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

Percentage Accurate: 99.7% → 99.7%

Time: 9.4s

Alternatives: 17

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 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+45} \lor \neg \left(y \leq 2 \cdot 10^{+24}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.3e+45) (not (<= y 2e+24)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.3e+45) || !(y <= 2e+24)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (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 ((y <= (-2.3d+45)) .or. (.not. (y <= 2d+24))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.3e+45) || !(y <= 2e+24)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.3e+45) or not (y <= 2e+24):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.3e+45) || !(y <= 2e+24))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.3e+45) || ~((y <= 2e+24)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.3e+45], N[Not[LessEqual[y, 2e+24]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000012e45 or 2e24 < y

   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. Taylor expanded in x around inf 93.5%

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

   if -2.30000000000000012e45 < y < 2e24

   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* 99.7%

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

      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 98.2%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

      1. expm1-log1p-u 94.5%

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

      2. expm1-undefine 94.5%

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

      3. log1p-undefine 94.5%

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

      4. add-exp-log 98.3%

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

      5. add-sqr-sqrt 98.2%

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

      6. sqrt-unprod 78.4%

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

      7. frac-times 78.4%

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

      8. metadata-eval 78.4%

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

      9. metadata-eval 78.4%

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

      10. frac-times 78.4%

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

      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 49.9%

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

   9. Applied egg-rr 49.9%

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

       1. +-commutative 49.9%

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

       2. associate--l+ 49.9%

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

       3. metadata-eval 49.9%

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

   11. Simplified 49.9%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 78.4%

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

       3. +-rgt-identity 78.4%

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

       4. +-rgt-identity 78.4%

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

       5. frac-times 78.4%

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

       6. metadata-eval 78.4%

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

       7. metadata-eval 78.4%

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

       8. frac-times 78.4%

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

       9. sqrt-unprod 98.2%

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

       10. add-sqr-sqrt 98.3%

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

       11. clear-num 98.3%

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

       12. div-inv 98.4%

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

       13. metadata-eval 98.4%

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

   13. Applied egg-rr 98.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+45} \lor \neg \left(y \leq 2 \cdot 10^{+24}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+45} \lor \neg \left(y \leq 2.6 \cdot 10^{+24}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+45) (not (<= y 2.6e+24)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+45) || !(y <= 2.6e+24)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (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 ((y <= (-1.6d+45)) .or. (.not. (y <= 2.6d+24))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+45) || !(y <= 2.6e+24)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+45) or not (y <= 2.6e+24):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+45) || !(y <= 2.6e+24))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.6e+45) || ~((y <= 2.6e+24)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+45], N[Not[LessEqual[y, 2.6e+24]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001e45 or 2.5999999999999998e24 < y

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. *-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 99.5%

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

      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 93.4%

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

   if -1.6000000000000001e45 < y < 2.5999999999999998e24

   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* 99.7%

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

      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 98.2%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

      1. expm1-log1p-u 94.5%

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

      2. expm1-undefine 94.5%

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

      3. log1p-undefine 94.5%

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

      4. add-exp-log 98.3%

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

      5. add-sqr-sqrt 98.2%

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

      6. sqrt-unprod 78.4%

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

      7. frac-times 78.4%

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

      8. metadata-eval 78.4%

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

      9. metadata-eval 78.4%

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

      10. frac-times 78.4%

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

      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 49.9%

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

   9. Applied egg-rr 49.9%

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

       1. +-commutative 49.9%

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

       2. associate--l+ 49.9%

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

       3. metadata-eval 49.9%

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

   11. Simplified 49.9%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 78.4%

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

       3. +-rgt-identity 78.4%

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

       4. +-rgt-identity 78.4%

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

       5. frac-times 78.4%

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

       6. metadata-eval 78.4%

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

       7. metadata-eval 78.4%

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

       8. frac-times 78.4%

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

       9. sqrt-unprod 98.2%

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

       10. add-sqr-sqrt 98.3%

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

       11. clear-num 98.3%

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

       12. div-inv 98.4%

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

       13. metadata-eval 98.4%

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

   13. Applied egg-rr 98.4%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+45} \lor \neg \left(y \leq 2.6 \cdot 10^{+24}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+45}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e+45)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 2.6e+24)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e+45) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 2.6e+24) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d+45)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 2.6d+24) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    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 (y <= -2e+45) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 2.6e+24) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e+45:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 2.6e+24:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e+45)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 2.6e+24)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	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 (y <= -2e+45)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 2.6e+24)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e+45], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+24], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9999999999999999e45

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 99.7%

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

   4. Taylor expanded in x around inf 97.2%

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

   if -1.9999999999999999e45 < y < 2.5999999999999998e24

   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* 99.7%

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

      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 98.2%

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

      1. associate-*r/ 98.3%

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

      2. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

      1. expm1-log1p-u 94.5%

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

      2. expm1-undefine 94.5%

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

      3. log1p-undefine 94.5%

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

      4. add-exp-log 98.3%

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

      5. add-sqr-sqrt 98.2%

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

      6. sqrt-unprod 78.4%

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

      7. frac-times 78.4%

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

      8. metadata-eval 78.4%

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

      9. metadata-eval 78.4%

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

      10. frac-times 78.4%

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

      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 49.9%

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

   9. Applied egg-rr 49.9%

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

       1. +-commutative 49.9%

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

       2. associate--l+ 49.9%

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

       3. metadata-eval 49.9%

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

   11. Simplified 49.9%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 78.4%

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

       3. +-rgt-identity 78.4%

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

       4. +-rgt-identity 78.4%

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

       5. frac-times 78.4%

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

       6. metadata-eval 78.4%

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

       7. metadata-eval 78.4%

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

       8. frac-times 78.4%

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

       9. sqrt-unprod 98.2%

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

       10. add-sqr-sqrt 98.3%

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

       11. clear-num 98.3%

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

       12. div-inv 98.4%

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

       13. metadata-eval 98.4%

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

   13. Applied egg-rr 98.4%

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

   if 2.5999999999999998e24 < 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 91.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+45}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+71}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+62}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3e+71)
   (/ (* y -0.3333333333333333) (sqrt x))
   (if (<= y 1.05e+62)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* (* y -0.3333333333333333) (pow x -0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+71) {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	} else if (y <= 1.05e+62) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.3d+71)) then
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    else if (y <= 1.05d+62) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.3e+71) {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	} else if (y <= 1.05e+62) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.3e+71:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	elif y <= 1.05e+62:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) * math.pow(x, -0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.3e+71)
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	elseif (y <= 1.05e+62)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.3e+71)
		tmp = (y * -0.3333333333333333) / sqrt(x);
	elseif (y <= 1.05e+62)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.3e+71], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+62], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.29999999999999996e71

   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 99.5%

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

      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 inf 98.5%

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

   8. Taylor expanded in y around inf 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*l* 96.3%

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

   10. Simplified 96.3%

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

       1. *-commutative 96.3%

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

       2. sqrt-div 96.3%

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

       3. metadata-eval 96.3%

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

       4. un-div-inv 96.4%

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

   12. Applied egg-rr 96.4%

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

   if -1.29999999999999996e71 < y < 1.05e62

   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* 99.7%

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

      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 91.6%

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

      1. associate-*r/ 91.7%

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

      2. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

      1. expm1-log1p-u 88.2%

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

      2. expm1-undefine 88.2%

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

      3. log1p-undefine 88.2%

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

      4. add-exp-log 91.7%

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

      5. add-sqr-sqrt 91.5%

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

      6. sqrt-unprod 73.1%

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

      7. frac-times 73.1%

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

      8. metadata-eval 73.1%

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

      9. metadata-eval 73.1%

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

      10. frac-times 73.1%

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

      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 47.7%

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

   9. Applied egg-rr 47.7%

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

       1. +-commutative 47.7%

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

       2. associate--l+ 47.7%

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

       3. metadata-eval 47.7%

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

   11. Simplified 47.7%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 73.1%

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

       3. +-rgt-identity 73.1%

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

       4. +-rgt-identity 73.1%

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

       5. frac-times 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. frac-times 73.1%

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

       9. sqrt-unprod 91.5%

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

       10. add-sqr-sqrt 91.7%

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

       11. clear-num 91.7%

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

       12. div-inv 91.7%

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

       13. metadata-eval 91.7%

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

   13. Applied egg-rr 91.7%

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

   if 1.05e62 < y

   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 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 95.4%

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

   8. Taylor expanded in y around inf 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*l* 88.1%

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

   10. Simplified 88.1%

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

       1. *-un-lft-identity 88.1%

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

       2. inv-pow 88.1%

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

       3. sqrt-pow1 88.2%

          \[\leadsto \left(1 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot \left(y \cdot -0.3333333333333333\right) \]

       4. metadata-eval 88.2%

          \[\leadsto \left(1 \cdot {x}^{\color{blue}{-0.5}}\right) \cdot \left(y \cdot -0.3333333333333333\right) \]

   12. Applied egg-rr 88.2%

       \[\leadsto \color{blue}{\left(1 \cdot {x}^{-0.5}\right)} \cdot \left(y \cdot -0.3333333333333333\right) \]
   13. Step-by-step derivation

       1. *-lft-identity 88.2%

          \[\leadsto \color{blue}{{x}^{-0.5}} \cdot \left(y \cdot -0.3333333333333333\right) \]

   14. Simplified 88.2%

       \[\leadsto \color{blue}{{x}^{-0.5}} \cdot \left(y \cdot -0.3333333333333333\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+71}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+62}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+75} \lor \neg \left(y \leq 1.08 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7e+75) (not (<= y 1.08e+62)))
   (/ (* y -0.3333333333333333) (sqrt x))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7e+75) || !(y <= 1.08e+62)) {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	} else {
		tmp = 1.0 + (-1.0 / (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 ((y <= (-7d+75)) .or. (.not. (y <= 1.08d+62))) then
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7e+75) || !(y <= 1.08e+62)) {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7e+75) or not (y <= 1.08e+62):
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7e+75) || !(y <= 1.08e+62))
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7e+75) || ~((y <= 1.08e+62)))
		tmp = (y * -0.3333333333333333) / sqrt(x);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7e+75], N[Not[LessEqual[y, 1.08e+62]], $MachinePrecision]], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999997e75 or 1.0800000000000001e62 < y

   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. Taylor expanded in x around inf 96.9%

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

   8. Taylor expanded in y around inf 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*l* 92.0%

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

   10. Simplified 92.0%

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

       1. *-commutative 92.0%

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

       2. sqrt-div 91.9%

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

       3. metadata-eval 91.9%

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

       4. un-div-inv 92.0%

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

   12. Applied egg-rr 92.0%

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

   if -6.9999999999999997e75 < y < 1.0800000000000001e62

   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* 99.7%

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

      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 91.6%

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

      1. associate-*r/ 91.7%

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

      2. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

      1. expm1-log1p-u 88.2%

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

      2. expm1-undefine 88.2%

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

      3. log1p-undefine 88.2%

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

      4. add-exp-log 91.7%

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

      5. add-sqr-sqrt 91.5%

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

      6. sqrt-unprod 73.1%

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

      7. frac-times 73.1%

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

      8. metadata-eval 73.1%

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

      9. metadata-eval 73.1%

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

      10. frac-times 73.1%

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

      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 47.7%

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

   9. Applied egg-rr 47.7%

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

       1. +-commutative 47.7%

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

       2. associate--l+ 47.7%

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

       3. metadata-eval 47.7%

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

   11. Simplified 47.7%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 73.1%

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

       3. +-rgt-identity 73.1%

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

       4. +-rgt-identity 73.1%

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

       5. frac-times 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. frac-times 73.1%

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

       9. sqrt-unprod 91.5%

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

       10. add-sqr-sqrt 91.7%

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

       11. clear-num 91.7%

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

       12. div-inv 91.7%

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

       13. metadata-eval 91.7%

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

   13. Applied egg-rr 91.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+75} \lor \neg \left(y \leq 1.08 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+71} \lor \neg \left(y \leq 1.05 \cdot 10^{+62}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.2e+71) (not (<= y 1.05e+62)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+71) || !(y <= 1.05e+62)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (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 ((y <= (-7.2d+71)) .or. (.not. (y <= 1.05d+62))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+71) || !(y <= 1.05e+62)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.2e+71) or not (y <= 1.05e+62):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.2e+71) || !(y <= 1.05e+62))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -7.2e+71) || ~((y <= 1.05e+62)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.2e+71], N[Not[LessEqual[y, 1.05e+62]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.1999999999999999e71 or 1.05e62 < y

   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. Taylor expanded in x around inf 96.9%

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

   8. Taylor expanded in y around inf 92.0%

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

      1. *-commutative 92.0%

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

      2. associate-*l* 92.0%

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

   10. Simplified 92.0%

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

       1. *-commutative 92.0%

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

       2. sqrt-div 91.9%

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

       3. metadata-eval 91.9%

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

       4. un-div-inv 92.0%

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

   12. Applied egg-rr 92.0%

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

       1. associate-/l* 91.9%

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

   14. Simplified 91.9%

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

   if -7.1999999999999999e71 < y < 1.05e62

   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* 99.7%

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

      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 91.6%

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

      1. associate-*r/ 91.7%

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

      2. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

      1. expm1-log1p-u 88.2%

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

      2. expm1-undefine 88.2%

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

      3. log1p-undefine 88.2%

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

      4. add-exp-log 91.7%

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

      5. add-sqr-sqrt 91.5%

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

      6. sqrt-unprod 73.1%

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

      7. frac-times 73.1%

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

      8. metadata-eval 73.1%

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

      9. metadata-eval 73.1%

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

      10. frac-times 73.1%

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

      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 47.7%

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

   9. Applied egg-rr 47.7%

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

       1. +-commutative 47.7%

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

       2. associate--l+ 47.7%

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

       3. metadata-eval 47.7%

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

   11. Simplified 47.7%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 73.1%

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

       3. +-rgt-identity 73.1%

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

       4. +-rgt-identity 73.1%

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

       5. frac-times 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. frac-times 73.1%

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

       9. sqrt-unprod 91.5%

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

       10. add-sqr-sqrt 91.7%

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

       11. clear-num 91.7%

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

       12. div-inv 91.7%

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

       13. metadata-eval 91.7%

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

   13. Applied egg-rr 91.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+71} \lor \neg \left(y \leq 1.05 \cdot 10^{+62}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+62}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.2e+73)
   (/ (* y -0.3333333333333333) (sqrt x))
   (if (<= y 1.08e+62) (+ 1.0 (/ -1.0 (* x 9.0))) (/ (/ y -3.0) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.2e+73) {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	} else if (y <= 1.08e+62) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y / -3.0) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.2d+73)) then
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    else if (y <= 1.08d+62) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y / (-3.0d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.2e+73) {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	} else if (y <= 1.08e+62) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y / -3.0) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.2e+73:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	elif y <= 1.08e+62:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y / -3.0) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.2e+73)
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	elseif (y <= 1.08e+62)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y / -3.0) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.2e+73)
		tmp = (y * -0.3333333333333333) / sqrt(x);
	elseif (y <= 1.08e+62)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y / -3.0) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.2e+73], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.08e+62], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.1999999999999999e73

   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 99.5%

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

      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 inf 98.5%

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

   8. Taylor expanded in y around inf 96.3%

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

      1. *-commutative 96.3%

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

      2. associate-*l* 96.3%

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

   10. Simplified 96.3%

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

       1. *-commutative 96.3%

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

       2. sqrt-div 96.3%

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

       3. metadata-eval 96.3%

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

       4. un-div-inv 96.4%

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

   12. Applied egg-rr 96.4%

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

   if -6.1999999999999999e73 < y < 1.0800000000000001e62

   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* 99.7%

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

      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 91.6%

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

      1. associate-*r/ 91.7%

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

      2. metadata-eval 91.7%

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

   7. Simplified 91.7%

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

      1. expm1-log1p-u 88.2%

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

      2. expm1-undefine 88.2%

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

      3. log1p-undefine 88.2%

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

      4. add-exp-log 91.7%

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

      5. add-sqr-sqrt 91.5%

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

      6. sqrt-unprod 73.1%

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

      7. frac-times 73.1%

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

      8. metadata-eval 73.1%

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

      9. metadata-eval 73.1%

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

      10. frac-times 73.1%

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

      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 47.7%

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

   9. Applied egg-rr 47.7%

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

       1. +-commutative 47.7%

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

       2. associate--l+ 47.7%

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

       3. metadata-eval 47.7%

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

   11. Simplified 47.7%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 73.1%

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

       3. +-rgt-identity 73.1%

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

       4. +-rgt-identity 73.1%

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

       5. frac-times 73.1%

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

       6. metadata-eval 73.1%

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

       7. metadata-eval 73.1%

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

       8. frac-times 73.1%

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

       9. sqrt-unprod 91.5%

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

       10. add-sqr-sqrt 91.7%

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

       11. clear-num 91.7%

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

       12. div-inv 91.7%

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

       13. metadata-eval 91.7%

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

   13. Applied egg-rr 91.7%

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

   if 1.0800000000000001e62 < y

   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 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 95.4%

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

   8. Taylor expanded in y around inf 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*l* 88.1%

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

   10. Simplified 88.1%

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

       1. *-commutative 88.1%

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

       2. sqrt-div 88.0%

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

       3. metadata-eval 88.0%

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

       4. un-div-inv 88.1%

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

   12. Applied egg-rr 88.1%

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

       1. associate-*l/ 88.0%

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

       2. metadata-eval 88.0%

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

       3. times-frac 88.2%

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

       4. *-rgt-identity 88.2%

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

       5. *-commutative 88.2%

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

       6. associate-/r* 88.2%

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

   14. Simplified 88.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+62}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{-3}}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (- (/ -0.1111111111111111 x) (/ y (sqrt (* x 9.0))))
   (- 1.0 (/ y (* 3.0 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (-0.1111111111111111 / x) - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	}
	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) - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    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) - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = (-0.1111111111111111 / x) - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (-0.1111111111111111 / x) - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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 99.7%

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

      4. pow1/2 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 97.8%

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

   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. Taylor expanded in x around 0 99.8%

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

   4. Taylor expanded in x around inf 99.1%

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

Alternative 10: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - \frac{y}{3 \cdot \sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (/ y (* 3.0 (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (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 - (0.1111111111111111d0 / x)) - (y / (3.0d0 * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (y / (3.0 * Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (y / (3.0 * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64(y / Float64(3.0 * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - (y / (3.0 * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0 99.7%

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

Alternative 11: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 12: 64.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x \cdot 9\right) \cdot \left(\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e+161)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (-
    1.0
    (* (* x 9.0) (* (/ -0.1111111111111111 x) (/ -0.1111111111111111 x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+161) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((x * 9.0) * ((-0.1111111111111111 / x) * (-0.1111111111111111 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d+161) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((x * 9.0d0) * (((-0.1111111111111111d0) / x) * ((-0.1111111111111111d0) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e+161) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((x * 9.0) * ((-0.1111111111111111 / x) * (-0.1111111111111111 / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e+161:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((x * 9.0) * ((-0.1111111111111111 / x) * (-0.1111111111111111 / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e+161)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(x * 9.0) * Float64(Float64(-0.1111111111111111 / x) * Float64(-0.1111111111111111 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e+161)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((x * 9.0) * ((-0.1111111111111111 / x) * (-0.1111111111111111 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e+161], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x * 9.0), $MachinePrecision] * N[(N[(-0.1111111111111111 / x), $MachinePrecision] * N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000006e161

   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* 99.6%

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

      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 70.0%

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

      1. associate-*r/ 70.1%

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

      2. metadata-eval 70.1%

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

   7. Simplified 70.1%

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

      1. expm1-log1p-u 67.5%

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

      2. expm1-undefine 67.5%

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

      3. log1p-undefine 67.5%

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

      4. add-exp-log 70.1%

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

      5. add-sqr-sqrt 70.0%

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

      6. sqrt-unprod 56.2%

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

      7. frac-times 56.2%

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

      8. metadata-eval 56.2%

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

      9. metadata-eval 56.2%

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

      10. frac-times 56.2%

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

      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 37.6%

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

   9. Applied egg-rr 37.6%

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

       1. +-commutative 37.6%

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

       2. associate--l+ 37.6%

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

       3. metadata-eval 37.6%

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

   11. Simplified 37.6%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 56.2%

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

       3. +-rgt-identity 56.2%

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

       4. +-rgt-identity 56.2%

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

       5. frac-times 56.2%

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

       6. metadata-eval 56.2%

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

       7. metadata-eval 56.2%

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

       8. frac-times 56.2%

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

       9. sqrt-unprod 70.0%

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

       10. add-sqr-sqrt 70.1%

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

       11. clear-num 70.1%

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

       12. div-inv 70.1%

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

       13. metadata-eval 70.1%

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

   13. Applied egg-rr 70.1%

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

   if 1.50000000000000006e161 < 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* 99.6%

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

      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* 99.4%

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

      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 y around 0 3.1%

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

      1. associate-*r/ 3.1%

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

      2. metadata-eval 3.1%

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

   7. Simplified 3.1%

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

      1. expm1-log1p-u 3.1%

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

      2. expm1-undefine 3.1%

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

      3. log1p-undefine 3.1%

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

      4. add-exp-log 3.1%

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

      5. add-sqr-sqrt 3.1%

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

      6. sqrt-unprod 20.3%

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

      7. frac-times 20.3%

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

      8. metadata-eval 20.3%

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

      9. metadata-eval 20.3%

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

      10. frac-times 20.3%

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

      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 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. +-commutative 0.8%

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

       2. associate--l+ 0.8%

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

       3. metadata-eval 0.8%

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

   11. Simplified 0.8%

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

       1. --rgt-identity 0.8%

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

       2. flip-- 0.7%

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

       3. +-rgt-identity 0.7%

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

       4. div-inv 0.7%

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

       5. metadata-eval 0.7%

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

       6. --rgt-identity 0.7%

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

       7. +-rgt-identity 0.7%

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

       8. +-rgt-identity 0.7%

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

       9. frac-times 0.7%

          \[\leadsto 1 - \color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}} \cdot \frac{1}{\frac{-0.1111111111111111}{x} + 0} \]

       10. metadata-eval 0.7%

           \[\leadsto 1 - \frac{\color{blue}{0.012345679012345678}}{x \cdot x} \cdot \frac{1}{\frac{-0.1111111111111111}{x} + 0} \]

       11. pow2 0.7%

           \[\leadsto 1 - \frac{0.012345679012345678}{\color{blue}{{x}^{2}}} \cdot \frac{1}{\frac{-0.1111111111111111}{x} + 0} \]

       12. add-sqr-sqrt 0.0%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}}} \]

       13. sqrt-unprod 1.2%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}}} \]

       14. +-rgt-identity 1.2%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\sqrt{\color{blue}{\frac{-0.1111111111111111}{x}} \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

       15. +-rgt-identity 1.2%

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

       16. frac-times 1.2%

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

       17. metadata-eval 1.2%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}} \]

       18. metadata-eval 1.2%

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

       19. frac-times 1.2%

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

       20. sqrt-unprod 20.3%

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

       21. add-sqr-sqrt 20.3%

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

   13. Applied egg-rr 20.3%

       \[\leadsto 1 - \color{blue}{\frac{0.012345679012345678}{{x}^{2}} \cdot \left(x \cdot 9\right)} \]
   14. Step-by-step derivation

       1. metadata-eval 20.3%

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

       2. unpow2 20.3%

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

       3. frac-times 20.3%

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

   15. Applied egg-rr 20.3%

       \[\leadsto 1 - \color{blue}{\left(\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\right)} \cdot \left(x \cdot 9\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+161}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x \cdot 9\right) \cdot \left(\frac{-0.1111111111111111}{x} \cdot \frac{-0.1111111111111111}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+144}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x \cdot 9\right) \cdot \frac{0.012345679012345678}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6e+144)
   (+ 1.0 (/ -1.0 (* x 9.0)))
   (- 1.0 (* (* x 9.0) (/ 0.012345679012345678 (* x x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6e+144) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((x * 9.0) * (0.012345679012345678 / (x * 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 <= 6d+144) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 - ((x * 9.0d0) * (0.012345679012345678d0 / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6e+144) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 - ((x * 9.0) * (0.012345679012345678 / (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6e+144:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 - ((x * 9.0) * (0.012345679012345678 / (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6e+144)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 - Float64(Float64(x * 9.0) * Float64(0.012345679012345678 / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6e+144)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 - ((x * 9.0) * (0.012345679012345678 / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6e+144], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(x * 9.0), $MachinePrecision] * N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.9999999999999998e144

   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* 99.6%

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

      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 70.9%

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

      1. associate-*r/ 70.9%

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

      2. metadata-eval 70.9%

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

   7. Simplified 70.9%

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

      1. expm1-log1p-u 68.4%

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

      2. expm1-undefine 68.4%

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

      3. log1p-undefine 68.4%

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

      4. add-exp-log 70.9%

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

      5. add-sqr-sqrt 70.8%

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

      6. sqrt-unprod 56.9%

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

      7. frac-times 56.9%

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

      8. metadata-eval 56.9%

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

      9. metadata-eval 56.9%

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

      10. frac-times 56.9%

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

      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 38.0%

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

   9. Applied egg-rr 38.0%

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

       1. +-commutative 38.0%

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

       2. associate--l+ 38.0%

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

       3. metadata-eval 38.0%

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

   11. Simplified 38.0%

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

       1. add-sqr-sqrt 0.0%

          \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

       2. sqrt-unprod 56.9%

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

       3. +-rgt-identity 56.9%

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

       4. +-rgt-identity 56.9%

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

       5. frac-times 56.9%

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

       6. metadata-eval 56.9%

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

       7. metadata-eval 56.9%

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

       8. frac-times 56.9%

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

       9. sqrt-unprod 70.8%

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

       10. add-sqr-sqrt 70.9%

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

       11. clear-num 71.0%

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

       12. div-inv 71.0%

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

       13. metadata-eval 71.0%

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

   13. Applied egg-rr 71.0%

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

   if 5.9999999999999998e144 < 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* 99.5%

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

      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* 99.3%

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

      12. metadata-eval 99.3%

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

      13. *-commutative 99.3%

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

      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 0 3.2%

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

      1. associate-*r/ 3.2%

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

      2. metadata-eval 3.2%

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

   7. Simplified 3.2%

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

      1. expm1-log1p-u 3.2%

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

      2. expm1-undefine 3.2%

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

      3. log1p-undefine 3.2%

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

      4. add-exp-log 3.2%

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

      5. add-sqr-sqrt 3.2%

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

      6. sqrt-unprod 18.7%

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

      7. frac-times 18.7%

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

      8. metadata-eval 18.7%

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

      9. metadata-eval 18.7%

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

      10. frac-times 18.7%

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

      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 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. +-commutative 0.8%

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

       2. associate--l+ 0.8%

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

       3. metadata-eval 0.8%

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

   11. Simplified 0.8%

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

       1. --rgt-identity 0.8%

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

       2. flip-- 0.8%

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

       3. +-rgt-identity 0.8%

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

       4. div-inv 0.8%

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

       5. metadata-eval 0.8%

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

       6. --rgt-identity 0.8%

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

       7. +-rgt-identity 0.8%

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

       8. +-rgt-identity 0.8%

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

       9. frac-times 0.8%

          \[\leadsto 1 - \color{blue}{\frac{-0.1111111111111111 \cdot -0.1111111111111111}{x \cdot x}} \cdot \frac{1}{\frac{-0.1111111111111111}{x} + 0} \]

       10. metadata-eval 0.8%

           \[\leadsto 1 - \frac{\color{blue}{0.012345679012345678}}{x \cdot x} \cdot \frac{1}{\frac{-0.1111111111111111}{x} + 0} \]

       11. pow2 0.8%

           \[\leadsto 1 - \frac{0.012345679012345678}{\color{blue}{{x}^{2}}} \cdot \frac{1}{\frac{-0.1111111111111111}{x} + 0} \]

       12. add-sqr-sqrt 0.0%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}}} \]

       13. sqrt-unprod 1.5%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\color{blue}{\sqrt{\left(\frac{-0.1111111111111111}{x} + 0\right) \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}}} \]

       14. +-rgt-identity 1.5%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\sqrt{\color{blue}{\frac{-0.1111111111111111}{x}} \cdot \left(\frac{-0.1111111111111111}{x} + 0\right)}} \]

       15. +-rgt-identity 1.5%

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

       16. frac-times 1.5%

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

       17. metadata-eval 1.5%

           \[\leadsto 1 - \frac{0.012345679012345678}{{x}^{2}} \cdot \frac{1}{\sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}}} \]

       18. metadata-eval 1.5%

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

       19. frac-times 1.5%

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

       20. sqrt-unprod 18.7%

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

       21. add-sqr-sqrt 18.7%

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

   13. Applied egg-rr 18.7%

       \[\leadsto 1 - \color{blue}{\frac{0.012345679012345678}{{x}^{2}} \cdot \left(x \cdot 9\right)} \]
   14. Step-by-step derivation

       1. unpow2 18.7%

          \[\leadsto 1 - \frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right) \]

   15. Applied egg-rr 18.7%

       \[\leadsto 1 - \frac{0.012345679012345678}{\color{blue}{x \cdot x}} \cdot \left(x \cdot 9\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+144}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(x \cdot 9\right) \cdot \frac{0.012345679012345678}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% 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 99.6%

      \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
   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* 99.5%

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

      12. metadata-eval 99.5%

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

      13. *-commutative 99.5%

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

      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 97.7%

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

   6. Taylor expanded in y around 0 63.2%

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

   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 61.5%

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

      1. associate-*r/ 61.5%

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

      2. metadata-eval 61.5%

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

   7. Simplified 61.5%

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

   8. Taylor expanded in x around inf 60.7%

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

Alternative 15: 62.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x \cdot 9} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return 1.0 + (-1.0 / (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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 62.9%

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

   1. associate-*r/ 63.0%

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

   2. metadata-eval 63.0%

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

7. Simplified 63.0%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]
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 63.0%

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

   5. add-sqr-sqrt 62.9%

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

   6. sqrt-unprod 52.4%

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

   7. frac-times 52.4%

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

   8. metadata-eval 52.4%

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

   9. metadata-eval 52.4%

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

   10. frac-times 52.4%

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

   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 33.7%

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

9. Applied egg-rr 33.7%

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

    1. +-commutative 33.7%

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

    2. associate--l+ 33.7%

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

    3. metadata-eval 33.7%

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

11. Simplified 33.7%

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

    1. add-sqr-sqrt 0.0%

       \[\leadsto 1 - \color{blue}{\sqrt{\frac{-0.1111111111111111}{x} + 0} \cdot \sqrt{\frac{-0.1111111111111111}{x} + 0}} \]

    2. sqrt-unprod 52.4%

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

    3. +-rgt-identity 52.4%

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

    4. +-rgt-identity 52.4%

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

    5. frac-times 52.4%

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

    6. metadata-eval 52.4%

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

    7. metadata-eval 52.4%

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

    8. frac-times 52.4%

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

    9. sqrt-unprod 62.9%

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

    10. add-sqr-sqrt 63.0%

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

    11. clear-num 63.0%

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

    12. div-inv 63.0%

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

    13. metadata-eval 63.0%

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

13. Applied egg-rr 63.0%

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

14. Final simplification 63.0%

    \[\leadsto 1 + \frac{-1}{x \cdot 9} \]
15. Add Preprocessing

Alternative 16: 62.4% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 62.9%

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

   1. associate-*r/ 63.0%

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

   2. metadata-eval 63.0%

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

7. Simplified 63.0%

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

Alternative 17: 32.3% accurate, 113.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.7%

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

   1. 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* 99.6%

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

   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 62.9%

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

   1. associate-*r/ 63.0%

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

   2. metadata-eval 63.0%

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

7. Simplified 63.0%

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

8. Taylor expanded in x around inf 33.6%

   \[\leadsto \color{blue}{1} \]
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 2024163 
(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)))))