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

Percentage Accurate: 72.3% → 99.8%

Time: 9.3s

Alternatives: 13

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: 72.3% 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: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{-1 + y}\\ \mathbf{if}\;t\_0 \leq 0.8:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\frac{\frac{x - 1}{y} + \left(x - 1\right)}{y} + \left(x - 1\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (+ -1.0 y))))
   (if (<= t_0 0.8)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (+ (/ (+ (/ (- x 1.0) y) (- x 1.0)) y) (- x 1.0)) y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= 0.8) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log(((((((x - 1.0) / y) + (x - 1.0)) / y) + (x - 1.0)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / ((-1.0d0) + y)
    if (t_0 <= 0.8d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log(((((((x - 1.0d0) / y) + (x - 1.0d0)) / y) + (x - 1.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= 0.8) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log(((((((x - 1.0) / y) + (x - 1.0)) / y) + (x - 1.0)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (-1.0 + y)
	tmp = 0
	if t_0 <= 0.8:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log(((((((x - 1.0) / y) + (x - 1.0)) / y) + (x - 1.0)) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(-1.0 + y))
	tmp = 0.0
	if (t_0 <= 0.8)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(Float64(Float64(Float64(x - 1.0) / y) + Float64(x - 1.0)) / y) + Float64(x - 1.0)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / (-1.0 + y);
	tmp = 0.0;
	if (t_0 <= 0.8)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log(((((((x - 1.0) / y) + (x - 1.0)) / y) + (x - 1.0)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.8], N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(x - 1.0), $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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq 0.8:\\ \;\;\;\;1 - \log \left(1 - \frac{y - x}{-1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\frac{\frac{x - 1}{y} + \left(x - 1\right)}{y} + \left(x - 1\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{-1 + y}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \mathbf{elif}\;t\_0 \leq 0.3:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (+ -1.0 y))))
   (if (<= t_0 -20.0)
     (- 1.0 (log (/ x (+ -1.0 y))))
     (if (<= t_0 0.3)
       (- 1.0 (log1p (/ y (- 1.0 y))))
       (- 1.0 (log (/ (- x 1.0) y)))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = 1.0 - log((x / (-1.0 + y)));
	} else if (t_0 <= 0.3) {
		tmp = 1.0 - log1p((y / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = 1.0 - Math.log((x / (-1.0 + y)));
	} else if (t_0 <= 0.3) {
		tmp = 1.0 - Math.log1p((y / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (-1.0 + y)
	tmp = 0
	if t_0 <= -20.0:
		tmp = 1.0 - math.log((x / (-1.0 + y)))
	elif t_0 <= 0.3:
		tmp = 1.0 - math.log1p((y / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(-1.0 + y))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(1.0 - log(Float64(x / Float64(-1.0 + y))));
	elseif (t_0 <= 0.3)
		tmp = Float64(1.0 - log1p(Float64(y / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(1.0 - N[Log[N[(x / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.3], N[(1.0 - N[Log[1 + N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -20

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

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

   if -20 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.299999999999999989

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

   if 0.299999999999999989 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 9.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. lower--.f64 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq -20:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \mathbf{elif}\;\frac{y - x}{-1 + y} \leq 0.3:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{-1 + y}\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \mathbf{elif}\;t\_0 \leq 0.9999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (+ -1.0 y))))
   (if (<= t_0 -20.0)
     (- 1.0 (log (/ x (+ -1.0 y))))
     (if (<= t_0 0.9999998)
       (- 1.0 (log1p (/ y (- 1.0 y))))
       (- 1.0 (log (/ -1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = 1.0 - log((x / (-1.0 + y)));
	} else if (t_0 <= 0.9999998) {
		tmp = 1.0 - log1p((y / (1.0 - y)));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= -20.0) {
		tmp = 1.0 - Math.log((x / (-1.0 + y)));
	} else if (t_0 <= 0.9999998) {
		tmp = 1.0 - Math.log1p((y / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (-1.0 + y)
	tmp = 0
	if t_0 <= -20.0:
		tmp = 1.0 - math.log((x / (-1.0 + y)))
	elif t_0 <= 0.9999998:
		tmp = 1.0 - math.log1p((y / (1.0 - y)))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(-1.0 + y))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = Float64(1.0 - log(Float64(x / Float64(-1.0 + y))));
	elseif (t_0 <= 0.9999998)
		tmp = Float64(1.0 - log1p(Float64(y / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[(1.0 - N[Log[N[(x / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.9999998], N[(1.0 - N[Log[1 + N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -20

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.1%

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

   if -20 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999999799999999994

   1. Initial program 99.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 97.1

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

   5. Applied rewrites 97.1%

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

   if 0.999999799999999994 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 4.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 64.8%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq -20:\\ \;\;\;\;1 - \log \left(\frac{x}{-1 + y}\right)\\ \mathbf{elif}\;\frac{y - x}{-1 + y} \leq 0.9999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{-1 + y}\\ \mathbf{if}\;t\_0 \leq 0.8:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\frac{x - 1}{y} + \left(x - 1\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (+ -1.0 y))))
   (if (<= t_0 0.8)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (+ (/ (- x 1.0) y) (- x 1.0)) y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= 0.8) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log(((((x - 1.0) / y) + (x - 1.0)) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / ((-1.0d0) + y)
    if (t_0 <= 0.8d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log(((((x - 1.0d0) / y) + (x - 1.0d0)) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= 0.8) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log(((((x - 1.0) / y) + (x - 1.0)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (-1.0 + y)
	tmp = 0
	if t_0 <= 0.8:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log(((((x - 1.0) / y) + (x - 1.0)) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(-1.0 + y))
	tmp = 0.0
	if (t_0 <= 0.8)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(Float64(x - 1.0) / y) + Float64(x - 1.0)) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / (-1.0 + y);
	tmp = 0.0;
	if (t_0 <= 0.8)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log(((((x - 1.0) / y) + (x - 1.0)) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.8], N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + N[(x - 1.0), $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)) < 0.80000000000000004

   1. Initial program 100.0%

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

   if 0.80000000000000004 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.6%

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

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

      6. lower-/.f64 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 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq 0.8:\\ \;\;\;\;1 - \log \left(1 - \frac{y - x}{-1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\frac{x - 1}{y} + \left(x - 1\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{-1 + y}\\ \mathbf{if}\;t\_0 \leq 0.9999998:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (+ -1.0 y))))
   (if (<= t_0 0.9999998)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (- x 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= 0.9999998) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y - x) / ((-1.0d0) + y)
    if (t_0 <= 0.9999998d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log(((x - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (-1.0 + y);
	double tmp;
	if (t_0 <= 0.9999998) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (-1.0 + y)
	tmp = 0
	if t_0 <= 0.9999998:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(-1.0 + y))
	tmp = 0.0
	if (t_0 <= 0.9999998)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / (-1.0 + y);
	tmp = 0.0;
	if (t_0 <= 0.9999998)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log(((x - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.9999998], N[(1.0 - N[Log[N[(1.0 - t$95$0), $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)) < 0.999999799999999994

   1. Initial program 99.8%

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

   if 0.999999799999999994 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 4.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq 0.9999998:\\ \;\;\;\;1 - \log \left(1 - \frac{y - x}{-1 + y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \frac{y - x}{-1 + y} \leq 0.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (/ (- y x) (+ -1.0 y))) 0.5)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - ((y - x) / (-1.0 + y))) <= 0.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - ((y - x) / (-1.0 + y))) <= 0.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - ((y - x) / (-1.0 + y))) <= 0.5:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - Float64(Float64(y - x) / Float64(-1.0 + y))) <= 0.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-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))) < 0.5

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 6.1

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

   5. Applied rewrites 6.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 63.4%

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

   if 0.5 < (-.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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \frac{y - x}{-1 + y} \leq 0.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \frac{y - x}{-1 + y} \leq 2:\\ \;\;\;\;1 - \log 1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (/ (- y x) (+ -1.0 y))) 2.0)
   (- 1.0 (log 1.0))
   (- 1.0 (log (- x)))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((1.0d0 - ((y - x) / ((-1.0d0) + y))) <= 2.0d0) then
        tmp = 1.0d0 - log(1.0d0)
    else
        tmp = 1.0d0 - log(-x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((1.0 - ((y - x) / (-1.0 + y))) <= 2.0)
		tmp = 1.0 - log(1.0);
	else
		tmp = 1.0 - log(-x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(1.0 - N[Log[1.0], $MachinePrecision]), $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))) < 2

   1. Initial program 62.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 60.9

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 - \log 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 62.2%

      \[\leadsto 1 - \log 1 \]

   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 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 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 70.1%

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

   10. Applied rewrites 70.1%

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

4. Final simplification 64.4%

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

Alternative 8: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.92:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.92)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0)
     (- 1.0 (fma (fma 0.5 y 1.0) y (log1p (- x))))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.92) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - fma(fma(0.5, y, 1.0), y, log1p(-x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.92)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - fma(fma(0.5, y, 1.0), y, log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.92], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(N[(0.5 * y + 1.0), $MachinePrecision] * y + N[Log[1 + (-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 < -1.9199999999999999

   1. Initial program 26.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.0%

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

   if -1.9199999999999999 < 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 98.5%

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, y, 1\right), y, \mathsf{log1p}\left(-x\right)\right)} \]

   if 1 < y

   1. Initial program 44.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.7%

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

Alternative 9: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.92:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.92)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ (log1p (- x)) y)) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.92) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (log1p(-x) + y);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.92) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (Math.log1p(-x) + y);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.92:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (math.log1p(-x) + y)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.92)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + y));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.92], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.9199999999999999

   1. Initial program 26.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 62.0%

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

   if -1.9199999999999999 < 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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if 1 < y

   1. Initial program 44.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 93.7%

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

Alternative 10: 45.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq 0.3:\\ \;\;\;\;1 - \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (+ -1.0 y)) 0.3) (- 1.0 (- y x)) (- 1.0 (log 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / (-1.0 + y)) <= 0.3) {
		tmp = 1.0 - (y - x);
	} else {
		tmp = 1.0 - log(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((y - x) / ((-1.0d0) + y)) <= 0.3d0) then
        tmp = 1.0d0 - (y - x)
    else
        tmp = 1.0d0 - log(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((y - x) / (-1.0 + y)) <= 0.3) {
		tmp = 1.0 - (y - x);
	} else {
		tmp = 1.0 - Math.log(1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / (-1.0 + y)) <= 0.3:
		tmp = 1.0 - (y - x)
	else:
		tmp = 1.0 - math.log(1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(-1.0 + y)) <= 0.3)
		tmp = Float64(1.0 - Float64(y - x));
	else
		tmp = Float64(1.0 - log(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((y - x) / (-1.0 + y)) <= 0.3)
		tmp = 1.0 - (y - x);
	else
		tmp = 1.0 - log(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision], 0.3], N[(1.0 - N[(y - x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.299999999999999989

   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

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. div-add N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f64 N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-log1p.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 85.7

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.2%

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

   if 0.299999999999999989 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 6.0

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

   5. Applied rewrites 6.0%

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

   6. Taylor expanded in y around 0

      \[\leadsto 1 - \log 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 14.3%

      \[\leadsto 1 - \log 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{-1 + y} \leq 0.3:\\ \;\;\;\;1 - \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 73.3%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 63.6

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

5. Applied rewrites 63.6%

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

Alternative 12: 43.5% accurate, 20.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 45.1%

   \[\leadsto 1 - \left(-x\right) \]
9. Add Preprocessing

Alternative 13: 43.2% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ 1 - \left|x\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 45.1%

   \[\leadsto 1 - \left(-x\right) \]
9. Step-by-step derivation

10. Applied rewrites 45.1%

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

12. Applied rewrites 44.7%

    \[\leadsto 1 - \left|x\right| \]
13. 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 2024298 
(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))))))