leadsto 1 - e^{\mathsf{log1p}\left(\color{blue}{e^{\log x}} - 1\right) - \log x} \]

      6. expm1-define 0.0%

         \[\leadsto 1 - e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log x\right)}\right) - \log x} \]

      7. log1p-expm1-u 0.0%

         \[\leadsto 1 - e^{\color{blue}{\log x} - \log x} \]

      8. log1p-expm1-u 0.0%

         \[\leadsto 1 - e^{\log x - \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log x\right)\right)}} \]

      9. expm1-define 0.0%

         \[\leadsto 1 - e^{\log x - \mathsf{log1p}\left(\color{blue}{e^{\log x} - 1}\right)} \]

      10. add-exp-log 0.0%

          \[\leadsto 1 - e^{\log x - \mathsf{log1p}\left(\color{blue}{x} - 1\right)} \]

      11. log1p-define 0.0%

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

      12. associate--l+ 0.0%

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

      13. div-exp 0.0%

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

      14. add-exp-log 0.0%

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

      15. add-exp-log 3.1%

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

      16. flip-- 11.8%

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

      17. associate-/r/ 12.1%

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

   8. Applied egg-rr 12.1%

      \[\leadsto 1 - \color{blue}{\frac{x}{\left(x + 2\right) \cdot x} \cdot \left(x + 2\right)} \]

   if -1e20 < x < -3.999999999999988e-310

   1. Initial program 52.2%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 52.2%

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

      2. *-inverses 52.2%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 52.2%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 52.2%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 52.2%

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

      6. cancel-sign-sub 52.2%

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

      7. *-inverses 52.2%

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

      8. *-inverses 52.2%

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

      9. distribute-neg-frac 52.2%

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

      10. *-inverses 52.2%

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

      11. metadata-eval 52.2%

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

      12. associate-*l/ 52.2%

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

      13. neg-mul-1 52.2%

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

      14. distribute-neg-frac 52.2%

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

      15. distribute-neg-frac2 52.2%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 99.9%

         \[\leadsto 1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)}}{x} \]

      2. expm1-undefine 20.3%

         \[\leadsto 1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right|\right)} - 1}}{x} \]

      3. log1p-undefine 20.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 18.2%

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

      7. rem-exp-log 18.3%

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

   6. Applied egg-rr 18.3%

      \[\leadsto 1 - \frac{\color{blue}{\left(1 + x\right) - 1}}{x} \]
   7. Step-by-step derivation

      1. associate--l+ 18.4%

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

      2. +-commutative 18.4%

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

      3. sub-neg 18.4%

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

      4. metadata-eval 18.4%

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

   8. Applied egg-rr 18.4%

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

   if -3.999999999999988e-310 < x

   1. Initial program 48.7%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 48.7%

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

      2. *-inverses 48.7%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 48.7%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 48.7%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. *-inverses 48.7%

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

      8. *-inverses 48.7%

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

      9. distribute-neg-frac 48.7%

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

      10. *-inverses 48.7%

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

      11. metadata-eval 48.7%

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

      12. associate-*l/ 51.6%

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

      13. neg-mul-1 51.6%

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

      14. distribute-neg-frac 51.6%

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

      15. distribute-neg-frac2 51.6%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
   6. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

      2. rem-square-sqrt 49.1%

         \[\leadsto \frac{x}{x} - \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} \]

      3. fabs-sqr 49.1%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} \]

      4. rem-square-sqrt 100.0%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{x}}{x} \]

      5. *-inverses 100.0%

         \[\leadsto \color{blue}{1} - \frac{x}{x} \]

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+20}:\\ \;\;\;\;1 - \left(x + 2\right) \cdot \frac{x}{x \cdot \left(x + 2\right)}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{1 + \left(x + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{1 + \left(x + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310) (- 1.0 (/ (+ 1.0 (+ x -1.0)) x)) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-310)) then
        tmp = 1.0d0 - ((1.0d0 + (x + (-1.0d0))) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(1.0 - Float64(Float64(1.0 + Float64(x + -1.0)) / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = 1.0 - ((1.0 + (x + -1.0)) / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(1.0 - N[(N[(1.0 + N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 50.7%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 50.7%

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

      2. *-inverses 50.7%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 50.7%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 50.7%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 50.7%

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

      6. cancel-sign-sub 50.7%

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

      7. *-inverses 50.7%

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

      8. *-inverses 50.7%

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

      9. distribute-neg-frac 50.7%

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

      10. *-inverses 50.7%

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

      11. metadata-eval 50.7%

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

      12. associate-*l/ 50.7%

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

      13. neg-mul-1 50.7%

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

      14. distribute-neg-frac 50.7%

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

      15. distribute-neg-frac2 50.7%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 96.4%

         \[\leadsto 1 - \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right|\right)\right)}}{x} \]

      2. expm1-undefine 52.4%

         \[\leadsto 1 - \frac{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right|\right)} - 1}}{x} \]

      3. log1p-undefine 52.4%

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

      4. add-sqr-sqrt 0.0%

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

      5. fabs-sqr 0.0%

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

      6. add-sqr-sqrt 10.1%

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

      7. rem-exp-log 11.5%

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

   6. Applied egg-rr 11.5%

      \[\leadsto 1 - \frac{\color{blue}{\left(1 + x\right) - 1}}{x} \]
   7. Step-by-step derivation

      1. associate--l+ 11.6%

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

      2. +-commutative 11.6%

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

      3. sub-neg 11.6%

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

      4. metadata-eval 11.6%

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

   8. Applied egg-rr 11.6%

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

   if -3.999999999999988e-310 < x

   1. Initial program 48.7%

      \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
   2. Step-by-step derivation

      1. cancel-sign-sub-inv 48.7%

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

      2. *-inverses 48.7%

         \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

      3. distribute-frac-neg2 48.7%

         \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

      4. sqr-neg 48.7%

         \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

      5. *-inverses 48.7%

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

      6. cancel-sign-sub 48.7%

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

      7. *-inverses 48.7%

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

      8. *-inverses 48.7%

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

      9. distribute-neg-frac 48.7%

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

      10. *-inverses 48.7%

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

      11. metadata-eval 48.7%

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

      12. associate-*l/ 51.6%

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

      13. neg-mul-1 51.6%

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

      14. distribute-neg-frac 51.6%

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

      15. distribute-neg-frac2 51.6%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
   6. Step-by-step derivation

      1. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

      2. rem-square-sqrt 49.1%

         \[\leadsto \frac{x}{x} - \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} \]

      3. fabs-sqr 49.1%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} \]

      4. rem-square-sqrt 100.0%

         \[\leadsto \frac{x}{x} - \frac{\color{blue}{x}}{x} \]

      5. *-inverses 100.0%

         \[\leadsto \color{blue}{1} - \frac{x}{x} \]

      6. *-inverses 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;1 - \frac{1 + \left(x + -1\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.1% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 49.7%

   \[\frac{x}{x} - \frac{1}{x} \cdot \sqrt{x \cdot x} \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 49.7%

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

   2. *-inverses 49.7%

      \[\leadsto \frac{x}{x} + \left(-\frac{\color{blue}{\frac{x}{x}}}{x}\right) \cdot \sqrt{x \cdot x} \]

   3. distribute-frac-neg2 49.7%

      \[\leadsto \frac{x}{x} + \color{blue}{\frac{\frac{x}{x}}{-x}} \cdot \sqrt{x \cdot x} \]

   4. sqr-neg 49.7%

      \[\leadsto \frac{x}{x} + \frac{\frac{x}{x}}{-x} \cdot \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \]

   5. *-inverses 49.7%

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

   6. cancel-sign-sub 49.7%

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

   7. *-inverses 49.7%

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

   8. *-inverses 49.7%

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

   9. distribute-neg-frac 49.7%

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

   10. *-inverses 49.7%

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

   11. metadata-eval 49.7%

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

   12. associate-*l/ 51.2%

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

   13. neg-mul-1 51.2%

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

   14. distribute-neg-frac 51.2%

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

   15. distribute-neg-frac2 51.2%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{1 - \frac{\left|x\right|}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\frac{x - \left|x\right|}{x}} \]
6. Step-by-step derivation

   1. div-sub 100.0%

      \[\leadsto \color{blue}{\frac{x}{x} - \frac{\left|x\right|}{x}} \]

   2. rem-square-sqrt 25.5%

      \[\leadsto \frac{x}{x} - \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{x} \]

   3. fabs-sqr 25.5%

      \[\leadsto \frac{x}{x} - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{x} \]

   4. rem-square-sqrt 53.5%

      \[\leadsto \frac{x}{x} - \frac{\color{blue}{x}}{x} \]

   5. *-inverses 53.5%

      \[\leadsto \color{blue}{1} - \frac{x}{x} \]

   6. *-inverses 53.5%

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

   7. metadata-eval 53.5%

      \[\leadsto \color{blue}{0} \]

7. Simplified 53.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 100.0% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (< x 0.0) 2.0 0.0))

C

double code(double x) {
	double tmp;
	if (x < 0.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x < 0.0d0) then
        tmp = 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x < 0.0) {
		tmp = 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x < 0.0:
		tmp = 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x < 0.0)
		tmp = 2.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x < 0.0)
		tmp = 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[x, 0.0], 2.0, 0.0]

Reproduce

herbie shell --seed 2024179 
(FPCore (x)
  :name "sqrt sqr"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) 2 0))

  (- (/ x x) (* (/ 1.0 x) (sqrt (* x x)))))