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

Percentage Accurate: 71.1% → 100.0%

Time: 8.4s

Alternatives: 12

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: 71.1% 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: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y)
	return log(Float64(Float64(exp(1) / Float64(1.0 - x)) - Float64(exp(1) / Float64(Float64(1.0 - x) / y))))
end

MATLAB

function tmp = code(x, y)
	tmp = log(((2.71828182845904523536 / (1.0 - x)) - (2.71828182845904523536 / ((1.0 - x) / y))));
end

Wolfram

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

Derivation

1. Initial program 71.0%

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

   1. sub-neg 71.0%

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

   2. log1p-def 71.1%

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

   3. neg-sub0 71.1%

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

   4. div-sub 71.1%

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

   5. associate--r- 71.1%

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

   6. neg-sub0 71.1%

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

   7. +-commutative 71.1%

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

   8. sub-neg 71.1%

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

   9. div-sub 71.1%

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

3. Simplified 71.1%

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

   1. add-log-exp 71.1%

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

   2. exp-diff 71.0%

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

   3. exp-1-e 71.0%

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

   4. log1p-udef 71.0%

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

   5. add-exp-log 71.0%

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

5. Applied egg-rr 71.0%

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

6. Taylor expanded in y around 0 83.1%

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

   1. +-commutative 83.1%

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

   2. mul-1-neg 83.1%

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

   3. unsub-neg 83.1%

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

   4. associate-/l* 83.1%

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

   5. sub-neg 83.1%

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

   6. neg-mul-1 83.1%

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

   7. unpow2 83.1%

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

   8. *-commutative 83.1%

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

   9. times-frac 100.0%

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

   10. *-inverses 100.0%

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

   11. neg-mul-1 100.0%

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

   12. sub-neg 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(Float64(x + -1.0) / Float64(y * y)) + Float64(Float64(x + -1.0) / y))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.900000000000000022

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if 0.900000000000000022 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 8.6%

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

      1. sub-neg 8.6%

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

      2. log1p-def 8.6%

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

      3. neg-sub0 8.6%

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

      4. div-sub 8.7%

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

      5. associate--r- 8.7%

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

      6. neg-sub0 8.7%

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

      7. +-commutative 8.7%

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

      8. sub-neg 8.7%

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

      9. div-sub 8.6%

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

   3. Simplified 8.6%

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

      1. add-log-exp 8.6%

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

      2. exp-diff 8.6%

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

      3. exp-1-e 8.6%

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

      4. log1p-udef 8.6%

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

      5. add-exp-log 8.6%

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

   5. Applied egg-rr 8.6%

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

   6. Taylor expanded in y around -inf 99.6%

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

      1. +-commutative 99.6%

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

      2. associate--l+ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. sub-neg 99.6%

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

      5. neg-mul-1 99.6%

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

      6. mul-1-neg 99.6%

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

      7. neg-mul-1 99.6%

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

      8. sub-neg 99.6%

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

      9. unpow2 99.6%

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

      10. div-sub 99.6%

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

      11. sub-neg 99.6%

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

      12. metadata-eval 99.6%

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

   8. Simplified 99.6%

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

4. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 - x\right) \cdot \frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99998)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (* (- 1.0 x) (/ -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99998) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((1.0 - x) * (-1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99998) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((1.0 - x) * (-1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99998:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((1.0 - x) * (-1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99998)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(1.0 - x) * Float64(-1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.99998], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(1.0 - x), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.99997999999999998

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.8%

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

      3. neg-sub0 99.8%

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

      4. div-sub 99.8%

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

      5. associate--r- 99.8%

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

      6. neg-sub0 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

   if 0.99997999999999998 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 5.6%

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

      1. sub-neg 5.6%

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

      2. log1p-def 5.6%

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

      3. neg-sub0 5.6%

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

      4. div-sub 5.6%

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

      5. associate--r- 5.6%

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

      6. neg-sub0 5.6%

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

      7. +-commutative 5.6%

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

      8. sub-neg 5.6%

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

      9. div-sub 5.6%

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

   3. Simplified 5.6%

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

   4. Taylor expanded in y around -inf 86.8%

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

      1. sub-neg 86.8%

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

      2. metadata-eval 86.8%

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

      3. distribute-lft-in 86.8%

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

      4. metadata-eval 86.8%

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

      5. +-commutative 86.8%

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

      6. log1p-def 86.8%

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

      7. mul-1-neg 86.8%

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

   6. Simplified 86.8%

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

   7. Taylor expanded in y around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. log-rec 0.0%

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

      3. associate-+r+ 0.0%

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

      4. log-rec 0.0%

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

      5. unsub-neg 0.0%

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

      6. +-commutative 0.0%

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

      7. sub-neg 0.0%

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

      8. log1p-def 0.0%

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

      9. associate-+r- 0.0%

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

      10. log-div 86.8%

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

      11. rem-log-exp 86.8%

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

      12. exp-sum 86.8%

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

      13. neg-mul-1 86.8%

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

      14. log1p-def 86.8%

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

      15. rem-exp-log 86.8%

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

      16. neg-mul-1 86.8%

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

      17. sub-neg 86.8%

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

      18. rem-exp-log 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 - x\right) \cdot \frac{-1}{y}\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\left(1 - x\right) \cdot \frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6) (not (<= y 1.0)))
   (- 1.0 (log (* (- 1.0 x) (/ -1.0 y))))
   (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6) || !(y <= 1.0)) {
		tmp = 1.0 - log(((1.0 - x) * (-1.0 / y)));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.6) || !(y <= 1.0)) {
		tmp = 1.0 - Math.log(((1.0 - x) * (-1.0 / y)));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6) or not (y <= 1.0):
		tmp = 1.0 - math.log(((1.0 - x) * (-1.0 / y)))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(1.0 - x) * Float64(-1.0 / y))));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(1.0 - x), $MachinePrecision] * N[(-1.0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6000000000000001 or 1 < y

   1. Initial program 35.1%

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

      1. sub-neg 35.1%

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

      2. log1p-def 35.1%

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

      3. neg-sub0 35.1%

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

      4. div-sub 35.1%

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

      5. associate--r- 35.1%

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

      6. neg-sub0 35.1%

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

      7. +-commutative 35.1%

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

      8. sub-neg 35.1%

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

      9. div-sub 35.1%

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

   3. Simplified 35.1%

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

   4. Taylor expanded in y around -inf 66.9%

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

      1. sub-neg 66.9%

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

      2. metadata-eval 66.9%

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

      3. distribute-lft-in 66.9%

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

      4. metadata-eval 66.9%

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

      5. +-commutative 66.9%

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

      6. log1p-def 66.9%

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

      7. mul-1-neg 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in y around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. log-rec 0.0%

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

      3. associate-+r+ 0.0%

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

      4. log-rec 0.0%

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

      5. unsub-neg 0.0%

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

      6. +-commutative 0.0%

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

      7. sub-neg 0.0%

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

      8. log1p-def 0.0%

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

      9. associate-+r- 0.0%

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

      10. log-div 66.9%

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

      11. rem-log-exp 66.9%

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

      12. exp-sum 66.9%

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

      13. neg-mul-1 66.9%

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

      14. log1p-def 66.9%

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

      15. rem-exp-log 67.0%

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

      16. neg-mul-1 67.0%

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

      17. sub-neg 67.0%

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

      18. rem-exp-log 98.0%

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

   9. Simplified 98.0%

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

   if -1.6000000000000001 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.7%

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

      1. div-sub 98.7%

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

      2. mul-1-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. *-inverses 98.7%

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

      5. *-rgt-identity 98.7%

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

      6. log1p-def 98.7%

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

      7. mul-1-neg 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\left(1 - x\right) \cdot \frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000018

   1. Initial program 17.5%

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

      1. sub-neg 17.5%

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

      2. log1p-def 17.5%

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

      3. neg-sub0 17.5%

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

      4. div-sub 17.5%

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

      5. associate--r- 17.5%

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

      6. neg-sub0 17.5%

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

      7. +-commutative 17.5%

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

      8. sub-neg 17.5%

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

      9. div-sub 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in y around -inf 97.8%

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

      1. sub-neg 97.8%

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

      2. metadata-eval 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

      6. log1p-def 97.8%

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

      7. mul-1-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around 0 69.3%

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

   if -7.20000000000000018 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.7%

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

      1. div-sub 98.7%

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

      2. mul-1-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. *-inverses 98.7%

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

      5. *-rgt-identity 98.7%

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

      6. log1p-def 98.7%

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

      7. mul-1-neg 98.7%

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

   6. Simplified 98.7%

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

   if 1 < y

   1. Initial program 73.2%

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

      1. sub-neg 73.2%

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

      2. log1p-def 73.2%

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

      3. neg-sub0 73.2%

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

      4. div-sub 73.2%

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

      5. associate--r- 73.2%

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

      6. neg-sub0 73.2%

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

      7. +-commutative 73.2%

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

      8. sub-neg 73.2%

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

      9. div-sub 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in x around inf 68.4%

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

      1. neg-mul-1 68.4%

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

      2. distribute-neg-frac 68.4%

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

   6. Simplified 68.4%

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

   7. Taylor expanded in y around inf 65.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-\frac{x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- (/ x (- 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-(x / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-(x / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-(x / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-Float64(x / Float64(1.0 - y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000018

   1. Initial program 17.5%

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

      1. sub-neg 17.5%

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

      2. log1p-def 17.5%

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

      3. neg-sub0 17.5%

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

      4. div-sub 17.5%

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

      5. associate--r- 17.5%

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

      6. neg-sub0 17.5%

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

      7. +-commutative 17.5%

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

      8. sub-neg 17.5%

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

      9. div-sub 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in y around -inf 97.8%

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

      1. sub-neg 97.8%

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

      2. metadata-eval 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

      6. log1p-def 97.8%

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

      7. mul-1-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around 0 69.3%

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

   if -7.20000000000000018 < y

   1. Initial program 94.5%

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

      1. sub-neg 94.5%

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

      2. log1p-def 94.6%

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

      3. neg-sub0 94.6%

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

      4. div-sub 94.6%

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

      5. associate--r- 94.6%

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

      6. neg-sub0 94.6%

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

      7. +-commutative 94.6%

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

      8. sub-neg 94.6%

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

      9. div-sub 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in x around inf 91.5%

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

      1. neg-mul-1 91.5%

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

      2. distribute-neg-frac 91.5%

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

   6. Simplified 91.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-\frac{x}{1 - y}\right)\\ \end{array} \]

Alternative 7: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.9:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.9)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.5e-7) (- 1.0 (log1p (- x))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.9) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.5e-7) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.9) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.5e-7) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.9:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.5e-7:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.9)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.5e-7)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.9], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e-7], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9000000000000004

   1. Initial program 17.5%

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

      1. sub-neg 17.5%

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

      2. log1p-def 17.5%

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

      3. neg-sub0 17.5%

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

      4. div-sub 17.5%

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

      5. associate--r- 17.5%

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

      6. neg-sub0 17.5%

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

      7. +-commutative 17.5%

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

      8. sub-neg 17.5%

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

      9. div-sub 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in y around -inf 97.8%

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

      1. sub-neg 97.8%

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

      2. metadata-eval 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

      6. log1p-def 97.8%

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

      7. mul-1-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around 0 69.3%

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

   if -4.9000000000000004 < y < 1.4999999999999999e-7

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.2%

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

      1. log1p-def 98.2%

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

      2. mul-1-neg 98.2%

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

   6. Simplified 98.2%

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

   if 1.4999999999999999e-7 < y

   1. Initial program 74.6%

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

      1. sub-neg 74.6%

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

      2. log1p-def 74.7%

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

      3. neg-sub0 74.7%

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

      4. div-sub 74.6%

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

      5. associate--r- 74.6%

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

      6. neg-sub0 74.6%

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

      7. +-commutative 74.6%

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

      8. sub-neg 74.6%

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

      9. div-sub 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around inf 66.9%

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

      1. neg-mul-1 66.9%

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

      2. distribute-neg-frac 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in y around inf 64.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 8: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6

   1. Initial program 17.5%

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

      1. sub-neg 17.5%

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

      2. log1p-def 17.5%

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

      3. neg-sub0 17.5%

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

      4. div-sub 17.5%

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

      5. associate--r- 17.5%

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

      6. neg-sub0 17.5%

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

      7. +-commutative 17.5%

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

      8. sub-neg 17.5%

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

      9. div-sub 17.5%

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

   3. Simplified 17.5%

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

   4. Taylor expanded in y around -inf 97.8%

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

      1. sub-neg 97.8%

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

      2. metadata-eval 97.8%

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

      3. distribute-lft-in 97.8%

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

      4. metadata-eval 97.8%

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

      5. +-commutative 97.8%

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

      6. log1p-def 97.8%

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

      7. mul-1-neg 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in x around 0 69.3%

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

   if -6 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.4%

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

      1. log1p-def 97.4%

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

      2. mul-1-neg 97.4%

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

   6. Simplified 97.4%

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

   if 1 < y

   1. Initial program 73.2%

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

      1. sub-neg 73.2%

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

      2. log1p-def 73.2%

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

      3. neg-sub0 73.2%

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

      4. div-sub 73.2%

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

      5. associate--r- 73.2%

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

      6. neg-sub0 73.2%

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

      7. +-commutative 73.2%

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

      8. sub-neg 73.2%

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

      9. div-sub 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in y around 0 0.0%

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

      1. log1p-def 0.0%

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

      2. mul-1-neg 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. log-rec 0.0%

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

      4. remove-double-neg 0.0%

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

      5. log-prod 0.0%

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

      6. neg-mul-1 0.0%

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

   9. Simplified 0.0%

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

       1. sub-neg 0.0%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 4.1%

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

       4. sqr-neg 4.1%

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

       5. sqrt-unprod 12.7%

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

       6. add-sqr-sqrt 12.7%

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

   11. Applied egg-rr 12.7%

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

       1. sub-neg 12.7%

          \[\leadsto \color{blue}{1 - \log x} \]

   13. Simplified 12.7%

       \[\leadsto \color{blue}{1 - \log x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log x\\ \end{array} \]

Alternative 9: 61.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.82) (- 1.0 (log (- x))) (+ 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.82) {
		tmp = 1.0 - log(-x);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.82d0)) then
        tmp = 1.0d0 - log(-x)
    else
        tmp = 1.0d0 + x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.82) {
		tmp = 1.0 - Math.log(-x);
	} else {
		tmp = 1.0 + x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.82:
		tmp = 1.0 - math.log(-x)
	else:
		tmp = 1.0 + x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.82)
		tmp = Float64(1.0 - log(Float64(-x)));
	else
		tmp = Float64(1.0 + x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.82)
		tmp = 1.0 - log(-x);
	else
		tmp = 1.0 + x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.82], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.819999999999999951

   1. Initial program 80.0%

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

      1. sub-neg 80.0%

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

      2. log1p-def 80.0%

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

      3. neg-sub0 80.0%

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

      4. div-sub 80.0%

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

      5. associate--r- 80.0%

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

      6. neg-sub0 80.0%

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

      7. +-commutative 80.0%

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

      8. sub-neg 80.0%

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

      9. div-sub 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in y around 0 70.1%

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

      1. log1p-def 70.1%

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

      2. mul-1-neg 70.1%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. log-rec 0.0%

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

      4. remove-double-neg 0.0%

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

      5. log-prod 69.0%

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

      6. neg-mul-1 69.0%

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

   9. Simplified 69.0%

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

   if -0.819999999999999951 < x

   1. Initial program 66.9%

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

      1. sub-neg 66.9%

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

      2. log1p-def 66.9%

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

      3. neg-sub0 66.9%

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

      4. div-sub 66.9%

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

      5. associate--r- 66.9%

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

      6. neg-sub0 66.9%

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

      7. +-commutative 66.9%

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

      8. sub-neg 66.9%

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

      9. div-sub 66.9%

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

   3. Simplified 66.9%

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

   4. Taylor expanded in y around 0 51.8%

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

      1. log1p-def 51.8%

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

      2. mul-1-neg 51.8%

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

   6. Simplified 51.8%

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

   7. Taylor expanded in x around 0 52.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x\\ \end{array} \]

Alternative 10: 63.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 70.7%

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

      1. sub-neg 70.7%

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

      2. log1p-def 70.7%

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

      3. neg-sub0 70.7%

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

      4. div-sub 70.7%

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

      5. associate--r- 70.7%

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

      6. neg-sub0 70.7%

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

      7. +-commutative 70.7%

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

      8. sub-neg 70.7%

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

      9. div-sub 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in y around 0 67.0%

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

      1. log1p-def 67.0%

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

      2. mul-1-neg 67.0%

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

   6. Simplified 67.0%

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

   if 1 < y

   1. Initial program 73.2%

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

      1. sub-neg 73.2%

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

      2. log1p-def 73.2%

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

      3. neg-sub0 73.2%

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

      4. div-sub 73.2%

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

      5. associate--r- 73.2%

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

      6. neg-sub0 73.2%

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

      7. +-commutative 73.2%

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

      8. sub-neg 73.2%

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

      9. div-sub 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in y around 0 0.0%

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

      1. log1p-def 0.0%

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

      2. mul-1-neg 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. mul-1-neg 0.0%

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

      3. log-rec 0.0%

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

      4. remove-double-neg 0.0%

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

      5. log-prod 0.0%

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

      6. neg-mul-1 0.0%

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

   9. Simplified 0.0%

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

       1. sub-neg 0.0%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 4.1%

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

       4. sqr-neg 4.1%

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

       5. sqrt-unprod 12.7%

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

       6. add-sqr-sqrt 12.7%

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

   11. Applied egg-rr 12.7%

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

       1. sub-neg 12.7%

          \[\leadsto \color{blue}{1 - \log x} \]

   13. Simplified 12.7%

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

4. Final simplification 59.4%

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

Alternative 11: 42.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

   1. sub-neg 71.0%

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

   2. log1p-def 71.1%

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

   3. neg-sub0 71.1%

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

   4. div-sub 71.1%

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

   5. associate--r- 71.1%

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

   6. neg-sub0 71.1%

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

   7. +-commutative 71.1%

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

   8. sub-neg 71.1%

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

   9. div-sub 71.1%

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

3. Simplified 71.1%

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

4. Taylor expanded in y around 0 57.6%

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

   1. log1p-def 57.6%

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

   2. mul-1-neg 57.6%

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

6. Simplified 57.6%

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

7. Taylor expanded in x around 0 37.0%

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

8. Final simplification 37.0%

   \[\leadsto 1 + x \]

Alternative 12: 42.5% 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 71.0%

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

   1. sub-neg 71.0%

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

   2. log1p-def 71.1%

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

   3. neg-sub0 71.1%

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

   4. div-sub 71.1%

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

   5. associate--r- 71.1%

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

   6. neg-sub0 71.1%

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

   7. +-commutative 71.1%

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

   8. sub-neg 71.1%

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

   9. div-sub 71.1%

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

3. Simplified 71.1%

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

4. Taylor expanded in y around 0 57.6%

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

   1. log1p-def 57.6%

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

   2. mul-1-neg 57.6%

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

6. Simplified 57.6%

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

7. Taylor expanded in x around 0 36.9%

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

8. Final simplification 36.9%

   \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023192 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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