Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Percentage Accurate: 99.7% → 99.6%

Time: 10.5s

Alternatives: 16

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (1.0d0 / (x * 9.0d0))) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(1.0 / Float64(x * 9.0))) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (1.0 / (x * 9.0))) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{\frac{-1}{x}}{9}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (+ 1.0 (/ (/ -1.0 x) 9.0)) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 + ((-1.0 / x) / 9.0)) - (y / sqrt((x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + (((-1.0d0) / x) / 9.0d0)) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 + ((-1.0 / x) / 9.0)) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 + ((-1.0 / x) / 9.0)) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(Float64(-1.0 / x) / 9.0)) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + ((-1.0 / x) / 9.0)) - (y / sqrt((x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(N[(-1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

   2. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

   3. sqrt-prod 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

3. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
4. Step-by-step derivation

   1. unpow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

5. Simplified 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
7. Step-by-step derivation

   1. div-inv 99.7%

      \[\leadsto \left(1 - \color{blue}{0.1111111111111111 \cdot \frac{1}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   2. add-sqr-sqrt 99.6%

      \[\leadsto \left(1 - 0.1111111111111111 \cdot \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right)}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   3. associate-*r* 99.6%

      \[\leadsto \left(1 - \color{blue}{\left(0.1111111111111111 \cdot \sqrt{\frac{1}{x}}\right) \cdot \sqrt{\frac{1}{x}}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   4. inv-pow 99.6%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot \sqrt{\color{blue}{{x}^{-1}}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   5. sqrt-pow1 99.6%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   6. metadata-eval 99.6%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot {x}^{\color{blue}{-0.5}}\right) \cdot \sqrt{\frac{1}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   7. inv-pow 99.6%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot \sqrt{\color{blue}{{x}^{-1}}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   8. sqrt-pow1 99.6%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   9. metadata-eval 99.6%

      \[\leadsto \left(1 - \left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot {x}^{\color{blue}{-0.5}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

8. Applied egg-rr 99.6%

   \[\leadsto \left(1 - \color{blue}{\left(0.1111111111111111 \cdot {x}^{-0.5}\right) \cdot {x}^{-0.5}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]
9. Step-by-step derivation

   1. associate-*l* 99.6%

      \[\leadsto \left(1 - \color{blue}{0.1111111111111111 \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   2. metadata-eval 99.6%

      \[\leadsto \left(1 - \color{blue}{{9}^{-1}} \cdot \left({x}^{-0.5} \cdot {x}^{-0.5}\right)\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   3. pow-prod-up 99.7%

      \[\leadsto \left(1 - {9}^{-1} \cdot \color{blue}{{x}^{\left(-0.5 + -0.5\right)}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   4. metadata-eval 99.7%

      \[\leadsto \left(1 - {9}^{-1} \cdot {x}^{\color{blue}{-1}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   5. unpow-prod-down 99.7%

      \[\leadsto \left(1 - \color{blue}{{\left(9 \cdot x\right)}^{-1}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   6. *-commutative 99.7%

      \[\leadsto \left(1 - {\color{blue}{\left(x \cdot 9\right)}}^{-1}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   7. unpow-1 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{1}{x \cdot 9}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   8. associate-/r* 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

10. Applied egg-rr 99.7%

    \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{x}}{9}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

11. Final simplification 99.7%

    \[\leadsto \left(1 + \frac{\frac{-1}{x}}{9}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

Alternative 2: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91} \lor \neg \left(y \leq -6 \cdot 10^{+45}\right) \land \left(y \leq -1.15 \cdot 10^{+16} \lor \neg \left(y \leq 8.5 \cdot 10^{+25}\right)\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e+91)
         (and (not (<= y -6e+45)) (or (<= y -1.15e+16) (not (<= y 8.5e+25)))))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+91) || (!(y <= -6e+45) && ((y <= -1.15e+16) || !(y <= 8.5e+25)))) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.2d+91)) .or. (.not. (y <= (-6d+45))) .and. (y <= (-1.15d+16)) .or. (.not. (y <= 8.5d+25))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+91) || (!(y <= -6e+45) && ((y <= -1.15e+16) || !(y <= 8.5e+25)))) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e+91) or (not (y <= -6e+45) and ((y <= -1.15e+16) or not (y <= 8.5e+25))):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e+91) || (!(y <= -6e+45) && ((y <= -1.15e+16) || !(y <= 8.5e+25))))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e+91) || (~((y <= -6e+45)) && ((y <= -1.15e+16) || ~((y <= 8.5e+25)))))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e+91], And[N[Not[LessEqual[y, -6e+45]], $MachinePrecision], Or[LessEqual[y, -1.15e+16], N[Not[LessEqual[y, 8.5e+25]], $MachinePrecision]]]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999995e91 or -6.00000000000000021e45 < y < -1.15e16 or 8.5000000000000007e25 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.4%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 94.5%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)\right)} \]

      2. expm1-udef 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)} - 1\right)} \]

      3. *-commutative 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \sqrt{\frac{1}{x}}}\right)} - 1\right) \]

      4. sqrt-div 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} - 1\right) \]

      5. metadata-eval 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} - 1\right) \]

      6. div-inv 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\sqrt{x}}}\right)} - 1\right) \]

   6. Applied egg-rr 51.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 94.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   8. Simplified 94.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   if -6.19999999999999995e91 < y < -6.00000000000000021e45 or -1.15e16 < y < 8.5000000000000007e25

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 98.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91} \lor \neg \left(y \leq -6 \cdot 10^{+45}\right) \land \left(y \leq -1.15 \cdot 10^{+16} \lor \neg \left(y \leq 8.5 \cdot 10^{+25}\right)\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 3: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91} \lor \neg \left(y \leq -2.7 \cdot 10^{+51} \lor \neg \left(y \leq -1.08 \cdot 10^{+16}\right) \land y \leq 10^{+28}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.2e+91)
         (not (or (<= y -2.7e+51) (and (not (<= y -1.08e+16)) (<= y 1e+28)))))
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+91) || !((y <= -2.7e+51) || (!(y <= -1.08e+16) && (y <= 1e+28)))) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-6.2d+91)) .or. (.not. (y <= (-2.7d+51)) .or. (.not. (y <= (-1.08d+16))) .and. (y <= 1d+28))) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.2e+91) || !((y <= -2.7e+51) || (!(y <= -1.08e+16) && (y <= 1e+28)))) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.2e+91) or not ((y <= -2.7e+51) or (not (y <= -1.08e+16) and (y <= 1e+28))):
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.2e+91) || !((y <= -2.7e+51) || (!(y <= -1.08e+16) && (y <= 1e+28))))
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -6.2e+91) || ~(((y <= -2.7e+51) || (~((y <= -1.08e+16)) && (y <= 1e+28)))))
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.2e+91], N[Not[Or[LessEqual[y, -2.7e+51], And[N[Not[LessEqual[y, -1.08e+16]], $MachinePrecision], LessEqual[y, 1e+28]]]], $MachinePrecision]], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.19999999999999995e91 or -2.69999999999999992e51 < y < -1.08e16 or 9.99999999999999958e27 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.4%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 94.5%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)\right)} \]

      2. expm1-udef 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)} - 1\right)} \]

      3. *-commutative 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \sqrt{\frac{1}{x}}}\right)} - 1\right) \]

      4. sqrt-div 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} - 1\right) \]

      5. metadata-eval 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} - 1\right) \]

      6. div-inv 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\sqrt{x}}}\right)} - 1\right) \]

   6. Applied egg-rr 51.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 51.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 94.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   8. Simplified 94.7%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 94.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}} \]

      2. associate-*l/ 94.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot y} \]

      3. clear-num 94.7%

         \[\leadsto 1 + \color{blue}{\frac{1}{\frac{\sqrt{x}}{-0.3333333333333333}}} \cdot y \]

      4. div-inv 94.7%

         \[\leadsto 1 + \frac{1}{\color{blue}{\sqrt{x} \cdot \frac{1}{-0.3333333333333333}}} \cdot y \]

      5. metadata-eval 94.7%

         \[\leadsto 1 + \frac{1}{\sqrt{x} \cdot \color{blue}{-3}} \cdot y \]

      6. metadata-eval 94.7%

         \[\leadsto 1 + \frac{1}{\sqrt{x} \cdot \color{blue}{\left(-3\right)}} \cdot y \]

      7. distribute-rgt-neg-in 94.7%

         \[\leadsto 1 + \frac{1}{\color{blue}{-\sqrt{x} \cdot 3}} \cdot y \]

      8. *-commutative 94.7%

         \[\leadsto 1 + \frac{1}{-\color{blue}{3 \cdot \sqrt{x}}} \cdot y \]

      9. associate-*l/ 94.9%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot y}{-3 \cdot \sqrt{x}}} \]

      10. *-un-lft-identity 94.9%

          \[\leadsto 1 + \frac{\color{blue}{y}}{-3 \cdot \sqrt{x}} \]

      11. *-commutative 94.9%

          \[\leadsto 1 + \frac{y}{-\color{blue}{\sqrt{x} \cdot 3}} \]

      12. distribute-rgt-neg-in 94.9%

          \[\leadsto 1 + \frac{y}{\color{blue}{\sqrt{x} \cdot \left(-3\right)}} \]

      13. metadata-eval 94.9%

          \[\leadsto 1 + \frac{y}{\sqrt{x} \cdot \color{blue}{-3}} \]

   10. Applied egg-rr 94.9%

       \[\leadsto 1 + \color{blue}{\frac{y}{\sqrt{x} \cdot -3}} \]

   if -6.19999999999999995e91 < y < -2.69999999999999992e51 or -1.08e16 < y < 9.99999999999999958e27

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 98.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91} \lor \neg \left(y \leq -2.7 \cdot 10^{+51} \lor \neg \left(y \leq -1.08 \cdot 10^{+16}\right) \land y \leq 10^{+28}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-0.1111111111111111}{x}\\ t_1 := 1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -0.1111111111111111 x)))
        (t_1 (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))))
   (if (<= y -6.2e+91)
     t_1
     (if (<= y -1.65e+47)
       t_0
       (if (<= y -1.15e+16)
         t_1
         (if (<= y 2.5e+27)
           t_0
           (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.1111111111111111 / x);
	double t_1 = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	double tmp;
	if (y <= -6.2e+91) {
		tmp = t_1;
	} else if (y <= -1.65e+47) {
		tmp = t_0;
	} else if (y <= -1.15e+16) {
		tmp = t_1;
	} else if (y <= 2.5e+27) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.1111111111111111d0) / x)
    t_1 = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    if (y <= (-6.2d+91)) then
        tmp = t_1
    else if (y <= (-1.65d+47)) then
        tmp = t_0
    else if (y <= (-1.15d+16)) then
        tmp = t_1
    else if (y <= 2.5d+27) then
        tmp = t_0
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-0.1111111111111111 / x);
	double t_1 = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	double tmp;
	if (y <= -6.2e+91) {
		tmp = t_1;
	} else if (y <= -1.65e+47) {
		tmp = t_0;
	} else if (y <= -1.15e+16) {
		tmp = t_1;
	} else if (y <= 2.5e+27) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-0.1111111111111111 / x)
	t_1 = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	tmp = 0
	if y <= -6.2e+91:
		tmp = t_1
	elif y <= -1.65e+47:
		tmp = t_0
	elif y <= -1.15e+16:
		tmp = t_1
	elif y <= 2.5e+27:
		tmp = t_0
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.1111111111111111 / x))
	t_1 = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))))
	tmp = 0.0
	if (y <= -6.2e+91)
		tmp = t_1;
	elseif (y <= -1.65e+47)
		tmp = t_0;
	elseif (y <= -1.15e+16)
		tmp = t_1;
	elseif (y <= 2.5e+27)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-0.1111111111111111 / x);
	t_1 = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	tmp = 0.0;
	if (y <= -6.2e+91)
		tmp = t_1;
	elseif (y <= -1.65e+47)
		tmp = t_0;
	elseif (y <= -1.15e+16)
		tmp = t_1;
	elseif (y <= 2.5e+27)
		tmp = t_0;
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+91], t$95$1, If[LessEqual[y, -1.65e+47], t$95$0, If[LessEqual[y, -1.15e+16], t$95$1, If[LessEqual[y, 2.5e+27], t$95$0, N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.19999999999999995e91 or -1.65e47 < y < -1.15e16

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.3%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.3%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 95.6%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)\right)} \]

      2. expm1-udef 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)} - 1\right)} \]

      3. *-commutative 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \sqrt{\frac{1}{x}}}\right)} - 1\right) \]

      4. sqrt-div 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} - 1\right) \]

      5. metadata-eval 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} - 1\right) \]

      6. div-inv 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\sqrt{x}}}\right)} - 1\right) \]

   6. Applied egg-rr 6.0%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 95.9%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   8. Simplified 95.9%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   if -6.19999999999999995e91 < y < -1.65e47 or -1.15e16 < y < 2.4999999999999999e27

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 98.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   if 2.4999999999999999e27 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.5%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 93.7%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. *-commutative 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(y \cdot \sqrt{\frac{1}{x}}\right)} \]

      2. associate-*r* 93.7%

         \[\leadsto 1 + \color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot \sqrt{\frac{1}{x}}} \]

      3. sqrt-div 93.7%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

      4. metadata-eval 93.7%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

      5. div-inv 93.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}} \]

      6. pow1/2 93.7%

         \[\leadsto 1 + \frac{-0.3333333333333333 \cdot y}{\color{blue}{{x}^{0.5}}} \]

      7. metadata-eval 93.7%

         \[\leadsto 1 + \frac{-0.3333333333333333 \cdot y}{{x}^{\color{blue}{\left(0.25 + 0.25\right)}}} \]

      8. pow-prod-up 93.6%

         \[\leadsto 1 + \frac{-0.3333333333333333 \cdot y}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25}}} \]

      9. expm1-log1p-u 11.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot y}{{x}^{0.25} \cdot {x}^{0.25}}\right)\right)} \]

      10. associate-/l/ 11.1%

          \[\leadsto 1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{-0.3333333333333333 \cdot y}{{x}^{0.25}}}{{x}^{0.25}}}\right)\right) \]

      11. expm1-udef 11.1%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-0.3333333333333333 \cdot y}{{x}^{0.25}}}{{x}^{0.25}}\right)} - 1\right)} \]

   6. Applied egg-rr 11.1%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 11.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 93.8%

         \[\leadsto 1 + \color{blue}{y \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

   8. Simplified 93.8%

      \[\leadsto 1 + \color{blue}{y \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{+16}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 5: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-0.1111111111111111}{x}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -0.1111111111111111 x))))
   (if (<= y -6.2e+91)
     (+ 1.0 (/ -0.3333333333333333 (/ (sqrt x) y)))
     (if (<= y -2.15e+50)
       t_0
       (if (<= y -1.1e+16)
         (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
         (if (<= y 8.6e+27)
           t_0
           (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.1111111111111111 / x);
	double tmp;
	if (y <= -6.2e+91) {
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	} else if (y <= -2.15e+50) {
		tmp = t_0;
	} else if (y <= -1.1e+16) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 8.6e+27) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.1111111111111111d0) / x)
    if (y <= (-6.2d+91)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) / (sqrt(x) / y))
    else if (y <= (-2.15d+50)) then
        tmp = t_0
    else if (y <= (-1.1d+16)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 8.6d+27) then
        tmp = t_0
    else
        tmp = 1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-0.1111111111111111 / x);
	double tmp;
	if (y <= -6.2e+91) {
		tmp = 1.0 + (-0.3333333333333333 / (Math.sqrt(x) / y));
	} else if (y <= -2.15e+50) {
		tmp = t_0;
	} else if (y <= -1.1e+16) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 8.6e+27) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-0.1111111111111111 / x)
	tmp = 0
	if y <= -6.2e+91:
		tmp = 1.0 + (-0.3333333333333333 / (math.sqrt(x) / y))
	elif y <= -2.15e+50:
		tmp = t_0
	elif y <= -1.1e+16:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 8.6e+27:
		tmp = t_0
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -6.2e+91)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)));
	elseif (y <= -2.15e+50)
		tmp = t_0;
	elseif (y <= -1.1e+16)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 8.6e+27)
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= -6.2e+91)
		tmp = 1.0 + (-0.3333333333333333 / (sqrt(x) / y));
	elseif (y <= -2.15e+50)
		tmp = t_0;
	elseif (y <= -1.1e+16)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 8.6e+27)
		tmp = t_0;
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+91], N[(1.0 + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.15e+50], t$95$0, If[LessEqual[y, -1.1e+16], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+27], t$95$0, N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.19999999999999995e91

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.2%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 99.2%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 99.4%

         \[\leadsto 1 + \color{blue}{\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right) \cdot y} \]

      2. sqrt-div 99.3%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right) \cdot y \]

      3. metadata-eval 99.3%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right) \cdot y \]

      4. div-inv 99.4%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}}} \cdot y \]

      5. associate-/r/ 99.5%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}} \]

   6. Applied egg-rr 99.5%

      \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}} \]

   if -6.19999999999999995e91 < y < -2.1499999999999999e50 or -1.1e16 < y < 8.60000000000000017e27

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 98.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   if -2.1499999999999999e50 < y < -1.1e16

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.8%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.8%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.8%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.8%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.8%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.8%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 79.5%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)\right)} \]

      2. expm1-udef 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)} - 1\right)} \]

      3. *-commutative 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \sqrt{\frac{1}{x}}}\right)} - 1\right) \]

      4. sqrt-div 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} - 1\right) \]

      5. metadata-eval 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} - 1\right) \]

      6. div-inv 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\sqrt{x}}}\right)} - 1\right) \]

   6. Applied egg-rr 22.2%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 22.2%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 80.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   8. Simplified 80.0%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   if 8.60000000000000017e27 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.5%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 93.7%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. *-commutative 93.7%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(y \cdot \sqrt{\frac{1}{x}}\right)} \]

      2. associate-*r* 93.7%

         \[\leadsto 1 + \color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot \sqrt{\frac{1}{x}}} \]

      3. sqrt-div 93.7%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}} \]

      4. metadata-eval 93.7%

         \[\leadsto 1 + \left(-0.3333333333333333 \cdot y\right) \cdot \frac{\color{blue}{1}}{\sqrt{x}} \]

      5. div-inv 93.7%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}} \]

      6. pow1/2 93.7%

         \[\leadsto 1 + \frac{-0.3333333333333333 \cdot y}{\color{blue}{{x}^{0.5}}} \]

      7. metadata-eval 93.7%

         \[\leadsto 1 + \frac{-0.3333333333333333 \cdot y}{{x}^{\color{blue}{\left(0.25 + 0.25\right)}}} \]

      8. pow-prod-up 93.6%

         \[\leadsto 1 + \frac{-0.3333333333333333 \cdot y}{\color{blue}{{x}^{0.25} \cdot {x}^{0.25}}} \]

      9. expm1-log1p-u 11.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.3333333333333333 \cdot y}{{x}^{0.25} \cdot {x}^{0.25}}\right)\right)} \]

      10. associate-/l/ 11.1%

          \[\leadsto 1 + \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\frac{-0.3333333333333333 \cdot y}{{x}^{0.25}}}{{x}^{0.25}}}\right)\right) \]

      11. expm1-udef 11.1%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{-0.3333333333333333 \cdot y}{{x}^{0.25}}}{{x}^{0.25}}\right)} - 1\right)} \]

   6. Applied egg-rr 11.1%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 11.1%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 93.8%

         \[\leadsto 1 + \color{blue}{y \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]

   8. Simplified 93.8%

      \[\leadsto 1 + \color{blue}{y \cdot \frac{-0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+27}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 6: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-0.1111111111111111}{x}\\ t_1 := 1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{if}\;y \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -0.1111111111111111 x)))
        (t_1 (+ 1.0 (/ y (* (sqrt x) -3.0)))))
   (if (<= y -6.2e+91)
     t_1
     (if (<= y -1.4e+50)
       t_0
       (if (<= y -1.02e+16)
         t_1
         (if (<= y 3.8e+26) t_0 (- 1.0 (/ y (sqrt (* x 9.0))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-0.1111111111111111 / x);
	double t_1 = 1.0 + (y / (sqrt(x) * -3.0));
	double tmp;
	if (y <= -6.2e+91) {
		tmp = t_1;
	} else if (y <= -1.4e+50) {
		tmp = t_0;
	} else if (y <= -1.02e+16) {
		tmp = t_1;
	} else if (y <= 3.8e+26) {
		tmp = t_0;
	} else {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((-0.1111111111111111d0) / x)
    t_1 = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    if (y <= (-6.2d+91)) then
        tmp = t_1
    else if (y <= (-1.4d+50)) then
        tmp = t_0
    else if (y <= (-1.02d+16)) then
        tmp = t_1
    else if (y <= 3.8d+26) then
        tmp = t_0
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-0.1111111111111111 / x);
	double t_1 = 1.0 + (y / (Math.sqrt(x) * -3.0));
	double tmp;
	if (y <= -6.2e+91) {
		tmp = t_1;
	} else if (y <= -1.4e+50) {
		tmp = t_0;
	} else if (y <= -1.02e+16) {
		tmp = t_1;
	} else if (y <= 3.8e+26) {
		tmp = t_0;
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-0.1111111111111111 / x)
	t_1 = 1.0 + (y / (math.sqrt(x) * -3.0))
	tmp = 0
	if y <= -6.2e+91:
		tmp = t_1
	elif y <= -1.4e+50:
		tmp = t_0
	elif y <= -1.02e+16:
		tmp = t_1
	elif y <= 3.8e+26:
		tmp = t_0
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-0.1111111111111111 / x))
	t_1 = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)))
	tmp = 0.0
	if (y <= -6.2e+91)
		tmp = t_1;
	elseif (y <= -1.4e+50)
		tmp = t_0;
	elseif (y <= -1.02e+16)
		tmp = t_1;
	elseif (y <= 3.8e+26)
		tmp = t_0;
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-0.1111111111111111 / x);
	t_1 = 1.0 + (y / (sqrt(x) * -3.0));
	tmp = 0.0;
	if (y <= -6.2e+91)
		tmp = t_1;
	elseif (y <= -1.4e+50)
		tmp = t_0;
	elseif (y <= -1.02e+16)
		tmp = t_1;
	elseif (y <= 3.8e+26)
		tmp = t_0;
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.2e+91], t$95$1, If[LessEqual[y, -1.4e+50], t$95$0, If[LessEqual[y, -1.02e+16], t$95$1, If[LessEqual[y, 3.8e+26], t$95$0, N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.19999999999999995e91 or -1.3999999999999999e50 < y < -1.02e16

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. associate--l- 99.6%

         \[\leadsto \color{blue}{1 - \left(\frac{1}{x \cdot 9} + \frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 - \color{blue}{\left(\frac{y}{3 \cdot \sqrt{x}} + \frac{1}{x \cdot 9}\right)} \]

      3. associate--r+ 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{y}{3 \cdot \sqrt{x}}\right) - \frac{1}{x \cdot 9}} \]

      4. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)\right)} - \frac{1}{x \cdot 9} \]

      5. distribute-frac-neg 99.6%

         \[\leadsto \left(1 + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}}\right) - \frac{1}{x \cdot 9} \]

      6. associate-+r- 99.6%

         \[\leadsto \color{blue}{1 + \left(\frac{-y}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right)} \]

      7. neg-mul-1 99.6%

         \[\leadsto 1 + \left(\frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} - \frac{1}{x \cdot 9}\right) \]

      8. associate-*l/ 99.3%

         \[\leadsto 1 + \left(\color{blue}{\frac{-1}{3 \cdot \sqrt{x}} \cdot y} - \frac{1}{x \cdot 9}\right) \]

      9. fma-neg 99.3%

         \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(\frac{-1}{3 \cdot \sqrt{x}}, y, -\frac{1}{x \cdot 9}\right)} \]

      10. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{3}}{\sqrt{x}}}, y, -\frac{1}{x \cdot 9}\right) \]

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, y, -\frac{1}{x \cdot 9}\right) \]

      12. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{-1}{x \cdot 9}}\right) \]

      13. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-1}}{x \cdot 9}\right) \]

      14. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-1}{\color{blue}{9 \cdot x}}\right) \]

      15. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \color{blue}{\frac{\frac{-1}{9}}{x}}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(\frac{-0.3333333333333333}{\sqrt{x}}, y, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 95.6%

      \[\leadsto 1 + \color{blue}{-0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot y\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)\right)} \]

      2. expm1-udef 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{x}} \cdot y\right)} - 1\right)} \]

      3. *-commutative 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{y \cdot \sqrt{\frac{1}{x}}}\right)} - 1\right) \]

      4. sqrt-div 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x}}}\right)} - 1\right) \]

      5. metadata-eval 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(y \cdot \frac{\color{blue}{1}}{\sqrt{x}}\right)} - 1\right) \]

      6. div-inv 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\sqrt{x}}}\right)} - 1\right) \]

   6. Applied egg-rr 6.0%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 6.0%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\sqrt{x}}\right)\right)} \]

      2. expm1-log1p 95.9%

         \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   8. Simplified 95.9%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 95.8%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333 \cdot y}{\sqrt{x}}} \]

      2. associate-*l/ 95.8%

         \[\leadsto 1 + \color{blue}{\frac{-0.3333333333333333}{\sqrt{x}} \cdot y} \]

      3. clear-num 95.7%

         \[\leadsto 1 + \color{blue}{\frac{1}{\frac{\sqrt{x}}{-0.3333333333333333}}} \cdot y \]

      4. div-inv 95.8%

         \[\leadsto 1 + \frac{1}{\color{blue}{\sqrt{x} \cdot \frac{1}{-0.3333333333333333}}} \cdot y \]

      5. metadata-eval 95.8%

         \[\leadsto 1 + \frac{1}{\sqrt{x} \cdot \color{blue}{-3}} \cdot y \]

      6. metadata-eval 95.8%

         \[\leadsto 1 + \frac{1}{\sqrt{x} \cdot \color{blue}{\left(-3\right)}} \cdot y \]

      7. distribute-rgt-neg-in 95.8%

         \[\leadsto 1 + \frac{1}{\color{blue}{-\sqrt{x} \cdot 3}} \cdot y \]

      8. *-commutative 95.8%

         \[\leadsto 1 + \frac{1}{-\color{blue}{3 \cdot \sqrt{x}}} \cdot y \]

      9. associate-*l/ 96.1%

         \[\leadsto 1 + \color{blue}{\frac{1 \cdot y}{-3 \cdot \sqrt{x}}} \]

      10. *-un-lft-identity 96.1%

          \[\leadsto 1 + \frac{\color{blue}{y}}{-3 \cdot \sqrt{x}} \]

      11. *-commutative 96.1%

          \[\leadsto 1 + \frac{y}{-\color{blue}{\sqrt{x} \cdot 3}} \]

      12. distribute-rgt-neg-in 96.1%

          \[\leadsto 1 + \frac{y}{\color{blue}{\sqrt{x} \cdot \left(-3\right)}} \]

      13. metadata-eval 96.1%

          \[\leadsto 1 + \frac{y}{\sqrt{x} \cdot \color{blue}{-3}} \]

   10. Applied egg-rr 96.1%

       \[\leadsto 1 + \color{blue}{\frac{y}{\sqrt{x} \cdot -3}} \]

   if -6.19999999999999995e91 < y < -1.3999999999999999e50 or -1.02e16 < y < 3.8000000000000002e26

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 98.9%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 98.9%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 98.9%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 98.9%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 98.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   if 3.8000000000000002e26 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      2. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

      3. sqrt-prod 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

      4. pow1/2 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   4. Step-by-step derivation

      1. unpow1/2 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   5. Simplified 99.8%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   6. Taylor expanded in x around 0 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   7. Taylor expanded in x around inf 94.1%

      \[\leadsto \color{blue}{1} - \frac{y}{\sqrt{x \cdot 9}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+91}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{+50}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]

Alternative 7: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (+ 1.0 (/ -1.0 (* x 9.0))) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 + (-1.0 / (x * 9.0))) - (y / sqrt((x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((-1.0d0) / (x * 9.0d0))) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 + (-1.0 / (x * 9.0))) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 + (-1.0 / (x * 9.0))) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(-1.0 / Float64(x * 9.0))) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (-1.0 / (x * 9.0))) - (y / sqrt((x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

   2. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

   3. sqrt-prod 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

3. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
4. Step-by-step derivation

   1. unpow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

5. Simplified 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

6. Final simplification 99.7%

   \[\leadsto \left(1 + \frac{-1}{x \cdot 9}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - t_0\\ \mathbf{else}:\\ \;\;\;\;1 - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (sqrt (* x 9.0)))))
   (if (<= x 0.112) (- (/ -0.1111111111111111 x) t_0) (- 1.0 t_0))))

C

double code(double x, double y) {
	double t_0 = y / sqrt((x * 9.0));
	double tmp;
	if (x <= 0.112) {
		tmp = (-0.1111111111111111 / x) - t_0;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / sqrt((x * 9.0d0))
    if (x <= 0.112d0) then
        tmp = ((-0.1111111111111111d0) / x) - t_0
    else
        tmp = 1.0d0 - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / Math.sqrt((x * 9.0));
	double tmp;
	if (x <= 0.112) {
		tmp = (-0.1111111111111111 / x) - t_0;
	} else {
		tmp = 1.0 - t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / math.sqrt((x * 9.0))
	tmp = 0
	if x <= 0.112:
		tmp = (-0.1111111111111111 / x) - t_0
	else:
		tmp = 1.0 - t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / sqrt(Float64(x * 9.0)))
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(-0.1111111111111111 / x) - t_0);
	else
		tmp = Float64(1.0 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / sqrt((x * 9.0));
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (-0.1111111111111111 / x) - t_0;
	else
		tmp = 1.0 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.112], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - t$95$0), $MachinePrecision], N[(1.0 - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      2. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

      3. sqrt-prod 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

      4. pow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   4. Step-by-step derivation

      1. unpow1/2 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   5. Simplified 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   7. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} - \frac{y}{\sqrt{x \cdot 9}} \]

   if 0.112000000000000002 < x

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

      2. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

      3. sqrt-prod 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

      4. pow1/2 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

   3. Applied egg-rr 99.8%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
   4. Step-by-step derivation

      1. unpow1/2 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   5. Simplified 99.8%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   6. Taylor expanded in x around 0 99.8%

      \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

   7. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{1} - \frac{y}{\sqrt{x \cdot 9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]

Alternative 9: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- 1.0 (/ 0.1111111111111111 x)) (* -0.3333333333333333 (/ y (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) * (y / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) + Float64(-0.3333333333333333 * Float64(y / sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   4. associate-/r* 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   5. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

4. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

Alternative 10: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- 1.0 (/ 0.1111111111111111 x)) (/ -0.3333333333333333 (/ (sqrt x) y))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (sqrt(x) / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) / (sqrt(x) / y))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (Math.sqrt(x) / y));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (math.sqrt(x) / y))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) + Float64(-0.3333333333333333 / Float64(sqrt(x) / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 / (sqrt(x) / y));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 / N[(N[Sqrt[x], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   4. associate-/r* 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   5. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]
4. Step-by-step derivation

   1. clear-num 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \color{blue}{\frac{1}{\frac{\sqrt{x}}{y}}} \]

   2. un-div-inv 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}} \]

5. Applied egg-rr 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}}} \]

6. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{-0.3333333333333333}{\frac{\sqrt{x}}{y}} \]

Alternative 11: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - (y / sqrt((x * 9.0d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / Math.sqrt((x * 9.0)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / math.sqrt((x * 9.0)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / sqrt(Float64(x * 9.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / sqrt((x * 9.0)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x} \cdot 3}} \]

   2. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\sqrt{x} \cdot \color{blue}{\sqrt{9}}} \]

   3. sqrt-prod 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

   4. pow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]

3. Applied egg-rr 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{{\left(x \cdot 9\right)}^{0.5}}} \]
4. Step-by-step derivation

   1. unpow1/2 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

5. Simplified 99.7%

   \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{\color{blue}{\sqrt{x \cdot 9}}} \]

6. Taylor expanded in x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

7. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{\sqrt{x \cdot 9}} \]

Alternative 12: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+147}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 - \frac{0.037037037037037035}{x}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e+147)
   (+ 1.0 (/ -0.1111111111111111 x))
   (pow (- 1.0 (/ 0.037037037037037035 x)) 3.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e+147) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = pow((1.0 - (0.037037037037037035 / x)), 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d+147) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - (0.037037037037037035d0 / x)) ** 3.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e+147) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = Math.pow((1.0 - (0.037037037037037035 / x)), 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e+147:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = math.pow((1.0 - (0.037037037037037035 / x)), 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e+147)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(0.037037037037037035 / x)) ^ 3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e+147)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - (0.037037037037037035 / x)) ^ 3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e+147], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[Power[N[(1.0 - N[(0.037037037037037035 / x), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999999e147

   1. Initial program 99.8%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.8%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.8%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 71.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 71.9%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 71.9%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 71.9%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 71.9%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 71.9%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 71.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   if 3.9999999999999999e147 < y

   1. Initial program 99.5%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.5%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.5%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.5%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.3%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.3%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 3.4%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 3.4%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 3.4%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 3.4%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 3.4%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 3.4%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 3.4%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 3.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x} + 1} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x} + 1}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x} + 1}} \]

      2. pow3 3.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x} + 1}\right)}^{3}} \]

   8. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x} + 1}\right)}^{3}} \]

   9. Taylor expanded in x around inf 24.4%

      \[\leadsto {\color{blue}{\left(1 - 0.037037037037037035 \cdot \frac{1}{x}\right)}}^{3} \]
   10. Step-by-step derivation

       1. associate-*r/ 24.4%

          \[\leadsto {\left(1 - \color{blue}{\frac{0.037037037037037035 \cdot 1}{x}}\right)}^{3} \]

       2. metadata-eval 24.4%

          \[\leadsto {\left(1 - \frac{\color{blue}{0.037037037037037035}}{x}\right)}^{3} \]

   11. Simplified 24.4%

       \[\leadsto {\color{blue}{\left(1 - \frac{0.037037037037037035}{x}\right)}}^{3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+147}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 - \frac{0.037037037037037035}{x}\right)}^{3}\\ \end{array} \]

Alternative 13: 64.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{0.1111111111111111}{x}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} - \frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 0.1111111111111111 x))))
   (if (<= y 1.35e+154)
     (+ 1.0 (/ -0.1111111111111111 x))
     (-
      (/ 1.0 t_0)
      (/ (* (/ 0.1111111111111111 x) (/ 0.1111111111111111 x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (0.1111111111111111 / x);
	double tmp;
	if (y <= 1.35e+154) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 / t_0) - (((0.1111111111111111 / x) * (0.1111111111111111 / x)) / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (0.1111111111111111d0 / x)
    if (y <= 1.35d+154) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 / t_0) - (((0.1111111111111111d0 / x) * (0.1111111111111111d0 / x)) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (0.1111111111111111 / x);
	double tmp;
	if (y <= 1.35e+154) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 / t_0) - (((0.1111111111111111 / x) * (0.1111111111111111 / x)) / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (0.1111111111111111 / x)
	tmp = 0
	if y <= 1.35e+154:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 / t_0) - (((0.1111111111111111 / x) * (0.1111111111111111 / x)) / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= 1.35e+154)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 / t_0) - Float64(Float64(Float64(0.1111111111111111 / x) * Float64(0.1111111111111111 / x)) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (0.1111111111111111 / x);
	tmp = 0.0;
	if (y <= 1.35e+154)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 / t_0) - (((0.1111111111111111 / x) * (0.1111111111111111 / x)) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e+154], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] - N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e154

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.7%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 71.0%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 71.0%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 71.0%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 71.0%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 71.0%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 71.0%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   if 1.35000000000000003e154 < y

   1. Initial program 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.6%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.6%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.6%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.6%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.3%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.3%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 3.3%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 3.3%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 3.3%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 3.3%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 3.3%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 3.3%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 3.3%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 3.3%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{-0.1111111111111111}{x} + 1} \cdot \sqrt[3]{\frac{-0.1111111111111111}{x} + 1}\right) \cdot \sqrt[3]{\frac{-0.1111111111111111}{x} + 1}} \]

      2. pow3 3.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x} + 1}\right)}^{3}} \]

   8. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{-0.1111111111111111}{x} + 1}\right)}^{3}} \]
   9. Step-by-step derivation

      1. rem-cube-cbrt 3.3%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

      2. +-commutative 3.3%

         \[\leadsto \color{blue}{1 + \frac{-0.1111111111111111}{x}} \]

      3. div-inv 3.3%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111 \cdot \frac{1}{x}} \]

      4. metadata-eval 3.3%

         \[\leadsto 1 + \color{blue}{\left(-0.1111111111111111\right)} \cdot \frac{1}{x} \]

      5. cancel-sign-sub-inv 3.3%

         \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      6. div-inv 3.3%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

      7. flip-- 19.7%

         \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 + \frac{0.1111111111111111}{x}}} \]

      8. metadata-eval 19.7%

         \[\leadsto \frac{\color{blue}{1} - \frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 + \frac{0.1111111111111111}{x}} \]

      9. frac-times 19.7%

         \[\leadsto \frac{1 - \color{blue}{\frac{0.1111111111111111 \cdot 0.1111111111111111}{x \cdot x}}}{1 + \frac{0.1111111111111111}{x}} \]

      10. metadata-eval 19.7%

          \[\leadsto \frac{1 - \frac{\color{blue}{0.012345679012345678}}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}} \]

      11. metadata-eval 19.7%

          \[\leadsto \frac{1 - \frac{\color{blue}{-0.1111111111111111 \cdot -0.1111111111111111}}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}} \]

      12. frac-times 19.7%

          \[\leadsto \frac{1 - \color{blue}{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}}{1 + \frac{0.1111111111111111}{x}} \]

      13. div-sub 19.7%

          \[\leadsto \color{blue}{\frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}}{1 + \frac{0.1111111111111111}{x}}} \]

      14. frac-times 19.7%

          \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}}}{1 + \frac{0.1111111111111111}{x}} \]

      15. metadata-eval 19.7%

          \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}} \]

      16. pow2 19.7%

          \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{0.012345679012345678}{\color{blue}{{x}^{2}}}}{1 + \frac{0.1111111111111111}{x}} \]

   10. Applied egg-rr 19.7%

       \[\leadsto \color{blue}{\frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{0.012345679012345678}{{x}^{2}}}{1 + \frac{0.1111111111111111}{x}}} \]
   11. Step-by-step derivation

       1. metadata-eval 19.7%

          \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{\color{blue}{0.1111111111111111 \cdot 0.1111111111111111}}{{x}^{2}}}{1 + \frac{0.1111111111111111}{x}} \]

       2. unpow2 19.7%

          \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{0.1111111111111111 \cdot 0.1111111111111111}{\color{blue}{x \cdot x}}}{1 + \frac{0.1111111111111111}{x}} \]

       3. frac-times 19.7%

          \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}}{1 + \frac{0.1111111111111111}{x}} \]

   12. Applied egg-rr 19.7%

       \[\leadsto \frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\color{blue}{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}}{1 + \frac{0.1111111111111111}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{0.1111111111111111}{x}} - \frac{\frac{0.1111111111111111}{x} \cdot \frac{0.1111111111111111}{x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 14: 61.3% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.7%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.6%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.6%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.6%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.5%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in y around 0 62.7%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 62.7%

         \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

      2. metadata-eval 62.7%

         \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

      3. associate-*r/ 62.8%

         \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

      4. metadata-eval 62.8%

         \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

      5. +-commutative 62.8%

         \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   6. Simplified 62.8%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

   7. Taylor expanded in x around 0 62.1%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]

   if 0.112000000000000002 < x

   1. Initial program 99.7%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

      2. distribute-frac-neg 99.7%

         \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

      3. *-commutative 99.7%

         \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      4. associate-/r* 99.7%

         \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      5. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

      6. neg-mul-1 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

      7. times-frac 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

      8. metadata-eval 99.7%

         \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

   4. Taylor expanded in x around inf 60.7%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 62.3% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ -0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((-0.1111111111111111d0) / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 + (-0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 + (-0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 + Float64(-0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (-0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   4. associate-/r* 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   5. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

4. Taylor expanded in y around 0 62.0%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. cancel-sign-sub-inv 62.0%

      \[\leadsto \color{blue}{1 + \left(-0.1111111111111111\right) \cdot \frac{1}{x}} \]

   2. metadata-eval 62.0%

      \[\leadsto 1 + \color{blue}{-0.1111111111111111} \cdot \frac{1}{x} \]

   3. associate-*r/ 62.0%

      \[\leadsto 1 + \color{blue}{\frac{-0.1111111111111111 \cdot 1}{x}} \]

   4. metadata-eval 62.0%

      \[\leadsto 1 + \frac{\color{blue}{-0.1111111111111111}}{x} \]

   5. +-commutative 62.0%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

6. Simplified 62.0%

   \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x} + 1} \]

7. Final simplification 62.0%

   \[\leadsto 1 + \frac{-0.1111111111111111}{x} \]

Alternative 16: 31.3% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Step-by-step derivation

   1. sub-neg 99.7%

      \[\leadsto \color{blue}{\left(1 - \frac{1}{x \cdot 9}\right) + \left(-\frac{y}{3 \cdot \sqrt{x}}\right)} \]

   2. distribute-frac-neg 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) + \color{blue}{\frac{-y}{3 \cdot \sqrt{x}}} \]

   3. *-commutative 99.7%

      \[\leadsto \left(1 - \frac{1}{\color{blue}{9 \cdot x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   4. associate-/r* 99.7%

      \[\leadsto \left(1 - \color{blue}{\frac{\frac{1}{9}}{x}}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   5. metadata-eval 99.7%

      \[\leadsto \left(1 - \frac{\color{blue}{0.1111111111111111}}{x}\right) + \frac{-y}{3 \cdot \sqrt{x}} \]

   6. neg-mul-1 99.7%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \frac{\color{blue}{-1 \cdot y}}{3 \cdot \sqrt{x}} \]

   7. times-frac 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{\frac{-1}{3} \cdot \frac{y}{\sqrt{x}}} \]

   8. metadata-eval 99.6%

      \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + \color{blue}{-0.3333333333333333} \cdot \frac{y}{\sqrt{x}} \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{\left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}} \]

4. Taylor expanded in x around inf 34.2%

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

5. Final simplification 34.2%

   \[\leadsto 1 \]

Developer target: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{\frac{1}{x}}{9}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - ((1.0d0 / x) / 9.0d0)) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(Float64(1.0 / x) / 9.0)) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - ((1.0 / x) / 9.0)) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023308 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (- (- 1.0 (/ (/ 1.0 x) 9.0)) (/ y (* 3.0 (sqrt x))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))