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

Percentage Accurate: 72.5% → 99.2%

Time: 8.1s

Alternatives: 7

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: 72.5% 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: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.002:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.002)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (log (/ (* y E) (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.002) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = log(((y * ((double) M_E)) / (x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.002) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = Math.log(((y * Math.E) / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.002:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = math.log(((y * math.e) / (x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.002)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(Float64(y * exp(1)) / Float64(x + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.002], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 2e-3

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if 2e-3 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.7%

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

      1. sub-neg 6.7%

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

      2. log1p-def 6.7%

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

      3. neg-sub0 6.7%

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

      4. div-sub 6.7%

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

      5. associate--r- 6.7%

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

      6. neg-sub0 6.7%

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

      7. +-commutative 6.7%

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

      8. sub-neg 6.7%

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

      9. div-sub 6.7%

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

   3. Simplified 6.7%

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

      1. add-log-exp 6.7%

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

      2. exp-diff 6.7%

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

      3. exp-1-e 6.7%

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

      4. log1p-udef 6.7%

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

      5. add-exp-log 6.7%

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

   5. Applied egg-rr 6.7%

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

   6. Taylor expanded in y around -inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.002:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \end{array} \]

Alternative 2: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(1.0 - y))
	tmp = 0.0
	if (Float64(1.0 + t_0) <= 0.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - log1p(t_0));
	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[N[(1.0 + t$95$0), $MachinePrecision], 0.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-def 3.1%

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

      3. neg-sub0 3.1%

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

      4. div-sub 3.1%

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

      5. associate--r- 3.1%

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

      6. neg-sub0 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

      9. div-sub 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 3.1%

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

      1. log1p-def 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 77.0%

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

   9. Simplified 77.0%

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

       1. sub-neg 77.0%

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

       2. neg-log 77.0%

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

       3. clear-num 77.0%

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

       4. div-inv 77.0%

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

       5. metadata-eval 77.0%

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

   11. Applied egg-rr 77.0%

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

       1. *-commutative 77.0%

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

       2. neg-mul-1 77.0%

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

   13. Simplified 77.0%

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

   if 0.0 < (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. log1p-def 99.4%

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

      3. neg-sub0 99.4%

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

      4. div-sub 99.4%

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

      5. associate--r- 99.4%

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

      6. neg-sub0 99.4%

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

      7. +-commutative 99.4%

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

      8. sub-neg 99.4%

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

      9. div-sub 99.4%

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

   3. Simplified 99.4%

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

4. Final simplification 92.9%

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

Alternative 3: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+23}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.22e+23) (+ 1.0 (log (- y))) (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+23) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.22e+23) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.22e+23:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - math.log1p((-x / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.22e+23)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(-x) / Float64(1.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.22e+23], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.22e23

   1. Initial program 12.3%

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

      1. sub-neg 12.3%

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

      2. log1p-def 12.3%

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

      3. neg-sub0 12.3%

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

      4. div-sub 12.3%

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

      5. associate--r- 12.3%

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

      6. neg-sub0 12.3%

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

      7. +-commutative 12.3%

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

      8. sub-neg 12.3%

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

      9. div-sub 12.3%

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

   3. Simplified 12.3%

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

   4. Taylor expanded in x around 0 3.0%

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

      1. log1p-def 3.0%

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

   6. Simplified 3.0%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 78.8%

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

   9. Simplified 78.8%

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

       1. sub-neg 78.8%

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

       2. neg-log 78.8%

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

       3. clear-num 78.8%

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

       4. div-inv 78.8%

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

       5. metadata-eval 78.8%

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

   11. Applied egg-rr 78.8%

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

       1. *-commutative 78.8%

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

       2. neg-mul-1 78.8%

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

   13. Simplified 78.8%

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

   if -1.22e23 < y

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. log1p-def 95.2%

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

      3. neg-sub0 95.2%

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

      4. div-sub 95.2%

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

      5. associate--r- 95.2%

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

      6. neg-sub0 95.2%

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

      7. +-commutative 95.2%

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

      8. sub-neg 95.2%

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

      9. div-sub 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in x around inf 93.8%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{-1 \cdot \frac{x}{1 - y}}\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 93.8%

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

      2. distribute-neg-frac 93.8%

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

   6. Simplified 93.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+23}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternative 4: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -19.5) (+ 1.0 (log (- y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -19.5) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -19.5) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -19.5:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -19.5)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -19.5], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -19.5

   1. Initial program 14.7%

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

      1. sub-neg 14.7%

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

      2. log1p-def 14.7%

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

      3. neg-sub0 14.7%

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

      4. div-sub 14.7%

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

      5. associate--r- 14.7%

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

      6. neg-sub0 14.7%

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

      7. +-commutative 14.7%

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

      8. sub-neg 14.7%

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

      9. div-sub 14.7%

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

   3. Simplified 14.7%

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

   4. Taylor expanded in x around 0 2.9%

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

      1. log1p-def 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 76.7%

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

   9. Simplified 76.7%

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

       1. sub-neg 76.7%

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

       2. neg-log 76.7%

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

       3. clear-num 76.7%

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

       4. div-inv 76.7%

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

       5. metadata-eval 76.7%

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

   11. Applied egg-rr 76.7%

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

       1. *-commutative 76.7%

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

       2. neg-mul-1 76.7%

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

   13. Simplified 76.7%

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

   if -19.5 < y

   1. Initial program 95.1%

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

      1. sub-neg 95.1%

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

      2. log1p-def 95.2%

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

      3. neg-sub0 95.2%

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

      4. div-sub 95.2%

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

      5. associate--r- 95.2%

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

      6. neg-sub0 95.2%

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

      7. +-commutative 95.2%

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

      8. sub-neg 95.2%

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

      9. div-sub 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around 0 86.2%

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

      1. div-sub 86.2%

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

      2. mul-1-neg 86.2%

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

      3. sub-neg 86.2%

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

      4. *-inverses 86.2%

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

      5. *-rgt-identity 86.2%

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

      6. log1p-def 86.2%

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

      7. mul-1-neg 86.2%

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

   6. Simplified 86.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -19.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 5: 58.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.5 + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+20) (+ 1.0 (log (- y))) (+ 1.0 (* y (+ (* y -0.5) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+20) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 + (y * ((y * -0.5) + -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 <= (-2.9d+20)) then
        tmp = 1.0d0 + log(-y)
    else
        tmp = 1.0d0 + (y * ((y * (-0.5d0)) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+20) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 + (y * ((y * -0.5) + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+20:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 + (y * ((y * -0.5) + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+20)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(Float64(y * -0.5) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.9e+20)
		tmp = 1.0 + log(-y);
	else
		tmp = 1.0 + (y * ((y * -0.5) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+20], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e20

   1. Initial program 12.3%

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

      1. sub-neg 12.3%

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

      2. log1p-def 12.3%

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

      3. neg-sub0 12.3%

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

      4. div-sub 12.3%

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

      5. associate--r- 12.3%

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

      6. neg-sub0 12.3%

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

      7. +-commutative 12.3%

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

      8. sub-neg 12.3%

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

      9. div-sub 12.3%

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

   3. Simplified 12.3%

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

   4. Taylor expanded in x around 0 3.0%

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

      1. log1p-def 3.0%

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

   6. Simplified 3.0%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 78.8%

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

   9. Simplified 78.8%

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

       1. sub-neg 78.8%

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

       2. neg-log 78.8%

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

       3. clear-num 78.8%

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

       4. div-inv 78.8%

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

       5. metadata-eval 78.8%

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

   11. Applied egg-rr 78.8%

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

       1. *-commutative 78.8%

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

       2. neg-mul-1 78.8%

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

   13. Simplified 78.8%

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

   if -2.9e20 < y

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. log1p-def 95.2%

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

      3. neg-sub0 95.2%

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

      4. div-sub 95.2%

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

      5. associate--r- 95.2%

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

      6. neg-sub0 95.2%

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

      7. +-commutative 95.2%

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

      8. sub-neg 95.2%

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

      9. div-sub 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in x around 0 59.5%

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

      1. log1p-def 59.5%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in y around 0 60.2%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{y + {y}^{2}}\right) \]
   8. Step-by-step derivation

      1. unpow2 60.2%

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

   9. Simplified 60.2%

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

   10. Taylor expanded in y around 0 59.5%

       \[\leadsto \color{blue}{1 + \left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)} \]
   11. Step-by-step derivation

       1. unpow2 59.5%

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

       2. associate-*r* 59.5%

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

       3. distribute-rgt-out 59.5%

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

       4. *-commutative 59.5%

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

   12. Simplified 59.5%

       \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot -0.5 + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.5 + -1\right)\\ \end{array} \]

Alternative 6: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+20) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+20) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+20) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+20:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+20)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+20], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e20

   1. Initial program 12.3%

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

      1. sub-neg 12.3%

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

      2. log1p-def 12.3%

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

      3. neg-sub0 12.3%

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

      4. div-sub 12.3%

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

      5. associate--r- 12.3%

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

      6. neg-sub0 12.3%

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

      7. +-commutative 12.3%

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

      8. sub-neg 12.3%

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

      9. div-sub 12.3%

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

   3. Simplified 12.3%

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

   4. Taylor expanded in x around 0 3.0%

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

      1. log1p-def 3.0%

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

   6. Simplified 3.0%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 78.8%

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

   9. Simplified 78.8%

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

       1. sub-neg 78.8%

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

       2. neg-log 78.8%

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

       3. clear-num 78.8%

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

       4. div-inv 78.8%

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

       5. metadata-eval 78.8%

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

   11. Applied egg-rr 78.8%

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

       1. *-commutative 78.8%

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

       2. neg-mul-1 78.8%

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

   13. Simplified 78.8%

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

   if -2.9e20 < y

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. log1p-def 95.2%

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

      3. neg-sub0 95.2%

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

      4. div-sub 95.2%

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

      5. associate--r- 95.2%

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

      6. neg-sub0 95.2%

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

      7. +-commutative 95.2%

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

      8. sub-neg 95.2%

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

      9. div-sub 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around 0 85.0%

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

      1. log1p-def 85.0%

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

      2. mul-1-neg 85.0%

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

   6. Simplified 85.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 7: 42.4% 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 71.5%

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

   1. sub-neg 71.5%

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

   2. log1p-def 71.6%

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

   3. neg-sub0 71.6%

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

   4. div-sub 71.6%

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

   5. associate--r- 71.6%

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

   6. neg-sub0 71.6%

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

   7. +-commutative 71.6%

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

   8. sub-neg 71.6%

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

   9. div-sub 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in x around 0 43.4%

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

   1. log1p-def 43.4%

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

6. Simplified 43.4%

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

7. Taylor expanded in y around 0 43.6%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{y + {y}^{2}}\right) \]
8. Step-by-step derivation

   1. unpow2 43.6%

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

9. Simplified 43.6%

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

10. Taylor expanded in y around 0 46.2%

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

11. Final simplification 46.2%

    \[\leadsto 1 \]

Developer target: 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 2023230 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

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