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

Percentage Accurate: 72.8% → 99.9%

Time: 12.0s

Alternatives: 10

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: 72.8% 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, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.4:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{\left(x + -1\right) + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.4)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (-
    1.0
    (log
     (/
      (+ (+ x -1.0) (/ (+ (+ x -1.0) (/ (+ -1.0 (+ x (/ (+ x -1.0) y))) y)) y))
      y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.4) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((((x + -1.0) + (((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y)) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.4) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((((x + -1.0) + (((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.4:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((((x + -1.0) + (((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.4)
		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(Float64(x + -1.0) + Float64(Float64(-1.0 + Float64(x + Float64(Float64(x + -1.0) / y))) / y)) / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.4], 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[(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] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.40000000000000002

   1. Initial program 100.0%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.2%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\left(1 + -1 \cdot \frac{\left(x + -1 \cdot \frac{1 + \left(-1 \cdot x + -1 \cdot \frac{x - 1}{y}\right)}{y}\right) - 1}{y}\right) - x}{y}\right)}\right)\right) \]

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.4:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{\left(x + -1\right) + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}}{y}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9999)
   (- 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)) <= 0.9999) {
		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)) <= 0.9999) {
		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)) <= 0.9999:
		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)) <= 0.9999)
		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], 0.9999], 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)) < 0.99990000000000001

   1. Initial program 99.9%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   if 0.99990000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.1%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\left(1 + -1 \cdot \frac{x - 1}{y}\right) - x}{y}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(-1 \cdot \frac{\left(1 + -1 \cdot \frac{x - 1}{y}\right) + \left(\mathsf{neg}\left(x\right)\right)}{y}\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(-1 \cdot \frac{\left(1 + -1 \cdot \frac{x - 1}{y}\right) + -1 \cdot x}{y}\right)\right)\right) \]

      3. associate-+r+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(-1 \cdot \frac{1 + \left(-1 \cdot \frac{x - 1}{y} + -1 \cdot x\right)}{y}\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(-1 \cdot \frac{1 + \left(-1 \cdot x + -1 \cdot \frac{x - 1}{y}\right)}{y}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{-1 \cdot \left(1 + \left(-1 \cdot x + -1 \cdot \frac{x - 1}{y}\right)\right)}{y}\right)\right)\right) \]

   5. Simplified 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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9999)
   (- 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)) <= 0.9999) {
		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)) <= 0.9999) {
		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)) <= 0.9999:
		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)) <= 0.9999)
		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], 0.9999], 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)) < 0.99990000000000001

   1. Initial program 99.9%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   if 0.99990000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{y}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(\frac{1 + -1 \cdot x}{y}\right)\right)\right)\right) \]

      2. distribute-frac-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(1 + -1 \cdot x\right)\right)}{y}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(-1 \cdot x + 1\right)\right)}{y}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{y}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{y}\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{x + \left(\mathsf{neg}\left(1\right)\right)}{y}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{x - 1}{y}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), y\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), y\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(-1 + x\right), y\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, x\right), y\right)\right)\right) \]

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ (+ x -1.0) y)))))
   (if (<= y -1.0) t_0 (if (<= y 1.0) (- 1.0 (log1p (- 0.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x + -1.0) / y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x + -1.0) / y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x + -1.0) / y))
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p((0.0 - x))
	else:
		tmp = t_0
	return tmp

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{1 + -1 \cdot x}{y}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(\frac{1 + -1 \cdot x}{y}\right)\right)\right)\right) \]

      2. distribute-frac-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(1 + -1 \cdot x\right)\right)}{y}\right)\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(-1 \cdot x + 1\right)\right)}{y}\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{y}\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}{y}\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{x + \left(\mathsf{neg}\left(1\right)\right)}{y}\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{x - 1}{y}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x - 1\right), y\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + \left(\mathsf{neg}\left(1\right)\right)\right), y\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x + -1\right), y\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(-1 + x\right), y\right)\right)\right) \]

      12. +-lowering-+.f64 99.4%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(-1, x\right), y\right)\right)\right) \]

   5. Simplified 99.4%

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

   if -1 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(0 - x\right)\right)\right) \]

      3. --lowering--.f64 97.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 97.9%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 97.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   9. Applied egg-rr 97.9%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -580:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -580.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (log1p (- 0.0 x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -580.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p((0.0 - x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -580.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p((0.0 - x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -580.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p((0.0 - x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -580.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(0.0 - x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -580.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -580

   1. Initial program 22.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 + \frac{y}{1 - y}\right)}\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{y}{1 - y}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(1 - y\right)\right)\right)\right) \]

      3. --lowering--.f64 4.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, y\right)\right)\right)\right) \]

   5. Simplified 4.3%

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

   6. Taylor expanded in y around -inf

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(\frac{-1}{y}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{-1}{y}\right)\right)\right) \]

      7. /-lowering-/.f64 68.8%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, y\right)\right)\right) \]

   8. Simplified 68.8%

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

   if -580 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(0 - x\right)\right)\right) \]

      3. --lowering--.f64 97.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 97.9%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 97.9%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   9. Applied egg-rr 97.9%

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

   if 1 < y

   1. Initial program 40.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{x}{1 - y}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(\frac{x}{1 - y}\right)\right)\right)\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{x}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(1 + -1 \cdot y\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\left(-1 \cdot y + 1\right)\right)\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(-1 \cdot y\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 91.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   5. Simplified 91.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\color{blue}{\left(\frac{x}{y}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 91.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right) \]

   8. Simplified 91.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -580:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -28:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -28.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- 0.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -28.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((0.0 - x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -28.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((0.0 - x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -28.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((0.0 - x))
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[y, -28.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -28

   1. Initial program 22.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 + \frac{y}{1 - y}\right)}\right) \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{y}{1 - y}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(1 - y\right)\right)\right)\right) \]

      3. --lowering--.f64 4.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, y\right)\right)\right)\right) \]

   5. Simplified 4.3%

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

   6. Taylor expanded in y around -inf

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(\frac{-1}{y}\right)}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(\frac{1}{y}\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{y}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\left(\frac{-1}{y}\right)\right)\right) \]

      7. /-lowering-/.f64 68.8%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, y\right)\right)\right) \]

   8. Simplified 68.8%

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

   if -28 < y

   1. Initial program 92.9%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      2. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

      8. distribute-neg-in N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      11. +-lowering-+.f64 92.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(0 - x\right)\right)\right) \]

      3. --lowering--.f64 86.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

   7. Simplified 86.2%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 86.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   9. Applied egg-rr 86.2%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -28:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(0 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p (- 0.0 x))))

C

double code(double x, double y) {
	return 1.0 - log1p((0.0 - x));
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p((0.0 - x));
}

Python

def code(x, y):
	return 1.0 - math.log1p((0.0 - x))

Julia

function code(x, y)
	return Float64(1.0 - log1p(Float64(0.0 - x)))
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.0%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   2. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

   11. +-lowering-+.f64 71.0%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

4. Applied egg-rr 71.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(0 - x\right)\right)\right) \]

   3. --lowering--.f64 63.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

7. Simplified 63.4%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
8. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 63.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

9. Applied egg-rr 63.4%

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

10. Final simplification 63.4%

    \[\leadsto 1 - \mathsf{log1p}\left(0 - x\right) \]
11. Add Preprocessing

Alternative 8: 43.4% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{1}{\frac{-1 - y \cdot -0.5}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (/ 1.0 (/ (- -1.0 (* y -0.5)) y))))

C

double code(double x, double y) {
	return 1.0 + (1.0 / ((-1.0 - (y * -0.5)) / y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (1.0d0 / (((-1.0d0) - (y * (-0.5d0))) / y))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (1.0 / ((-1.0 - (y * -0.5)) / y));
}

Python

def code(x, y):
	return 1.0 + (1.0 / ((-1.0 - (y * -0.5)) / y))

Julia

function code(x, y)
	return Float64(1.0 + Float64(1.0 / Float64(Float64(-1.0 - Float64(y * -0.5)) / y)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (1.0 / ((-1.0 - (y * -0.5)) / y));
end

Wolfram

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

Derivation

1. Initial program 71.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 + \frac{y}{1 - y}\right)}\right) \]
4. Step-by-step derivation

   1. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{y}{1 - y}\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(1 - y\right)\right)\right)\right) \]

   3. --lowering--.f64 39.9%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, y\right)\right)\right)\right) \]

5. Simplified 39.9%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(y \cdot \left(1 + \frac{1}{2} \cdot y\right)\right)}\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + \frac{1}{2} \cdot y\right)}\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{2} \cdot y\right)}\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 38.3%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \]

8. Simplified 38.3%

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

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(1 \cdot y + \color{blue}{\left(y \cdot \frac{1}{2}\right) \cdot y}\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(y + \color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot y\right)\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{y \cdot y - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)}{\color{blue}{y - \left(y \cdot \frac{1}{2}\right) \cdot y}}\right)\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{y - \left(y \cdot \frac{1}{2}\right) \cdot y}{y \cdot y - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)}}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{y - \left(y \cdot \frac{1}{2}\right) \cdot y}{y \cdot y - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)}\right)}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(y - \left(y \cdot \frac{1}{2}\right) \cdot y\right), \color{blue}{\left(y \cdot y - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)}\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right), \left(\color{blue}{y \cdot y} - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \left(y \cdot \left(y \cdot \frac{1}{2}\right)\right)\right), \left(y \cdot \color{blue}{y} - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{2}\right)\right)\right), \left(y \cdot \color{blue}{y} - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \left(y \cdot y - \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)\right)\right)\right) \]

   11. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\left(y \cdot y\right), \color{blue}{\left(\left(\left(y \cdot \frac{1}{2}\right) \cdot y\right) \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)}\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)} \cdot \left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left(\left(y \cdot \frac{1}{2}\right) \cdot y\right), \color{blue}{\left(\left(y \cdot \frac{1}{2}\right) \cdot y\right)}\right)\right)\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\left(y \cdot \left(y \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot y\right)\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \frac{1}{2}\right)\right), \left(\color{blue}{\left(y \cdot \frac{1}{2}\right)} \cdot y\right)\right)\right)\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \left(\left(y \cdot \color{blue}{\frac{1}{2}}\right) \cdot y\right)\right)\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{2}\right)}\right)\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f64 38.1%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \]

10. Applied egg-rr 38.1%

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

11. Taylor expanded in y around 0

    \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 + \frac{-1}{2} \cdot y}{y}\right)}\right)\right) \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(1 + \frac{-1}{2} \cdot y\right), \color{blue}{y}\right)\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot y\right)\right), y\right)\right)\right) \]

    3. *-lowering-*.f64 42.5%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{2}, y\right)\right), y\right)\right)\right) \]

13. Simplified 42.5%

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

14. Final simplification 42.5%

    \[\leadsto 1 + \frac{1}{\frac{-1 - y \cdot -0.5}{y}} \]
15. Add Preprocessing

Alternative 9: 42.8% accurate, 37.0× 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 71.0%

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

   1. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   2. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   3. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)\right)\right)\right)\right)\right) \]

   8. distribute-neg-in N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

   11. +-lowering-+.f64 71.0%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

4. Applied egg-rr 71.0%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(0 - x\right)\right)\right) \]

   3. --lowering--.f64 63.4%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right) \]

7. Simplified 63.4%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]

8. Taylor expanded in x around 0

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

   1. +-lowering-+.f64 42.0%

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{x}\right) \]

10. Simplified 42.0%

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

11. Final simplification 42.0%

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

Alternative 10: 42.6% accurate, 111.0× 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 71.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 + \frac{y}{1 - y}\right)}\right) \]
4. Step-by-step derivation

   1. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{y}{1 - y}\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(1 - y\right)\right)\right)\right) \]

   3. --lowering--.f64 39.9%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, y\right)\right)\right)\right) \]

5. Simplified 39.9%

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

6. Taylor expanded in y around 0

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

8. Simplified 41.7%

   \[\leadsto \color{blue}{1} \]
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 2024185 
(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))))))