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

Percentage Accurate: 99.7% → 99.7%

Time: 10.1s

Alternatives: 14

Speedup: 1.0×

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. Step-by-step derivation

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+81} \lor \neg \left(y \leq 9.5 \cdot 10^{+91}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.6e+81) (not (<= y 9.5e+91)))
   (* -0.3333333333333333 (* y (sqrt (/ 1.0 x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.6e+81) || !(y <= 9.5e+91)) {
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / 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 <= (-4.6d+81)) .or. (.not. (y <= 9.5d+91))) then
        tmp = (-0.3333333333333333d0) * (y * sqrt((1.0d0 / 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 <= -4.6e+81) || !(y <= 9.5e+91)) {
		tmp = -0.3333333333333333 * (y * Math.sqrt((1.0 / x)));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.6e+81) or not (y <= 9.5e+91):
		tmp = -0.3333333333333333 * (y * math.sqrt((1.0 / x)))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.6e+81) || !(y <= 9.5e+91))
		tmp = Float64(-0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / 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 <= -4.6e+81) || ~((y <= 9.5e+91)))
		tmp = -0.3333333333333333 * (y * sqrt((1.0 / x)));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.6e+81], N[Not[LessEqual[y, 9.5e+91]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / 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 < -4.5999999999999998e81 or 9.5000000000000001e91 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.3%

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

   if -4.5999999999999998e81 < y < 9.5000000000000001e91

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.7%

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

      9. fma-neg 99.7%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. cancel-sign-sub-inv 93.7%

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

      2. metadata-eval 93.7%

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

      3. associate-*r/ 93.7%

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

      4. metadata-eval 93.7%

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

      5. +-commutative 93.7%

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

   6. Simplified 93.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.0%

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

      3. frac-times 43.0%

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

      4. metadata-eval 43.0%

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

      5. metadata-eval 43.0%

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

      6. frac-times 43.0%

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

      7. sqrt-unprod 43.0%

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

      8. add-sqr-sqrt 43.0%

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

      9. clear-num 43.0%

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

      10. inv-pow 43.0%

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

      11. div-inv 43.0%

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

      12. metadata-eval 43.0%

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

   8. Applied egg-rr 43.0%

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

      1. *-commutative 43.0%

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

      2. metadata-eval 43.0%

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

      3. add-sqr-sqrt 43.0%

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

      4. swap-sqr 43.0%

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

      5. pow-prod-down 43.0%

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

      6. inv-pow 43.0%

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

      7. frac-2neg 43.0%

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

      8. metadata-eval 43.0%

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

      9. inv-pow 43.0%

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

      10. clear-num 43.0%

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

      11. frac-times 43.0%

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

      12. metadata-eval 43.0%

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

      13. /-rgt-identity 43.0%

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

      14. distribute-lft-neg-in 43.0%

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

      15. metadata-eval 43.0%

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

      16. metadata-eval 43.0%

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

      17. sqrt-prod 43.0%

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

      18. *-commutative 43.0%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 93.6%

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

      21. swap-sqr 93.6%

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

   10. Applied egg-rr 93.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+81} \lor \neg \left(y \leq 9.5 \cdot 10^{+91}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 3: 92.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+81} \lor \neg \left(y \leq 9.2 \cdot 10^{+91}\right):\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.4e+81) (not (<= y 9.2e+91)))
   (* (sqrt (/ 1.0 x)) (* y -0.3333333333333333))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.4e+81) || !(y <= 9.2e+91)) {
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} 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 <= (-2.4d+81)) .or. (.not. (y <= 9.2d+91))) then
        tmp = sqrt((1.0d0 / x)) * (y * (-0.3333333333333333d0))
    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 <= -2.4e+81) || !(y <= 9.2e+91)) {
		tmp = Math.sqrt((1.0 / x)) * (y * -0.3333333333333333);
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.4e+81) or not (y <= 9.2e+91):
		tmp = math.sqrt((1.0 / x)) * (y * -0.3333333333333333)
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.4e+81) || !(y <= 9.2e+91))
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(y * -0.3333333333333333));
	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 <= -2.4e+81) || ~((y <= 9.2e+91)))
		tmp = sqrt((1.0 / x)) * (y * -0.3333333333333333);
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.4e+81], N[Not[LessEqual[y, 9.2e+91]], $MachinePrecision]], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * -0.3333333333333333), $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 < -2.3999999999999999e81 or 9.19999999999999965e91 < y

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. unpow1/2 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l* 94.5%

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

   8. Simplified 94.5%

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

   if -2.3999999999999999e81 < y < 9.19999999999999965e91

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.7%

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

      9. fma-neg 99.7%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 93.7%

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

      1. cancel-sign-sub-inv 93.7%

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

      2. metadata-eval 93.7%

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

      3. associate-*r/ 93.7%

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

      4. metadata-eval 93.7%

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

      5. +-commutative 93.7%

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

   6. Simplified 93.7%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.0%

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

      3. frac-times 43.0%

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

      4. metadata-eval 43.0%

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

      5. metadata-eval 43.0%

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

      6. frac-times 43.0%

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

      7. sqrt-unprod 43.0%

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

      8. add-sqr-sqrt 43.0%

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

      9. clear-num 43.0%

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

      10. inv-pow 43.0%

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

      11. div-inv 43.0%

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

      12. metadata-eval 43.0%

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

   8. Applied egg-rr 43.0%

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

      1. *-commutative 43.0%

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

      2. metadata-eval 43.0%

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

      3. add-sqr-sqrt 43.0%

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

      4. swap-sqr 43.0%

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

      5. pow-prod-down 43.0%

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

      6. inv-pow 43.0%

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

      7. frac-2neg 43.0%

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

      8. metadata-eval 43.0%

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

      9. inv-pow 43.0%

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

      10. clear-num 43.0%

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

      11. frac-times 43.0%

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

      12. metadata-eval 43.0%

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

      13. /-rgt-identity 43.0%

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

      14. distribute-lft-neg-in 43.0%

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

      15. metadata-eval 43.0%

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

      16. metadata-eval 43.0%

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

      17. sqrt-prod 43.0%

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

      18. *-commutative 43.0%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 93.6%

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

      21. swap-sqr 93.6%

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

   10. Applied egg-rr 93.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+81} \lor \neg \left(y \leq 9.2 \cdot 10^{+91}\right):\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 4: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57} \lor \neg \left(y \leq 2.7 \cdot 10^{+40}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.8e+57) (not (<= y 2.7e+40)))
   (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+57) || !(y <= 2.7e+40)) {
		tmp = 1.0 + (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 <= (-4.8d+57)) .or. (.not. (y <= 2.7d+40))) then
        tmp = 1.0d0 + (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 <= -4.8e+57) || !(y <= 2.7e+40)) {
		tmp = 1.0 + (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 <= -4.8e+57) or not (y <= 2.7e+40):
		tmp = 1.0 + (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 <= -4.8e+57) || !(y <= 2.7e+40))
		tmp = Float64(1.0 + 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 <= -4.8e+57) || ~((y <= 2.7e+40)))
		tmp = 1.0 + (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, -4.8e+57], N[Not[LessEqual[y, 2.7e+40]], $MachinePrecision]], N[(1.0 + N[(y * N[(-0.3333333333333333 / 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 < -4.80000000000000009e57 or 2.70000000000000009e40 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-*l* 96.2%

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

      3. *-commutative 96.2%

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

   6. Simplified 96.2%

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

      1. pow1 96.2%

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

      2. *-commutative 96.2%

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

      3. *-commutative 96.2%

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

      4. inv-pow 96.2%

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

      5. sqrt-pow1 96.2%

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

      6. metadata-eval 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. unpow1 96.2%

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

      2. associate-*l* 96.1%

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

   10. Simplified 96.1%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 5.3%

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

       3. swap-sqr 5.3%

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

       4. metadata-eval 5.3%

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

       5. metadata-eval 5.3%

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

       6. pow-prod-up 5.3%

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

       7. metadata-eval 5.3%

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

       8. unpow-prod-down 5.3%

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

       9. metadata-eval 5.3%

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

       10. add-sqr-sqrt 5.3%

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

       11. swap-sqr 5.3%

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

       12. pow-prod-down 5.3%

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

       13. inv-pow 5.3%

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

       14. inv-pow 5.3%

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

       15. sqrt-unprod 0.0%

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

       16. add-sqr-sqrt 96.2%

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

       17. expm1-log1p-u 45.5%

           \[\leadsto 1 + y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{-3 \cdot \sqrt{x}}\right)\right)} \]

       18. expm1-udef 7.2%

           \[\leadsto 1 + y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{-3 \cdot \sqrt{x}}\right)} - 1\right)} \]

       19. associate-/r* 7.2%

           \[\leadsto 1 + y \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{-3}}{\sqrt{x}}}\right)} - 1\right) \]

       20. metadata-eval 7.2%

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

   12. Applied egg-rr 7.2%

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

       1. expm1-def 45.4%

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

       2. expm1-log1p 96.1%

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

   14. Simplified 96.1%

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

   if -4.80000000000000009e57 < y < 2.70000000000000009e40

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.8%

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

      1. cancel-sign-sub-inv 97.8%

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

      2. metadata-eval 97.8%

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

      3. associate-*r/ 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

   6. Simplified 97.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.7%

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

      3. frac-times 43.7%

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

      4. metadata-eval 43.7%

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

      5. metadata-eval 43.7%

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

      6. frac-times 43.7%

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

      7. sqrt-unprod 43.8%

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

      8. add-sqr-sqrt 43.8%

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

      9. clear-num 43.8%

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

      10. inv-pow 43.8%

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

      11. div-inv 43.8%

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

      12. metadata-eval 43.8%

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

   8. Applied egg-rr 43.8%

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

      1. *-commutative 43.8%

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

      2. metadata-eval 43.8%

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

      3. add-sqr-sqrt 43.8%

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

      4. swap-sqr 43.8%

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

      5. pow-prod-down 43.8%

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

      6. inv-pow 43.8%

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

      7. frac-2neg 43.8%

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

      8. metadata-eval 43.8%

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

      9. inv-pow 43.8%

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

      10. clear-num 43.8%

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

      11. frac-times 43.8%

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

      12. metadata-eval 43.8%

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

      13. /-rgt-identity 43.8%

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

      14. distribute-lft-neg-in 43.8%

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

      15. metadata-eval 43.8%

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

      16. metadata-eval 43.8%

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

      17. sqrt-prod 43.8%

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

      18. *-commutative 43.8%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 97.7%

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

      21. swap-sqr 97.6%

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

   10. Applied egg-rr 97.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57} \lor \neg \left(y \leq 2.7 \cdot 10^{+40}\right):\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 5: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57} \lor \neg \left(y \leq 3.1 \cdot 10^{+45}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.8e+57) (not (<= y 3.1e+45)))
   (+ 1.0 (/ y (* (sqrt x) -3.0)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.8e+57) || !(y <= 3.1e+45)) {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	} 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 <= (-4.8d+57)) .or. (.not. (y <= 3.1d+45))) then
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    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 <= -4.8e+57) || !(y <= 3.1e+45)) {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.8e+57) or not (y <= 3.1e+45):
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.8e+57) || !(y <= 3.1e+45))
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	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 <= -4.8e+57) || ~((y <= 3.1e+45)))
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.8e+57], N[Not[LessEqual[y, 3.1e+45]], $MachinePrecision]], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $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 < -4.80000000000000009e57 or 3.09999999999999988e45 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 96.1%

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

      1. *-commutative 96.1%

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

      2. associate-*l* 96.2%

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

      3. *-commutative 96.2%

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

   6. Simplified 96.2%

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

      1. pow1 96.2%

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

      2. *-commutative 96.2%

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

      3. *-commutative 96.2%

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

      4. inv-pow 96.2%

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

      5. sqrt-pow1 96.2%

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

      6. metadata-eval 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. unpow1 96.2%

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

      2. associate-*l* 96.1%

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

   10. Simplified 96.1%

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

       1. add-sqr-sqrt 44.7%

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

       2. sqrt-unprod 30.5%

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

       3. swap-sqr 25.9%

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

       4. swap-sqr 25.9%

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

       5. metadata-eval 25.9%

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

       6. metadata-eval 25.9%

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

       7. pow-prod-up 26.0%

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

       8. metadata-eval 26.0%

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

       9. unpow-prod-down 25.9%

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

       10. *-commutative 25.9%

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

       11. add-sqr-sqrt 25.9%

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

       12. unpow-prod-down 25.9%

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

       13. inv-pow 25.9%

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

       14. inv-pow 25.9%

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

       15. swap-sqr 30.5%

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

       16. div-inv 30.6%

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

       17. div-inv 30.6%

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

   12. Applied egg-rr 96.3%

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

       1. *-commutative 96.3%

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

   14. Simplified 96.3%

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

   if -4.80000000000000009e57 < y < 3.09999999999999988e45

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.8%

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

      1. cancel-sign-sub-inv 97.8%

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

      2. metadata-eval 97.8%

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

      3. associate-*r/ 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

   6. Simplified 97.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.7%

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

      3. frac-times 43.7%

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

      4. metadata-eval 43.7%

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

      5. metadata-eval 43.7%

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

      6. frac-times 43.7%

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

      7. sqrt-unprod 43.8%

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

      8. add-sqr-sqrt 43.8%

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

      9. clear-num 43.8%

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

      10. inv-pow 43.8%

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

      11. div-inv 43.8%

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

      12. metadata-eval 43.8%

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

   8. Applied egg-rr 43.8%

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

      1. *-commutative 43.8%

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

      2. metadata-eval 43.8%

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

      3. add-sqr-sqrt 43.8%

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

      4. swap-sqr 43.8%

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

      5. pow-prod-down 43.8%

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

      6. inv-pow 43.8%

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

      7. frac-2neg 43.8%

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

      8. metadata-eval 43.8%

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

      9. inv-pow 43.8%

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

      10. clear-num 43.8%

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

      11. frac-times 43.8%

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

      12. metadata-eval 43.8%

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

      13. /-rgt-identity 43.8%

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

      14. distribute-lft-neg-in 43.8%

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

      15. metadata-eval 43.8%

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

      16. metadata-eval 43.8%

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

      17. sqrt-prod 43.8%

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

      18. *-commutative 43.8%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 97.7%

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

      21. swap-sqr 97.6%

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

   10. Applied egg-rr 97.9%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+57} \lor \neg \left(y \leq 3.1 \cdot 10^{+45}\right):\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 6: 95.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+57}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6e+57)
   (+ 1.0 (/ (/ y -3.0) (sqrt x)))
   (if (<= y 1.1e+45)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (+ 1.0 (/ y (* (sqrt x) -3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+57) {
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	} else if (y <= 1.1e+45) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.6d+57)) then
        tmp = 1.0d0 + ((y / (-3.0d0)) / sqrt(x))
    else if (y <= 1.1d+45) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = 1.0d0 + (y / (sqrt(x) * (-3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6e+57) {
		tmp = 1.0 + ((y / -3.0) / Math.sqrt(x));
	} else if (y <= 1.1e+45) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = 1.0 + (y / (Math.sqrt(x) * -3.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6e+57:
		tmp = 1.0 + ((y / -3.0) / math.sqrt(x))
	elif y <= 1.1e+45:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = 1.0 + (y / (math.sqrt(x) * -3.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6e+57)
		tmp = Float64(1.0 + Float64(Float64(y / -3.0) / sqrt(x)));
	elseif (y <= 1.1e+45)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(1.0 + Float64(y / Float64(sqrt(x) * -3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.6e+57)
		tmp = 1.0 + ((y / -3.0) / sqrt(x));
	elseif (y <= 1.1e+45)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = 1.0 + (y / (sqrt(x) * -3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6e+57], N[(1.0 + N[(N[(y / -3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+45], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5999999999999998e57

   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. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. sub-neg 99.5%

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

      5. distribute-frac-neg 99.5%

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

      6. associate-+r- 99.5%

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

      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. associate-*l/ 99.5%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

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

      12. distribute-neg-frac 99.4%

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

      13. metadata-eval 99.4%

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

      14. *-commutative 99.4%

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

      15. associate-/r* 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 96.0%

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

      1. *-commutative 96.0%

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

      2. associate-*l* 96.1%

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

      3. *-commutative 96.1%

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

   6. Simplified 96.1%

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

      1. pow1 96.1%

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

      2. *-commutative 96.1%

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

      3. *-commutative 96.1%

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

      4. inv-pow 96.1%

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

      5. sqrt-pow1 96.0%

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

      6. metadata-eval 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. unpow1 96.0%

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

      2. associate-*l* 96.1%

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

   10. Simplified 96.1%

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

       1. add-sqr-sqrt 95.8%

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

       2. sqrt-unprod 61.7%

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

       3. swap-sqr 52.0%

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

       4. swap-sqr 52.0%

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

       5. metadata-eval 52.0%

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

       6. metadata-eval 52.0%

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

       7. pow-prod-up 52.1%

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

       8. metadata-eval 52.1%

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

       9. unpow-prod-down 52.0%

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

       10. *-commutative 52.0%

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

       11. add-sqr-sqrt 52.0%

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

       12. unpow-prod-down 52.0%

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

       13. inv-pow 52.0%

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

       14. inv-pow 52.0%

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

       15. swap-sqr 61.7%

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

       16. div-inv 61.8%

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

       17. div-inv 61.8%

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

   12. Applied egg-rr 96.3%

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

   if -4.5999999999999998e57 < y < 1.1e45

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.7%

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

      16. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 97.8%

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

      1. cancel-sign-sub-inv 97.8%

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

      2. metadata-eval 97.8%

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

      3. associate-*r/ 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

   6. Simplified 97.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 43.7%

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

      3. frac-times 43.7%

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

      4. metadata-eval 43.7%

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

      5. metadata-eval 43.7%

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

      6. frac-times 43.7%

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

      7. sqrt-unprod 43.8%

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

      8. add-sqr-sqrt 43.8%

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

      9. clear-num 43.8%

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

      10. inv-pow 43.8%

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

      11. div-inv 43.8%

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

      12. metadata-eval 43.8%

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

   8. Applied egg-rr 43.8%

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

      1. *-commutative 43.8%

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

      2. metadata-eval 43.8%

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

      3. add-sqr-sqrt 43.8%

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

      4. swap-sqr 43.8%

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

      5. pow-prod-down 43.8%

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

      6. inv-pow 43.8%

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

      7. frac-2neg 43.8%

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

      8. metadata-eval 43.8%

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

      9. inv-pow 43.8%

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

      10. clear-num 43.8%

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

      11. frac-times 43.8%

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

      12. metadata-eval 43.8%

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

      13. /-rgt-identity 43.8%

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

      14. distribute-lft-neg-in 43.8%

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

      15. metadata-eval 43.8%

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

      16. metadata-eval 43.8%

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

      17. sqrt-prod 43.8%

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

      18. *-commutative 43.8%

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

      19. add-sqr-sqrt 0.0%

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

      20. sqrt-unprod 97.7%

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

      21. swap-sqr 97.6%

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

   10. Applied egg-rr 97.9%

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

   if 1.1e45 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. associate--r+ 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-frac-neg 99.6%

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

      6. associate-+r- 99.6%

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

      7. neg-mul-1 99.6%

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

      8. associate-*l/ 99.4%

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

      9. fma-neg 99.5%

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

      10. associate-/r* 99.3%

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

      11. metadata-eval 99.3%

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

      12. distribute-neg-frac 99.3%

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

      13. metadata-eval 99.3%

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

      14. *-commutative 99.3%

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

      15. associate-/r* 99.3%

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

      16. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around inf 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-*l* 96.3%

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

      3. *-commutative 96.3%

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

   6. Simplified 96.3%

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

      1. pow1 96.3%

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

      2. *-commutative 96.3%

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

      3. *-commutative 96.3%

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

      4. inv-pow 96.3%

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

      5. sqrt-pow1 96.3%

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

      6. metadata-eval 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. unpow1 96.3%

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

      2. associate-*l* 96.2%

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

   10. Simplified 96.2%

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

       1. add-sqr-sqrt 0.0%

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

       2. sqrt-unprod 3.2%

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

       3. swap-sqr 3.1%

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

       4. swap-sqr 3.1%

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

       5. metadata-eval 3.1%

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

       6. metadata-eval 3.1%

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

       7. pow-prod-up 3.1%

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

       8. metadata-eval 3.1%

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

       9. unpow-prod-down 3.1%

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

       10. *-commutative 3.1%

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

       11. add-sqr-sqrt 3.1%

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

       12. unpow-prod-down 3.1%

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

       13. inv-pow 3.1%

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

       14. inv-pow 3.1%

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

       15. swap-sqr 3.2%

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

       16. div-inv 3.2%

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

       17. div-inv 3.2%

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

   12. Applied egg-rr 96.5%

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

       1. *-commutative 96.5%

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

   14. Simplified 96.5%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+57}:\\ \;\;\;\;1 + \frac{\frac{y}{-3}}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y}{\sqrt{x} \cdot -3}\\ \end{array} \]

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 8: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + ((y * -0.3333333333333333) / 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 * (-0.3333333333333333d0)) / sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. neg-mul-1 99.6%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. associate-*r/ 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. metadata-eval 99.7%

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

   3. sqrt-prod 99.7%

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

   4. pow1/2 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unpow1/2 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in x around 0 99.7%

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

7. Final simplification 99.7%

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

Alternative 10: 61.0% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (* (/ 1.0 x) -0.1111111111111111) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} 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 <= 0.11d0) then
        tmp = (1.0d0 / x) * (-0.1111111111111111d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (1.0 / x) * -0.1111111111111111
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(1.0 / x) * -0.1111111111111111);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = (1.0 / x) * -0.1111111111111111;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(1.0 / x), $MachinePrecision] * -0.1111111111111111), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 57.6%

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

      1. clear-num 57.6%

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

      2. associate-/r/ 57.6%

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

   8. Applied egg-rr 57.6%

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

   if 0.110000000000000001 < 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. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 62.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11: 62.0% 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. +-commutative 99.7%

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

   3. associate--r+ 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. associate-+r- 99.7%

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

   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. associate-*l/ 99.6%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. *-commutative 99.6%

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

   15. associate-/r* 99.5%

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

   16. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 61.7%

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

   1. cancel-sign-sub-inv 61.7%

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

   2. metadata-eval 61.7%

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

   3. associate-*r/ 61.7%

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

   4. metadata-eval 61.7%

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

   5. +-commutative 61.7%

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

6. Simplified 61.7%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 31.6%

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

   3. frac-times 31.6%

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

   4. metadata-eval 31.6%

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

   5. metadata-eval 31.6%

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

   6. frac-times 31.6%

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

   7. sqrt-unprod 29.0%

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

   8. add-sqr-sqrt 29.0%

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

   9. clear-num 29.0%

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

   10. inv-pow 29.0%

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

   11. div-inv 29.0%

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

   12. metadata-eval 29.0%

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

8. Applied egg-rr 29.0%

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

   1. *-commutative 29.0%

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

   2. metadata-eval 29.0%

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

   3. add-sqr-sqrt 29.0%

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

   4. swap-sqr 29.0%

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

   5. pow-prod-down 29.0%

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

   6. inv-pow 29.0%

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

   7. frac-2neg 29.0%

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

   8. metadata-eval 29.0%

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

   9. inv-pow 29.0%

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

   10. clear-num 29.0%

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

   11. frac-times 29.0%

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

   12. metadata-eval 29.0%

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

   13. /-rgt-identity 29.0%

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

   14. distribute-lft-neg-in 29.0%

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

   15. metadata-eval 29.0%

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

   16. metadata-eval 29.0%

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

   17. sqrt-prod 29.0%

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

   18. *-commutative 29.0%

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

   19. add-sqr-sqrt 0.0%

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

   20. sqrt-unprod 61.6%

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

   21. swap-sqr 61.6%

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

10. Applied egg-rr 61.8%

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

11. Final simplification 61.8%

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

Alternative 12: 61.0% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		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 <= 0.11d0) 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 <= 0.11) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		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 <= 0.11)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. unpow1/2 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 57.6%

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

   if 0.110000000000000001 < 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. +-commutative 99.8%

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

      3. associate--r+ 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-frac-neg 99.8%

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

      6. associate-+r- 99.8%

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

      7. neg-mul-1 99.8%

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

      8. associate-*l/ 99.8%

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

      9. fma-neg 99.8%

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

      10. associate-/r* 99.8%

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

      11. metadata-eval 99.8%

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

      12. distribute-neg-frac 99.8%

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

      13. metadata-eval 99.8%

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

      14. *-commutative 99.8%

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

      15. associate-/r* 99.8%

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

      16. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 62.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 62.0% accurate, 22.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. +-commutative 99.7%

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

   3. associate--r+ 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. associate-+r- 99.7%

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

   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. associate-*l/ 99.6%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. *-commutative 99.6%

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

   15. associate-/r* 99.5%

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

   16. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 61.7%

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

   1. cancel-sign-sub-inv 61.7%

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

   2. metadata-eval 61.7%

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

   3. associate-*r/ 61.7%

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

   4. metadata-eval 61.7%

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

   5. +-commutative 61.7%

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

6. Simplified 61.7%

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

7. Final simplification 61.7%

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

Alternative 14: 31.9% 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. +-commutative 99.7%

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

   3. associate--r+ 99.7%

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

   4. sub-neg 99.7%

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

   5. distribute-frac-neg 99.7%

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

   6. associate-+r- 99.7%

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

   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. associate-*l/ 99.6%

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

   9. fma-neg 99.7%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

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

   12. distribute-neg-frac 99.6%

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

   13. metadata-eval 99.6%

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

   14. *-commutative 99.6%

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

   15. associate-/r* 99.5%

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

   16. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around inf 28.9%

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

5. Final simplification 28.9%

   \[\leadsto 1 \]

Developer target: 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 2023320 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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