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

Percentage Accurate: 72.2% → 99.9%

Time: 10.3s

Alternatives: 11

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ (+ x -1.0) (/ (+ -1.0 (+ x (/ (+ x -1.0) y))) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x + -1.0) + Float64(Float64(-1.0 + Float64(x + Float64(Float64(x + -1.0) / y))) / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.9], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(x + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.900000000000000022

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 0.900000000000000022 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x - y}{y + -1}\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y + -1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- x y) (+ y -1.0)))))
   (if (<= t_0 0.2)
     (- 1.0 (log (/ (+ x -1.0) y)))
     (if (<= t_0 2.0)
       (- (- 1.0 y) (log1p (- x)))
       (+ 1.0 (log (/ (+ y -1.0) x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x - y) / (y + -1.0));
	double tmp;
	if (t_0 <= 0.2) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (t_0 <= 2.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 + log(((y + -1.0) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x - y) / (y + -1.0));
	double tmp;
	if (t_0 <= 0.2) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (t_0 <= 2.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 + Math.log(((y + -1.0) / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x - y) / (y + -1.0))
	tmp = 0
	if t_0 <= 0.2:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif t_0 <= 2.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 + math.log(((y + -1.0) / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.2)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 + log(Float64(Float64(y + -1.0) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.2], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.20000000000000001

   1. Initial program 9.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.20000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 2

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.4%

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

   if 2 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 - \log \color{blue}{\left(\mathsf{neg}\left(\frac{x}{1 - y}\right)\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 1 - \log \color{blue}{\left(\frac{x}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 - \log \color{blue}{\left(\frac{x}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)} \]

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. neg-log N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. lower-log.f64 N/A

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

      12. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 0.2:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;1 + \frac{x - y}{y + -1} \leq 2:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y + -1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ -1.0 (+ x (/ (+ x -1.0) y))) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((-1.0 + (x + ((x + -1.0) / y))) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((-1.0 + (x + ((x + -1.0) / y))) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((-1.0 + (x + ((x + -1.0) / y))) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + Float64(x + Float64(Float64(x + -1.0) / y))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.9], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(-1.0 + N[(x + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.900000000000000022

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   if 0.900000000000000022 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99999995)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999995) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999995) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99999995:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99999995)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99999995], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999999949999999971

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-lft-identity N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. neg-mul-1 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 0.999999949999999971 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-1}{\mathsf{fma}\left(y, -0.5, 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- x y) (+ y -1.0))) 50000000000.0)
   (fma y (/ -1.0 (fma y -0.5 1.0)) 1.0)
   (- 1.0 (log (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((x - y) / (y + -1.0))) <= 50000000000.0) {
		tmp = fma(y, (-1.0 / fma(y, -0.5, 1.0)), 1.0);
	} else {
		tmp = 1.0 - log(-x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0))) <= 50000000000.0)
		tmp = fma(y, Float64(-1.0 / fma(y, -0.5, 1.0)), 1.0);
	else
		tmp = Float64(1.0 - log(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 50000000000.0], N[(y * N[(-1.0 / N[(y * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 5e10

   1. Initial program 64.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{-1}{2} \cdot y - 1, 1\right)} \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, y \cdot \frac{-1}{2} + \color{blue}{-1}, 1\right) \]

      6. lower-fma.f64 58.6

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, -0.5, -1\right)}, 1\right) \]

   8. Applied rewrites 58.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, -0.5, -1\right), 1\right)} \]
   9. Step-by-step derivation

      1. flip-+ N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\left(y \cdot \frac{-1}{2}\right) \cdot \left(y \cdot \frac{-1}{2}\right) - -1 \cdot -1}{y \cdot \frac{-1}{2} - -1}}, 1\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot \frac{-1}{2}\right) \cdot \left(y \cdot \frac{-1}{2}\right) - -1 \cdot -1\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}}, 1\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot \frac{-1}{2}\right) \cdot \left(y \cdot \frac{-1}{2}\right) - -1 \cdot -1\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}}, 1\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\left(y \cdot \frac{-1}{2}\right) \cdot \left(y \cdot \frac{-1}{2}\right) - \color{blue}{1}\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\left(y \cdot \frac{-1}{2}\right) \cdot \left(y \cdot \frac{-1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      6. swap-sqr N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\color{blue}{\left(y \cdot y\right) \cdot \left(\frac{-1}{2} \cdot \frac{-1}{2}\right)} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(y, \left(\left(y \cdot y\right) \cdot \left(\frac{-1}{2} \cdot \frac{-1}{2}\right) + \color{blue}{-1}\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y \cdot y, \frac{-1}{2} \cdot \frac{-1}{2}, -1\right)} \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{y \cdot y}, \frac{-1}{2} \cdot \frac{-1}{2}, -1\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \color{blue}{\frac{1}{4}}, -1\right) \cdot \frac{1}{y \cdot \frac{-1}{2} - -1}, 1\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \frac{1}{4}, -1\right) \cdot \color{blue}{\frac{1}{y \cdot \frac{-1}{2} - -1}}, 1\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \frac{1}{4}, -1\right) \cdot \frac{1}{\color{blue}{y \cdot \frac{-1}{2} + \left(\mathsf{neg}\left(-1\right)\right)}}, 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, \frac{1}{4}, -1\right) \cdot \frac{1}{y \cdot \frac{-1}{2} + \color{blue}{1}}, 1\right) \]

      14. lower-fma.f64 58.6

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, 0.25, -1\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(y, -0.5, 1\right)}}, 1\right) \]

   10. Applied rewrites 58.6%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y \cdot y, 0.25, -1\right) \cdot \frac{1}{\mathsf{fma}\left(y, -0.5, 1\right)}}, 1\right) \]

   11. Taylor expanded in y around 0

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1} \cdot \frac{1}{\mathsf{fma}\left(y, \frac{-1}{2}, 1\right)}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 64.0%

       \[\leadsto \mathsf{fma}\left(y, \color{blue}{-1} \cdot \frac{1}{\mathsf{fma}\left(y, -0.5, 1\right)}, 1\right) \]

   if 5e10 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around -inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. distribute-neg-frac2 N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 76.1

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

   8. Applied rewrites 76.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 50000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-1}{\mathsf{fma}\left(y, -0.5, 1\right)}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.1) (- 1.0 (log1p (- x))) (+ 1.0 (log (- y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.1) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 + log(-y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.1) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 + Math.log(-y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.1:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 + math.log(-y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.1)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 + log(Float64(-y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.1], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.10000000000000001

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if 0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 9.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in y around -inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \log \left(\frac{-1}{y}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \log \left(\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - \log \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto 1 - \log \color{blue}{\left(\frac{1}{\mathsf{neg}\left(y\right)}\right)} \]

      5. mul-1-neg N/A

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

      6. log-rec N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(\log \left(-1 \cdot y\right)\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(\log \left(-1 \cdot y\right)\right)\right)} \]

      8. lower-log.f64 N/A

         \[\leadsto 1 - \left(\mathsf{neg}\left(\color{blue}{\log \left(-1 \cdot y\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto 1 - \left(\mathsf{neg}\left(\log \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      10. lower-neg.f64 67.9

          \[\leadsto 1 - \left(-\log \color{blue}{\left(-y\right)}\right) \]

   8. Applied rewrites 67.9%

      \[\leadsto \color{blue}{1 - \left(-\log \left(-y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -28:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -28.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (+ 1.0 (log (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -28.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 + log((y / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -28.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 + Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -28.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 + math.log((y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -28.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 + log(Float64(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -28.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -28

   1. Initial program 21.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in y around -inf

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

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{1 - \log \left(\frac{-1}{y}\right)} \]

      2. metadata-eval N/A

         \[\leadsto 1 - \log \left(\frac{\color{blue}{\mathsf{neg}\left(1\right)}}{y}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto 1 - \log \color{blue}{\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)} \]

      4. distribute-neg-frac2 N/A

         \[\leadsto 1 - \log \color{blue}{\left(\frac{1}{\mathsf{neg}\left(y\right)}\right)} \]

      5. mul-1-neg N/A

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

      6. log-rec N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(\log \left(-1 \cdot y\right)\right)\right)} \]

      7. lower-neg.f64 N/A

         \[\leadsto 1 - \color{blue}{\left(\mathsf{neg}\left(\log \left(-1 \cdot y\right)\right)\right)} \]

      8. lower-log.f64 N/A

         \[\leadsto 1 - \left(\mathsf{neg}\left(\color{blue}{\log \left(-1 \cdot y\right)}\right)\right) \]

      9. mul-1-neg N/A

         \[\leadsto 1 - \left(\mathsf{neg}\left(\log \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right) \]

      10. lower-neg.f64 75.6

          \[\leadsto 1 - \left(-\log \color{blue}{\left(-y\right)}\right) \]

   8. Applied rewrites 75.6%

      \[\leadsto \color{blue}{1 - \left(-\log \left(-y\right)\right)} \]

   if -28 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 99.1%

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

   if 1 < y

   1. Initial program 49.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto 1 - \log \color{blue}{\left(\mathsf{neg}\left(\frac{x}{1 - y}\right)\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto 1 - \log \color{blue}{\left(\frac{x}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 1 - \log \color{blue}{\left(\frac{x}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)} \]

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 98.6

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

   8. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. sub-neg N/A

         \[\leadsto \color{blue}{1 + \left(\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)\right) + 1} \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{x}{y}\right)\right)\right) + 1} \]

      6. lift-log.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\log \left(\frac{x}{y}\right)}\right)\right) + 1 \]

      7. neg-log N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. lower-log.f64 N/A

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

      11. lower-/.f64 98.6

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

   10. Applied rewrites 98.6%

       \[\leadsto \color{blue}{\log \left(\frac{y}{x}\right) + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -28:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(-x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p (- x))))

C

double code(double x, double y) {
	return 1.0 - log1p(-x);
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p(-x);
}

Python

def code(x, y):
	return 1.0 - math.log1p(-x)

Julia

function code(x, y)
	return Float64(1.0 - log1p(Float64(-x)))
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 66.9

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

5. Applied rewrites 66.9%

   \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]
6. Add Preprocessing

Alternative 9: 42.7% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 45.5

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

8. Applied rewrites 45.5%

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

   1. flip3-+ N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. clear-num N/A

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

   5. flip3-+ N/A

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

   6. lift-+.f64 N/A

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

   7. lower-/.f64 45.5

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

10. Applied rewrites 45.5%

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

11. Final simplification 45.5%

    \[\leadsto \frac{1}{\frac{1}{x + 1}} \]
12. Add Preprocessing

Alternative 10: 42.7% accurate, 31.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ x 1.0))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x + 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in x around 0

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

   1. lower-+.f64 45.5

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

8. Applied rewrites 45.5%

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

9. Final simplification 45.5%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 11: 42.4% accurate, 124.0× 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 75.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 66.9

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

5. Applied rewrites 66.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 44.8%

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

Developer Target 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))