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

Percentage Accurate: 72.6% → 99.0%

Time: 9.7s

Alternatives: 9

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: 72.6% 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: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 2e-5)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (* E (/ y (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-5) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((((double) M_E) * (y / (x + -1.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-5) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((Math.E * (y / (x + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 2e-5:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((math.e * (y / (x + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 2e-5)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(exp(1) * Float64(y / 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], 2e-5], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E * N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 2.00000000000000016e-5

   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-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 2.00000000000000016e-5 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 7.9%

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

      1. sub-neg 7.9%

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

      2. log1p-define 7.9%

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

      3. distribute-neg-frac2 7.9%

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

      4. neg-sub0 7.9%

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

      5. associate--r- 7.9%

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

      6. metadata-eval 7.9%

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

      7. +-commutative 7.9%

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

   3. Simplified 7.9%

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

   5. Taylor expanded in y around -inf 85.5%

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

      1. sub-neg 85.5%

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

      2. metadata-eval 85.5%

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

      3. distribute-lft-in 85.5%

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

      4. metadata-eval 85.5%

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

      5. +-commutative 85.5%

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

      6. log1p-define 85.5%

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

      7. mul-1-neg 85.5%

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

   7. Simplified 85.5%

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

      1. add-log-exp 85.5%

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

      2. sub-neg 85.5%

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

      3. exp-sum 85.5%

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

      4. add-log-exp 85.5%

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

      5. neg-log 85.5%

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

      6. exp-sum 85.5%

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

      7. add-exp-log 85.7%

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

      8. log1p-undefine 85.7%

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

      9. rem-exp-log 99.3%

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

      10. metadata-eval 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. +-commutative 99.3%

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

      13. frac-2neg 99.3%

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

      14. metadata-eval 99.3%

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

      15. div-inv 99.3%

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

      16. frac-2neg 99.3%

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

   9. Applied egg-rr 99.3%

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

       1. exp-1-e 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -510:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 40000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -510.0)
   (log (* E (/ y (+ x -1.0))))
   (if (<= y 40000000000000.0)
     (- 1.0 (log1p (/ x (+ y -1.0))))
     (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -510.0) {
		tmp = log((((double) M_E) * (y / (x + -1.0))));
	} else if (y <= 40000000000000.0) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -510.0) {
		tmp = Math.log((Math.E * (y / (x + -1.0))));
	} else if (y <= 40000000000000.0) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -510.0:
		tmp = math.log((math.e * (y / (x + -1.0))))
	elif y <= 40000000000000.0:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -510.0)
		tmp = log(Float64(exp(1) * Float64(y / Float64(x + -1.0))));
	elseif (y <= 40000000000000.0)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -510.0], N[Log[N[(E * N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 40000000000000.0], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -510

   1. Initial program 24.8%

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

      1. sub-neg 24.8%

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

      2. log1p-define 24.8%

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

      3. distribute-neg-frac2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. metadata-eval 24.8%

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

      7. +-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. log1p-define 98.5%

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

      7. mul-1-neg 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 98.5%

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

      4. add-log-exp 98.5%

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

      5. neg-log 98.5%

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

      6. exp-sum 98.5%

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

      7. add-exp-log 98.5%

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

      8. log1p-undefine 98.5%

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

      9. rem-exp-log 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. +-commutative 98.9%

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

      13. frac-2neg 98.9%

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

      14. metadata-eval 98.9%

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

      15. div-inv 98.9%

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

      16. frac-2neg 98.9%

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

   9. Applied egg-rr 99.0%

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

       1. exp-1-e 99.0%

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

   11. Simplified 99.0%

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

   if -510 < y < 4e13

   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-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.9%

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

   if 4e13 < y

   1. Initial program 63.3%

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

      1. sub-neg 63.3%

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

      2. log1p-define 63.3%

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

      3. distribute-neg-frac2 63.3%

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

      4. neg-sub0 63.3%

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

      5. associate--r- 63.3%

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

      6. metadata-eval 63.3%

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

      7. +-commutative 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-define 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-log-exp 0.0%

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

      5. neg-log 0.0%

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

      6. exp-sum 0.0%

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

      7. add-exp-log 0.0%

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

      8. log1p-undefine 0.0%

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

      9. rem-exp-log 99.9%

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

      10. metadata-eval 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. +-commutative 99.9%

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

      13. frac-2neg 99.9%

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

      14. metadata-eval 99.9%

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

      15. div-inv 99.9%

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

      16. frac-2neg 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. exp-1-e 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in x around inf 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 40000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13.5)
   (log (* y (- E)))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13.5) {
		tmp = log((y * -((double) M_E)));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -13.5) {
		tmp = Math.log((y * -Math.E));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13.5:
		tmp = math.log((y * -math.e))
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13.5)
		tmp = log(Float64(y * Float64(-exp(1))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13.5], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -13.5

   1. Initial program 24.8%

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

      1. sub-neg 24.8%

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

      2. log1p-define 24.8%

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

      3. distribute-neg-frac2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. metadata-eval 24.8%

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

      7. +-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. log1p-define 98.5%

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

      7. mul-1-neg 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 98.5%

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

      4. add-log-exp 98.5%

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

      5. neg-log 98.5%

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

      6. exp-sum 98.5%

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

      7. add-exp-log 98.5%

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

      8. log1p-undefine 98.5%

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

      9. rem-exp-log 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. +-commutative 98.9%

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

      13. frac-2neg 98.9%

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

      14. metadata-eval 98.9%

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

      15. div-inv 98.9%

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

      16. frac-2neg 98.9%

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

   9. Applied egg-rr 99.0%

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

       1. exp-1-e 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in x around 0 66.6%

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

       1. *-commutative 66.6%

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

       2. neg-mul-1 66.6%

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

       3. distribute-lft-neg-in 66.6%

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

   14. Simplified 66.6%

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

   if -13.5 < y < 1

   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-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.7%

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

   7. Simplified 99.7%

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

   if 1 < y

   1. Initial program 64.8%

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

      1. sub-neg 64.8%

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

      2. log1p-define 64.8%

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

      3. distribute-neg-frac2 64.8%

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

      4. neg-sub0 64.8%

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

      5. associate--r- 64.8%

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

      6. metadata-eval 64.8%

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

      7. +-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-define 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-log-exp 0.0%

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

      5. neg-log 0.0%

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

      6. exp-sum 0.0%

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

      7. add-exp-log 0.0%

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

      8. log1p-undefine 0.0%

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

      9. rem-exp-log 98.7%

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

      10. metadata-eval 98.7%

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

      11. distribute-neg-in 98.7%

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

      12. +-commutative 98.7%

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

      13. frac-2neg 98.7%

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

      14. metadata-eval 98.7%

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

      15. div-inv 98.8%

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

      16. frac-2neg 98.8%

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

   9. Applied egg-rr 98.8%

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

       1. exp-1-e 98.8%

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

   11. Simplified 98.8%

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

   12. Taylor expanded in x around inf 98.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.76:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.76)
   (log (* E (/ y (+ x -1.0))))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.76) {
		tmp = log((((double) M_E) * (y / (x + -1.0))));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.76) {
		tmp = Math.log((Math.E * (y / (x + -1.0))));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.76:
		tmp = math.log((math.e * (y / (x + -1.0))))
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.76)
		tmp = log(Float64(exp(1) * Float64(y / Float64(x + -1.0))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.76], N[Log[N[(E * N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.76000000000000001

   1. Initial program 24.8%

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

      1. sub-neg 24.8%

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

      2. log1p-define 24.8%

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

      3. distribute-neg-frac2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. metadata-eval 24.8%

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

      7. +-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. log1p-define 98.5%

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

      7. mul-1-neg 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 98.5%

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

      4. add-log-exp 98.5%

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

      5. neg-log 98.5%

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

      6. exp-sum 98.5%

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

      7. add-exp-log 98.5%

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

      8. log1p-undefine 98.5%

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

      9. rem-exp-log 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. +-commutative 98.9%

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

      13. frac-2neg 98.9%

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

      14. metadata-eval 98.9%

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

      15. div-inv 98.9%

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

      16. frac-2neg 98.9%

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

   9. Applied egg-rr 99.0%

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

       1. exp-1-e 99.0%

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

   11. Simplified 99.0%

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

   if -1.76000000000000001 < y < 1

   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-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.7%

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

   7. Simplified 99.7%

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

   if 1 < y

   1. Initial program 64.8%

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

      1. sub-neg 64.8%

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

      2. log1p-define 64.8%

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

      3. distribute-neg-frac2 64.8%

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

      4. neg-sub0 64.8%

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

      5. associate--r- 64.8%

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

      6. metadata-eval 64.8%

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

      7. +-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-define 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-log-exp 0.0%

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

      5. neg-log 0.0%

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

      6. exp-sum 0.0%

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

      7. add-exp-log 0.0%

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

      8. log1p-undefine 0.0%

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

      9. rem-exp-log 98.7%

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

      10. metadata-eval 98.7%

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

      11. distribute-neg-in 98.7%

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

      12. +-commutative 98.7%

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

      13. frac-2neg 98.7%

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

      14. metadata-eval 98.7%

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

      15. div-inv 98.8%

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

      16. frac-2neg 98.8%

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

   9. Applied egg-rr 98.8%

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

       1. exp-1-e 98.8%

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

   11. Simplified 98.8%

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

   12. Taylor expanded in x around inf 98.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.76:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -22:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -22.0)
   (log (* y (- E)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (log (/ (* y E) x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -22.0) {
		tmp = log((y * -((double) M_E)));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = log(((y * ((double) M_E)) / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -22.0) {
		tmp = Math.log((y * -Math.E));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = Math.log(((y * Math.E) / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -22.0:
		tmp = math.log((y * -math.e))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = math.log(((y * math.e) / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -22.0)
		tmp = log(Float64(y * Float64(-exp(1))));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y * exp(1)) / x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -22.0], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -22

   1. Initial program 24.8%

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

      1. sub-neg 24.8%

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

      2. log1p-define 24.8%

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

      3. distribute-neg-frac2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. metadata-eval 24.8%

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

      7. +-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. log1p-define 98.5%

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

      7. mul-1-neg 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 98.5%

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

      4. add-log-exp 98.5%

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

      5. neg-log 98.5%

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

      6. exp-sum 98.5%

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

      7. add-exp-log 98.5%

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

      8. log1p-undefine 98.5%

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

      9. rem-exp-log 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. +-commutative 98.9%

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

      13. frac-2neg 98.9%

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

      14. metadata-eval 98.9%

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

      15. div-inv 98.9%

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

      16. frac-2neg 98.9%

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

   9. Applied egg-rr 99.0%

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

       1. exp-1-e 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in x around 0 66.6%

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

       1. *-commutative 66.6%

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

       2. neg-mul-1 66.6%

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

       3. distribute-lft-neg-in 66.6%

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

   14. Simplified 66.6%

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

   if -22 < y < 1

   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-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. log1p-define 99.6%

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

      2. mul-1-neg 99.6%

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

   7. Simplified 99.6%

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

   if 1 < y

   1. Initial program 64.8%

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

      1. sub-neg 64.8%

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

      2. log1p-define 64.8%

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

      3. distribute-neg-frac2 64.8%

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

      4. neg-sub0 64.8%

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

      5. associate--r- 64.8%

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

      6. metadata-eval 64.8%

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

      7. +-commutative 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in y around -inf 0.0%

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

      1. sub-neg 0.0%

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

      2. metadata-eval 0.0%

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

      3. distribute-lft-in 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

      6. log1p-define 0.0%

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

      7. mul-1-neg 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. sub-neg 0.0%

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

      3. exp-sum 0.0%

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

      4. add-log-exp 0.0%

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

      5. neg-log 0.0%

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

      6. exp-sum 0.0%

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

      7. add-exp-log 0.0%

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

      8. log1p-undefine 0.0%

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

      9. rem-exp-log 98.7%

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

      10. metadata-eval 98.7%

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

      11. distribute-neg-in 98.7%

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

      12. +-commutative 98.7%

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

      13. frac-2neg 98.7%

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

      14. metadata-eval 98.7%

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

      15. div-inv 98.8%

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

      16. frac-2neg 98.8%

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

   9. Applied egg-rr 98.8%

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

       1. exp-1-e 98.8%

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

   11. Simplified 98.8%

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

   12. Taylor expanded in x around inf 98.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -22:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.35) (log (* y (- E))) (+ 1.0 (/ x (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.35) {
		tmp = log((y * -((double) M_E)));
	} else {
		tmp = 1.0 + (x / (1.0 - y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.35) {
		tmp = Math.log((y * -Math.E));
	} else {
		tmp = 1.0 + (x / (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.35:
		tmp = math.log((y * -math.e))
	else:
		tmp = 1.0 + (x / (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.35)
		tmp = log(Float64(y * Float64(-exp(1))));
	else
		tmp = Float64(1.0 + Float64(x / Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.35)
		tmp = log((y * -2.71828182845904523536));
	else
		tmp = 1.0 + (x / (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.35], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001

   1. Initial program 24.8%

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

      1. sub-neg 24.8%

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

      2. log1p-define 24.8%

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

      3. distribute-neg-frac2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. metadata-eval 24.8%

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

      7. +-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. log1p-define 98.5%

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

      7. mul-1-neg 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 98.5%

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

      4. add-log-exp 98.5%

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

      5. neg-log 98.5%

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

      6. exp-sum 98.5%

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

      7. add-exp-log 98.5%

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

      8. log1p-undefine 98.5%

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

      9. rem-exp-log 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. +-commutative 98.9%

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

      13. frac-2neg 98.9%

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

      14. metadata-eval 98.9%

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

      15. div-inv 98.9%

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

      16. frac-2neg 98.9%

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

   9. Applied egg-rr 99.0%

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

       1. exp-1-e 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in x around 0 66.6%

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

       1. *-commutative 66.6%

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

       2. neg-mul-1 66.6%

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

       3. distribute-lft-neg-in 66.6%

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

   14. Simplified 66.6%

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

   if -1.3500000000000001 < y

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. log1p-define 95.0%

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

      3. distribute-neg-frac2 95.0%

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

      4. neg-sub0 95.0%

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

      5. associate--r- 95.0%

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

      6. metadata-eval 95.0%

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

      7. +-commutative 95.0%

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

   3. Simplified 95.0%

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

      1. add-cbrt-cube 70.4%

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

      2. pow1/3 53.3%

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

      3. pow3 53.3%

         \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x - y}{y + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   6. Applied egg-rr 53.3%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{x - y}{y + -1}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

   7. Taylor expanded in x around inf 53.8%

      \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left(\frac{{x}^{3}}{{\left(y - 1\right)}^{3}}\right)}}^{0.3333333333333333}\right) \]
   8. Step-by-step derivation

      1. sub-neg 53.8%

         \[\leadsto 1 - \mathsf{log1p}\left({\left(\frac{{x}^{3}}{{\color{blue}{\left(y + \left(-1\right)\right)}}^{3}}\right)}^{0.3333333333333333}\right) \]

      2. metadata-eval 53.8%

         \[\leadsto 1 - \mathsf{log1p}\left({\left(\frac{{x}^{3}}{{\left(y + \color{blue}{-1}\right)}^{3}}\right)}^{0.3333333333333333}\right) \]

      3. cube-div 58.4%

         \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x}{y + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

      4. +-commutative 58.4%

         \[\leadsto 1 - \mathsf{log1p}\left({\left({\left(\frac{x}{\color{blue}{-1 + y}}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

   9. Simplified 58.4%

      \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x}{-1 + y}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   10. Taylor expanded in x around 0 58.3%

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

       1. mul-1-neg 58.3%

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

       2. sub-neg 58.3%

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

       3. metadata-eval 58.3%

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

       4. distribute-neg-frac2 58.3%

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

       5. distribute-neg-in 58.3%

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

       6. metadata-eval 58.3%

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

       7. +-commutative 58.3%

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

       8. unsub-neg 58.3%

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

   12. Simplified 58.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -54:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -54.0) (log (* y (- E))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -54.0) {
		tmp = log((y * -((double) M_E)));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -54.0) {
		tmp = Math.log((y * -Math.E));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -54.0:
		tmp = math.log((y * -math.e))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -54.0)
		tmp = log(Float64(y * Float64(-exp(1))));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -54.0], N[Log[N[(y * (-E)), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -54

   1. Initial program 24.8%

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

      1. sub-neg 24.8%

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

      2. log1p-define 24.8%

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

      3. distribute-neg-frac2 24.8%

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

      4. neg-sub0 24.8%

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

      5. associate--r- 24.8%

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

      6. metadata-eval 24.8%

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

      7. +-commutative 24.8%

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

   3. Simplified 24.8%

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

   5. Taylor expanded in y around -inf 98.5%

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

      1. sub-neg 98.5%

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

      2. metadata-eval 98.5%

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

      3. distribute-lft-in 98.5%

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

      4. metadata-eval 98.5%

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

      5. +-commutative 98.5%

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

      6. log1p-define 98.5%

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

      7. mul-1-neg 98.5%

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

   7. Simplified 98.5%

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

      1. add-log-exp 98.5%

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

      2. sub-neg 98.5%

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

      3. exp-sum 98.5%

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

      4. add-log-exp 98.5%

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

      5. neg-log 98.5%

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

      6. exp-sum 98.5%

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

      7. add-exp-log 98.5%

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

      8. log1p-undefine 98.5%

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

      9. rem-exp-log 98.9%

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

      10. metadata-eval 98.9%

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

      11. distribute-neg-in 98.9%

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

      12. +-commutative 98.9%

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

      13. frac-2neg 98.9%

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

      14. metadata-eval 98.9%

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

      15. div-inv 98.9%

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

      16. frac-2neg 98.9%

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

   9. Applied egg-rr 99.0%

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

       1. exp-1-e 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in x around 0 66.6%

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

       1. *-commutative 66.6%

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

       2. neg-mul-1 66.6%

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

       3. distribute-lft-neg-in 66.6%

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

   14. Simplified 66.6%

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

   if -54 < y

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. log1p-define 95.0%

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

      3. distribute-neg-frac2 95.0%

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

      4. neg-sub0 95.0%

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

      5. associate--r- 95.0%

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

      6. metadata-eval 95.0%

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

      7. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in y around 0 85.5%

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

      1. log1p-define 85.5%

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

      2. mul-1-neg 85.5%

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

   7. Simplified 85.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -54:\\ \;\;\;\;\log \left(y \cdot \left(-e\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.1% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

   1. sub-neg 73.4%

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

   2. log1p-define 73.4%

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

   3. distribute-neg-frac2 73.4%

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

   4. neg-sub0 73.4%

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

   5. associate--r- 73.4%

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

   6. metadata-eval 73.4%

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

   7. +-commutative 73.4%

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

3. Simplified 73.4%

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

   1. add-cbrt-cube 52.9%

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

   2. pow1/3 39.3%

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

   3. pow3 39.3%

      \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x - y}{y + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

6. Applied egg-rr 39.3%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{x - y}{y + -1}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

7. Taylor expanded in x around inf 41.3%

   \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left(\frac{{x}^{3}}{{\left(y - 1\right)}^{3}}\right)}}^{0.3333333333333333}\right) \]
8. Step-by-step derivation

   1. sub-neg 41.3%

      \[\leadsto 1 - \mathsf{log1p}\left({\left(\frac{{x}^{3}}{{\color{blue}{\left(y + \left(-1\right)\right)}}^{3}}\right)}^{0.3333333333333333}\right) \]

   2. metadata-eval 41.3%

      \[\leadsto 1 - \mathsf{log1p}\left({\left(\frac{{x}^{3}}{{\left(y + \color{blue}{-1}\right)}^{3}}\right)}^{0.3333333333333333}\right) \]

   3. cube-div 45.9%

      \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x}{y + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   4. +-commutative 45.9%

      \[\leadsto 1 - \mathsf{log1p}\left({\left({\left(\frac{x}{\color{blue}{-1 + y}}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

9. Simplified 45.9%

   \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x}{-1 + y}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

10. Taylor expanded in x around 0 44.2%

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

    1. mul-1-neg 44.2%

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

    2. sub-neg 44.2%

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

    3. metadata-eval 44.2%

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

    4. distribute-neg-frac2 44.2%

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

    5. distribute-neg-in 44.2%

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

    6. metadata-eval 44.2%

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

    7. +-commutative 44.2%

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

    8. unsub-neg 44.2%

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

12. Simplified 44.2%

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

13. Final simplification 44.2%

    \[\leadsto 1 + \frac{x}{1 - y} \]
14. Add Preprocessing

Alternative 9: 43.5% 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 73.4%

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

   1. sub-neg 73.4%

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

   2. log1p-define 73.4%

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

   3. distribute-neg-frac2 73.4%

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

   4. neg-sub0 73.4%

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

   5. associate--r- 73.4%

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

   6. metadata-eval 73.4%

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

   7. +-commutative 73.4%

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

3. Simplified 73.4%

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

   1. add-cbrt-cube 52.9%

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

   2. pow1/3 39.3%

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

   3. pow3 39.3%

      \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x - y}{y + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

6. Applied egg-rr 39.3%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{{\left({\left(\frac{x - y}{y + -1}\right)}^{3}\right)}^{0.3333333333333333}}\right) \]

7. Taylor expanded in x around inf 41.3%

   \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left(\frac{{x}^{3}}{{\left(y - 1\right)}^{3}}\right)}}^{0.3333333333333333}\right) \]
8. Step-by-step derivation

   1. sub-neg 41.3%

      \[\leadsto 1 - \mathsf{log1p}\left({\left(\frac{{x}^{3}}{{\color{blue}{\left(y + \left(-1\right)\right)}}^{3}}\right)}^{0.3333333333333333}\right) \]

   2. metadata-eval 41.3%

      \[\leadsto 1 - \mathsf{log1p}\left({\left(\frac{{x}^{3}}{{\left(y + \color{blue}{-1}\right)}^{3}}\right)}^{0.3333333333333333}\right) \]

   3. cube-div 45.9%

      \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x}{y + -1}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

   4. +-commutative 45.9%

      \[\leadsto 1 - \mathsf{log1p}\left({\left({\left(\frac{x}{\color{blue}{-1 + y}}\right)}^{3}\right)}^{0.3333333333333333}\right) \]

9. Simplified 45.9%

   \[\leadsto 1 - \mathsf{log1p}\left({\color{blue}{\left({\left(\frac{x}{-1 + y}\right)}^{3}\right)}}^{0.3333333333333333}\right) \]

10. Taylor expanded in x around 0 42.6%

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

11. Final simplification 42.6%

    \[\leadsto 1 \]
12. Add Preprocessing

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 2024096 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (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))))))