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

Percentage Accurate: 99.7% → 99.7%

Time: 8.1s

Alternatives: 12

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: 99.7% 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
2. Add Preprocessing
3. 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 99.6%

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

   4. pow1/2 99.6%

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

4. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26} \lor \neg \left(y \leq 1.45 \cdot 10^{+56}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.6e+26) (not (<= y 1.45e+56)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (- 1.0 (/ 0.3333333333333333 (* x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.6e+26) || !(y <= 1.45e+56)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (0.3333333333333333 / (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 <= (-9.6d+26)) .or. (.not. (y <= 1.45d+56))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 / (x * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.6e+26) || !(y <= 1.45e+56)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.6e+26) or not (y <= 1.45e+56):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.6e+26) || !(y <= 1.45e+56))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.6e+26) || ~((y <= 1.45e+56)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.6e+26], N[Not[LessEqual[y, 1.45e+56]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.60000000000000018e26 or 1.45000000000000004e56 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 90.3%

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

   if -9.60000000000000018e26 < y < 1.45000000000000004e56

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fmm-def 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.3%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.3%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.3%

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

      4. add-exp-log 97.4%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 76.4%

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

      7. frac-times 76.5%

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

      8. metadata-eval 76.5%

         \[\leadsto 1 - \left(\left(1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}\right) - 1\right) \]

      9. metadata-eval 76.5%

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

      10. frac-times 76.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity 43.3%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 76.5%

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

       4. frac-times 76.5%

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

       5. metadata-eval 76.5%

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

       6. metadata-eval 76.5%

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

       7. frac-times 76.5%

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

       8. sqrt-unprod 97.1%

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

       9. add-sqr-sqrt 97.4%

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

       10. metadata-eval 97.4%

           \[\leadsto 1 - \frac{\color{blue}{\frac{1}{9}}}{x} \]

       11. associate-/r* 97.4%

           \[\leadsto 1 - \color{blue}{\frac{1}{9 \cdot x}} \]

       12. *-commutative 97.4%

           \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot 9}} \]

       13. add-sqr-sqrt 97.2%

           \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}}} \]

       14. associate-/r* 97.2%

           \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}}} \]

       15. sqrt-prod 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}}{\sqrt{x \cdot 9}} \]

       16. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\sqrt{x} \cdot \color{blue}{3}}}{\sqrt{x \cdot 9}} \]

       17. *-commutative 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{3 \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       18. associate-/r* 97.3%

           \[\leadsto 1 - \frac{\color{blue}{\frac{\frac{1}{3}}{\sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       19. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{\color{blue}{0.3333333333333333}}{\sqrt{x}}}{\sqrt{x \cdot 9}} \]

       20. sqrt-prod 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

       21. metadata-eval 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot \color{blue}{3}} \]

   13. Applied egg-rr 97.2%

       \[\leadsto 1 - \color{blue}{\frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot 3}} \]
   14. Step-by-step derivation

       1. associate-/l/ 97.2%

          \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\left(\sqrt{x} \cdot 3\right) \cdot \sqrt{x}}} \]

       2. *-commutative 97.2%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \sqrt{x}} \]

       3. associate-*l* 97.3%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{3 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

       4. rem-square-sqrt 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{3 \cdot \color{blue}{x}} \]

       5. *-commutative 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{x \cdot 3}} \]

   15. Simplified 97.5%

       \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{x \cdot 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26} \lor \neg \left(y \leq 1.45 \cdot 10^{+56}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+24} \lor \neg \left(y \leq 7.2 \cdot 10^{+52}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.85e+24) (not (<= y 7.2e+52)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (/ 0.3333333333333333 (* x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.85e+24) || !(y <= 7.2e+52)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - (0.3333333333333333 / (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 <= (-1.85d+24)) .or. (.not. (y <= 7.2d+52))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 / (x * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.85e+24) || !(y <= 7.2e+52)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.85e+24) or not (y <= 7.2e+52):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.85e+24) || !(y <= 7.2e+52))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.85e+24) || ~((y <= 7.2e+52)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.85e+24], N[Not[LessEqual[y, 7.2e+52]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85e24 or 7.2e52 < 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. *-commutative 99.5%

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

      3. associate-/r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. neg-mul-1 99.4%

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

      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. Add Preprocessing

   5. Taylor expanded in x around inf 90.2%

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

   if -1.85e24 < y < 7.2e52

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fmm-def 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.3%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.3%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.3%

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

      4. add-exp-log 97.4%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 76.4%

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

      7. frac-times 76.5%

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

      8. metadata-eval 76.5%

         \[\leadsto 1 - \left(\left(1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}\right) - 1\right) \]

      9. metadata-eval 76.5%

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

      10. frac-times 76.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity 43.3%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 76.5%

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

       4. frac-times 76.5%

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

       5. metadata-eval 76.5%

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

       6. metadata-eval 76.5%

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

       7. frac-times 76.5%

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

       8. sqrt-unprod 97.1%

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

       9. add-sqr-sqrt 97.4%

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

       10. metadata-eval 97.4%

           \[\leadsto 1 - \frac{\color{blue}{\frac{1}{9}}}{x} \]

       11. associate-/r* 97.4%

           \[\leadsto 1 - \color{blue}{\frac{1}{9 \cdot x}} \]

       12. *-commutative 97.4%

           \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot 9}} \]

       13. add-sqr-sqrt 97.2%

           \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}}} \]

       14. associate-/r* 97.2%

           \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}}} \]

       15. sqrt-prod 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}}{\sqrt{x \cdot 9}} \]

       16. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\sqrt{x} \cdot \color{blue}{3}}}{\sqrt{x \cdot 9}} \]

       17. *-commutative 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{3 \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       18. associate-/r* 97.3%

           \[\leadsto 1 - \frac{\color{blue}{\frac{\frac{1}{3}}{\sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       19. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{\color{blue}{0.3333333333333333}}{\sqrt{x}}}{\sqrt{x \cdot 9}} \]

       20. sqrt-prod 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

       21. metadata-eval 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot \color{blue}{3}} \]

   13. Applied egg-rr 97.2%

       \[\leadsto 1 - \color{blue}{\frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot 3}} \]
   14. Step-by-step derivation

       1. associate-/l/ 97.2%

          \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\left(\sqrt{x} \cdot 3\right) \cdot \sqrt{x}}} \]

       2. *-commutative 97.2%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \sqrt{x}} \]

       3. associate-*l* 97.3%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{3 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

       4. rem-square-sqrt 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{3 \cdot \color{blue}{x}} \]

       5. *-commutative 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{x \cdot 3}} \]

   15. Simplified 97.5%

       \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{x \cdot 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+24} \lor \neg \left(y \leq 7.2 \cdot 10^{+52}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.6e+26)
   (* -0.3333333333333333 (/ y (sqrt x)))
   (if (<= y 1.4e+59)
     (- 1.0 (/ 0.3333333333333333 (* x 3.0)))
     (* -0.3333333333333333 (* y (sqrt (/ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.6e+26) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else if (y <= 1.4e+59) {
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / 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 <= (-9.6d+26)) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else if (y <= 1.4d+59) then
        tmp = 1.0d0 - (0.3333333333333333d0 / (x * 3.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.6e+26) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else if (y <= 1.4e+59) {
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	} else {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.6e+26:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	elif y <= 1.4e+59:
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0))
	else:
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.6e+26)
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	elseif (y <= 1.4e+59)
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(x * 3.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.6e+26)
		tmp = -0.3333333333333333 * (y / sqrt(x));
	elseif (y <= 1.4e+59)
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	else
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.6e+26], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+59], N[(1.0 - N[(0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.60000000000000018e26

   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. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.5%

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

      10. fmm-def 99.5%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 82.5%

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

   6. Taylor expanded in x around inf 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. add-sqr-sqrt 69.2%

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

      3. times-frac 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-*r* 69.2%

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

   8. Applied egg-rr 69.2%

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

      1. associate-*l/ 69.3%

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

      2. *-lft-identity 69.3%

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

      3. associate-*l* 69.2%

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

      4. associate-*r/ 69.2%

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

      5. associate-/l* 69.3%

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

      6. associate-/l* 77.4%

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

      7. *-inverses 77.4%

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

      8. *-rgt-identity 77.4%

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

   10. Simplified 77.4%

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

   if -9.60000000000000018e26 < y < 1.3999999999999999e59

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fmm-def 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.3%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.3%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.3%

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

      4. add-exp-log 97.4%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 76.4%

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

      7. frac-times 76.5%

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

      8. metadata-eval 76.5%

         \[\leadsto 1 - \left(\left(1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}\right) - 1\right) \]

      9. metadata-eval 76.5%

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

      10. frac-times 76.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity 43.3%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 76.5%

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

       4. frac-times 76.5%

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

       5. metadata-eval 76.5%

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

       6. metadata-eval 76.5%

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

       7. frac-times 76.5%

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

       8. sqrt-unprod 97.1%

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

       9. add-sqr-sqrt 97.4%

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

       10. metadata-eval 97.4%

           \[\leadsto 1 - \frac{\color{blue}{\frac{1}{9}}}{x} \]

       11. associate-/r* 97.4%

           \[\leadsto 1 - \color{blue}{\frac{1}{9 \cdot x}} \]

       12. *-commutative 97.4%

           \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot 9}} \]

       13. add-sqr-sqrt 97.2%

           \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}}} \]

       14. associate-/r* 97.2%

           \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}}} \]

       15. sqrt-prod 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}}{\sqrt{x \cdot 9}} \]

       16. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\sqrt{x} \cdot \color{blue}{3}}}{\sqrt{x \cdot 9}} \]

       17. *-commutative 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{3 \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       18. associate-/r* 97.3%

           \[\leadsto 1 - \frac{\color{blue}{\frac{\frac{1}{3}}{\sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       19. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{\color{blue}{0.3333333333333333}}{\sqrt{x}}}{\sqrt{x \cdot 9}} \]

       20. sqrt-prod 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

       21. metadata-eval 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot \color{blue}{3}} \]

   13. Applied egg-rr 97.2%

       \[\leadsto 1 - \color{blue}{\frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot 3}} \]
   14. Step-by-step derivation

       1. associate-/l/ 97.2%

          \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\left(\sqrt{x} \cdot 3\right) \cdot \sqrt{x}}} \]

       2. *-commutative 97.2%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \sqrt{x}} \]

       3. associate-*l* 97.3%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{3 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

       4. rem-square-sqrt 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{3 \cdot \color{blue}{x}} \]

       5. *-commutative 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{x \cdot 3}} \]

   15. Simplified 97.5%

       \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{x \cdot 3}} \]

   if 1.3999999999999999e59 < y

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.3%

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

      10. fmm-def 99.3%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.3%

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

      15. distribute-neg-frac 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 90.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+59}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26} \lor \neg \left(y \leq 7.8 \cdot 10^{+58}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.6e+26) (not (<= y 7.8e+58)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (- 1.0 (/ 0.3333333333333333 (* x 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.6e+26) || !(y <= 7.8e+58)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 - (0.3333333333333333 / (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 <= (-9.6d+26)) .or. (.not. (y <= 7.8d+58))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 / (x * 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -9.6e+26) || !(y <= 7.8e+58)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.6e+26) or not (y <= 7.8e+58):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.6e+26) || !(y <= 7.8e+58))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 / Float64(x * 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -9.6e+26) || ~((y <= 7.8e+58)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.6e+26], N[Not[LessEqual[y, 7.8e+58]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.60000000000000018e26 or 7.8000000000000002e58 < 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. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. sub-neg 99.5%

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

      7. neg-mul-1 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/l* 99.4%

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

      10. fmm-def 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 82.5%

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

   6. Taylor expanded in x around inf 73.3%

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

      1. *-un-lft-identity 73.3%

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

      2. add-sqr-sqrt 73.2%

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

      3. times-frac 73.0%

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

      4. *-commutative 73.0%

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

      5. associate-*r* 73.2%

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

   8. Applied egg-rr 73.2%

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

      1. associate-*l/ 73.2%

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

      2. *-lft-identity 73.2%

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

      3. associate-*l* 73.1%

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

      4. associate-*r/ 73.2%

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

      5. associate-/l* 73.3%

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

      6. associate-/l* 83.2%

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

      7. *-inverses 83.2%

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

      8. *-rgt-identity 83.2%

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

   10. Simplified 83.2%

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

   if -9.60000000000000018e26 < y < 7.8000000000000002e58

   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. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-frac-neg 99.7%

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

      6. sub-neg 99.7%

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

      7. neg-mul-1 99.7%

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

      8. *-commutative 99.7%

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

      9. associate-/l* 99.7%

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

      10. fmm-def 99.7%

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

      11. associate-/r* 99.7%

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

      12. metadata-eval 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-/r* 99.7%

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

      15. distribute-neg-frac 99.7%

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

      16. metadata-eval 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. associate-*r/ 97.4%

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

      2. metadata-eval 97.4%

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

   7. Simplified 97.4%

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

      1. expm1-log1p-u 93.3%

         \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

      2. expm1-undefine 93.3%

         \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

      3. log1p-undefine 93.3%

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

      4. add-exp-log 97.4%

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

      5. add-sqr-sqrt 97.1%

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

      6. sqrt-unprod 76.4%

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

      7. frac-times 76.5%

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

      8. metadata-eval 76.5%

         \[\leadsto 1 - \left(\left(1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}\right) - 1\right) \]

      9. metadata-eval 76.5%

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

      10. frac-times 76.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 43.3%

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

   9. Applied egg-rr 43.3%

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

       1. +-commutative 43.3%

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

       2. associate--l+ 43.3%

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

       3. metadata-eval 43.3%

          \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

   11. Simplified 43.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
   12. Step-by-step derivation

       1. +-rgt-identity 43.3%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 76.5%

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

       4. frac-times 76.5%

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

       5. metadata-eval 76.5%

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

       6. metadata-eval 76.5%

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

       7. frac-times 76.5%

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

       8. sqrt-unprod 97.1%

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

       9. add-sqr-sqrt 97.4%

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

       10. metadata-eval 97.4%

           \[\leadsto 1 - \frac{\color{blue}{\frac{1}{9}}}{x} \]

       11. associate-/r* 97.4%

           \[\leadsto 1 - \color{blue}{\frac{1}{9 \cdot x}} \]

       12. *-commutative 97.4%

           \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot 9}} \]

       13. add-sqr-sqrt 97.2%

           \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}}} \]

       14. associate-/r* 97.2%

           \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}}} \]

       15. sqrt-prod 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}}{\sqrt{x \cdot 9}} \]

       16. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\sqrt{x} \cdot \color{blue}{3}}}{\sqrt{x \cdot 9}} \]

       17. *-commutative 97.3%

           \[\leadsto 1 - \frac{\frac{1}{\color{blue}{3 \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       18. associate-/r* 97.3%

           \[\leadsto 1 - \frac{\color{blue}{\frac{\frac{1}{3}}{\sqrt{x}}}}{\sqrt{x \cdot 9}} \]

       19. metadata-eval 97.3%

           \[\leadsto 1 - \frac{\frac{\color{blue}{0.3333333333333333}}{\sqrt{x}}}{\sqrt{x \cdot 9}} \]

       20. sqrt-prod 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

       21. metadata-eval 97.2%

           \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot \color{blue}{3}} \]

   13. Applied egg-rr 97.2%

       \[\leadsto 1 - \color{blue}{\frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot 3}} \]
   14. Step-by-step derivation

       1. associate-/l/ 97.2%

          \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\left(\sqrt{x} \cdot 3\right) \cdot \sqrt{x}}} \]

       2. *-commutative 97.2%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \sqrt{x}} \]

       3. associate-*l* 97.3%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{3 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

       4. rem-square-sqrt 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{3 \cdot \color{blue}{x}} \]

       5. *-commutative 97.5%

          \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{x \cdot 3}} \]

   15. Simplified 97.5%

       \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{x \cdot 3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+26} \lor \neg \left(y \leq 7.8 \cdot 10^{+58}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.3333333333333333}{x \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (/ (- (* -0.3333333333333333 (* y (sqrt x))) 0.1111111111111111) x)
   (- 1.0 (/ y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	} 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) :: tmp
    if (x <= 0.112d0) then
        tmp = (((-0.3333333333333333d0) * (y * sqrt(x))) - 0.1111111111111111d0) / x
    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 tmp;
	if (x <= 0.112) {
		tmp = ((-0.3333333333333333 * (y * Math.sqrt(x))) - 0.1111111111111111) / x;
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = ((-0.3333333333333333 * (y * math.sqrt(x))) - 0.1111111111111111) / x
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(Float64(-0.3333333333333333 * Float64(y * sqrt(x))) - 0.1111111111111111) / x);
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = ((-0.3333333333333333 * (y * sqrt(x))) - 0.1111111111111111) / x;
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(N[(-0.3333333333333333 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.1111111111111111), $MachinePrecision] / x), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.4%

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

      10. fmm-def 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 97.3%

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

   if 0.112000000000000002 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.9%

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

      4. pow1/2 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow1/2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 98.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \left(y \cdot \sqrt{x}\right) - 0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.5% 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 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in x around 0 97.3%

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

   if 0.112000000000000002 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. metadata-eval 99.8%

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

      3. sqrt-prod 99.9%

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

      4. pow1/2 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. unpow1/2 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around inf 98.7%

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

Alternative 8: 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. *-commutative 99.6%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

   5. distribute-frac-neg 99.6%

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

   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 99.6%

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

   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. Add Preprocessing
5. Add Preprocessing

Alternative 9: 61.9% accurate, 14.1× 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 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-frac-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. neg-mul-1 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.4%

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

      10. fmm-def 99.4%

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

      11. associate-/r* 99.4%

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

      12. metadata-eval 99.4%

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

      13. *-commutative 99.4%

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

      14. associate-/r* 99.4%

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

      15. distribute-neg-frac 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 97.3%

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

   6. Taylor expanded in y around 0 63.0%

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

   if 0.112000000000000002 < x

   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. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. sub-neg 99.8%

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

      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. *-commutative 99.8%

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

      9. associate-/l* 99.8%

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

      10. fmm-def 99.8%

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

      11. associate-/r* 99.8%

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

      12. metadata-eval 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-/r* 99.8%

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

      15. distribute-neg-frac 99.8%

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

      16. metadata-eval 99.8%

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

      17. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 61.4%

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

      1. associate-*r/ 61.4%

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

      2. metadata-eval 61.4%

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

   7. Simplified 61.4%

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

   8. Taylor expanded in x around inf 60.2%

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

Alternative 10: 63.0% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.3333333333333333}{x \cdot 3} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ 0.3333333333333333 (* x 3.0))))

C

double code(double x, double y) {
	return 1.0 - (0.3333333333333333 / (x * 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - (0.3333333333333333d0 / (x * 3.0d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 - (0.3333333333333333 / (x * 3.0));
}

Python

def code(x, y):
	return 1.0 - (0.3333333333333333 / (x * 3.0))

Julia

function code(x, y)
	return Float64(1.0 - Float64(0.3333333333333333 / Float64(x * 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (0.3333333333333333 / (x * 3.0));
end

Wolfram

code[x_, y_] := N[(1.0 - N[(0.3333333333333333 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. sub-neg 99.6%

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

   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. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. fmm-def 99.6%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 63.4%

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

   1. associate-*r/ 63.4%

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

   2. metadata-eval 63.4%

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

7. Simplified 63.4%

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

   1. expm1-log1p-u 60.7%

      \[\leadsto 1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)\right)} \]

   2. expm1-undefine 60.7%

      \[\leadsto 1 - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.1111111111111111}{x}\right)} - 1\right)} \]

   3. log1p-undefine 60.7%

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

   4. add-exp-log 63.4%

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

   5. add-sqr-sqrt 63.2%

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

   6. sqrt-unprod 49.5%

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

   7. frac-times 49.5%

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

   8. metadata-eval 49.5%

      \[\leadsto 1 - \left(\left(1 + \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}\right) - 1\right) \]

   9. metadata-eval 49.5%

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

   10. frac-times 49.5%

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

   11. sqrt-unprod 0.0%

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

   12. add-sqr-sqrt 28.9%

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

9. Applied egg-rr 28.9%

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

    1. +-commutative 28.9%

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

    2. associate--l+ 28.9%

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

    3. metadata-eval 28.9%

       \[\leadsto 1 - \left(\frac{-0.1111111111111111}{x} + \color{blue}{0}\right) \]

11. Simplified 28.9%

    \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} + 0\right)} \]
12. Step-by-step derivation

    1. +-rgt-identity 28.9%

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

    2. add-sqr-sqrt 0.0%

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

    3. sqrt-unprod 49.5%

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

    4. frac-times 49.5%

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

    5. metadata-eval 49.5%

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

    6. metadata-eval 49.5%

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

    7. frac-times 49.5%

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

    8. sqrt-unprod 63.2%

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

    9. add-sqr-sqrt 63.4%

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

    10. metadata-eval 63.4%

        \[\leadsto 1 - \frac{\color{blue}{\frac{1}{9}}}{x} \]

    11. associate-/r* 63.4%

        \[\leadsto 1 - \color{blue}{\frac{1}{9 \cdot x}} \]

    12. *-commutative 63.4%

        \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot 9}} \]

    13. add-sqr-sqrt 63.2%

        \[\leadsto 1 - \frac{1}{\color{blue}{\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}}} \]

    14. associate-/r* 63.3%

        \[\leadsto 1 - \color{blue}{\frac{\frac{1}{\sqrt{x \cdot 9}}}{\sqrt{x \cdot 9}}} \]

    15. sqrt-prod 63.3%

        \[\leadsto 1 - \frac{\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}}}{\sqrt{x \cdot 9}} \]

    16. metadata-eval 63.3%

        \[\leadsto 1 - \frac{\frac{1}{\sqrt{x} \cdot \color{blue}{3}}}{\sqrt{x \cdot 9}} \]

    17. *-commutative 63.3%

        \[\leadsto 1 - \frac{\frac{1}{\color{blue}{3 \cdot \sqrt{x}}}}{\sqrt{x \cdot 9}} \]

    18. associate-/r* 63.3%

        \[\leadsto 1 - \frac{\color{blue}{\frac{\frac{1}{3}}{\sqrt{x}}}}{\sqrt{x \cdot 9}} \]

    19. metadata-eval 63.3%

        \[\leadsto 1 - \frac{\frac{\color{blue}{0.3333333333333333}}{\sqrt{x}}}{\sqrt{x \cdot 9}} \]

    20. sqrt-prod 63.3%

        \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

    21. metadata-eval 63.3%

        \[\leadsto 1 - \frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot \color{blue}{3}} \]

13. Applied egg-rr 63.3%

    \[\leadsto 1 - \color{blue}{\frac{\frac{0.3333333333333333}{\sqrt{x}}}{\sqrt{x} \cdot 3}} \]
14. Step-by-step derivation

    1. associate-/l/ 63.2%

       \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{\left(\sqrt{x} \cdot 3\right) \cdot \sqrt{x}}} \]

    2. *-commutative 63.2%

       \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \sqrt{x}} \]

    3. associate-*l* 63.3%

       \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{3 \cdot \left(\sqrt{x} \cdot \sqrt{x}\right)}} \]

    4. rem-square-sqrt 63.4%

       \[\leadsto 1 - \frac{0.3333333333333333}{3 \cdot \color{blue}{x}} \]

    5. *-commutative 63.4%

       \[\leadsto 1 - \frac{0.3333333333333333}{\color{blue}{x \cdot 3}} \]

15. Simplified 63.4%

    \[\leadsto 1 - \color{blue}{\frac{0.3333333333333333}{x \cdot 3}} \]
16. Add Preprocessing

Alternative 11: 63.0% 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. sub-neg 99.6%

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

   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. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. fmm-def 99.6%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 63.4%

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

   1. associate-*r/ 63.4%

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

   2. metadata-eval 63.4%

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

7. Simplified 63.4%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
8. Add Preprocessing

Alternative 12: 32.4% 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 99.6%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-frac-neg 99.6%

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

   6. sub-neg 99.6%

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

   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. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. fmm-def 99.6%

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

   11. associate-/r* 99.6%

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

   12. metadata-eval 99.6%

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

   13. *-commutative 99.6%

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

   14. associate-/r* 99.6%

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

   15. distribute-neg-frac 99.6%

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

   16. metadata-eval 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in y around 0 63.4%

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

   1. associate-*r/ 63.4%

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

   2. metadata-eval 63.4%

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

7. Simplified 63.4%

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

8. Taylor expanded in x around inf 28.7%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Developer Target 1: 99.7% 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 2024180 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

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