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

Percentage Accurate: 72.5% → 100.0%

Time: 13.6s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{e}{\left(x + -1\right) \cdot \frac{1}{y}}\\ \mathbf{if}\;\frac{x - y}{1 - y} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(t\_0 - \frac{t\_0}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ E (* (+ x -1.0) (/ 1.0 y)))))
   (if (<= (/ (- x y) (- 1.0 y)) 5e-16)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (log (- t_0 (/ t_0 y))))))

C

double code(double x, double y) {
	double t_0 = ((double) M_E) / ((x + -1.0) * (1.0 / y));
	double tmp;
	if (((x - y) / (1.0 - y)) <= 5e-16) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = log((t_0 - (t_0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.E / ((x + -1.0) * (1.0 / y));
	double tmp;
	if (((x - y) / (1.0 - y)) <= 5e-16) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = Math.log((t_0 - (t_0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.e / ((x + -1.0) * (1.0 / y))
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 5e-16:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = math.log((t_0 - (t_0 / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(exp(1) / Float64(Float64(x + -1.0) * Float64(1.0 / y)))
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 5e-16)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(t_0 - Float64(t_0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(E / N[(N[(x + -1.0), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 5e-16], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(t$95$0 - N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 5.0000000000000004e-16 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 9.9%

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

      1. sub-neg 9.9%

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

      2. log1p-def 9.9%

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

      3. distribute-neg-frac 9.9%

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

      4. sub-neg 9.9%

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

      5. distribute-neg-in 9.9%

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

      6. remove-double-neg 9.9%

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

      7. +-commutative 9.9%

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

      8. sub-neg 9.9%

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

   3. Simplified 9.9%

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

      1. add-log-exp 9.9%

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

   6. Applied egg-rr 9.9%

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

   7. Taylor expanded in y around inf 14.0%

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

      1. mul-1-neg 14.0%

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

      2. unsub-neg 14.0%

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

      3. exp-diff 14.0%

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

      4. e-exp-1 14.0%

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

      5. exp-sum 14.0%

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

      6. rem-exp-log 14.0%

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

      7. sub-neg 14.0%

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

      8. metadata-eval 14.0%

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

      9. +-commutative 14.0%

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

      10. rem-exp-log 14.3%

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

   9. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\left(x + -1\right) \cdot \frac{1}{y}} - \frac{\frac{e}{\left(x + -1\right) \cdot \frac{1}{y}}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\left(1 - x\right) \cdot \frac{-1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 4e-5)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (log (/ E (* (- 1.0 x) (/ -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 4e-5) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = log((((double) M_E) / ((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)) <= 4e-5) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = Math.log((Math.E / ((1.0 - x) * (-1.0 / y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 4e-5:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = math.log((math.e / ((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)) <= 4e-5)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(exp(1) / 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], 4e-5], 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[(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)) < 4.00000000000000033e-5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 4.00000000000000033e-5 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 7.5%

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

      1. sub-neg 7.5%

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

      2. log1p-def 7.5%

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

      3. distribute-neg-frac 7.5%

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

      4. sub-neg 7.5%

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

      5. distribute-neg-in 7.5%

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

      6. remove-double-neg 7.5%

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

      7. +-commutative 7.5%

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

      8. sub-neg 7.5%

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

   3. Simplified 7.5%

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

      1. add-log-exp 7.5%

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

   6. Applied egg-rr 7.5%

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

   7. Taylor expanded in y around -inf 84.9%

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

      1. exp-diff 84.9%

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

      2. e-exp-1 84.9%

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

      3. exp-sum 84.9%

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

      4. sub-neg 84.9%

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

      5. metadata-eval 84.9%

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

      6. distribute-lft-in 84.9%

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

      7. metadata-eval 84.9%

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

      8. +-commutative 84.9%

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

      9. rem-exp-log 84.9%

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

      10. mul-1-neg 84.9%

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

      11. sub-neg 84.9%

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

      12. rem-exp-log 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\left(1 - x\right) \cdot \frac{-1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\log \left(\frac{e}{\left(1 - x\right) \cdot \frac{-1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7) (not (<= y 1.0)))
   (log (/ E (* (- 1.0 x) (/ -1.0 y))))
   (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7) || !(y <= 1.0)) {
		tmp = log((((double) M_E) / ((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 <= -1.7) || !(y <= 1.0)) {
		tmp = Math.log((Math.E / ((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 <= -1.7) or not (y <= 1.0):
		tmp = math.log((math.e / ((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 <= -1.7) || !(y <= 1.0))
		tmp = log(Float64(exp(1) / 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, -1.7], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[Log[N[(E / 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 < -1.69999999999999996 or 1 < y

   1. Initial program 30.6%

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

      1. sub-neg 30.6%

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

      2. log1p-def 30.6%

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

      3. distribute-neg-frac 30.6%

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

      4. sub-neg 30.6%

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

      5. distribute-neg-in 30.6%

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

      6. remove-double-neg 30.6%

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

      7. +-commutative 30.6%

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

      8. sub-neg 30.6%

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

   3. Simplified 30.6%

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

      1. add-log-exp 30.6%

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

   6. Applied egg-rr 30.6%

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

   7. Taylor expanded in y around -inf 74.1%

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

      1. exp-diff 74.1%

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

      2. e-exp-1 74.1%

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

      3. exp-sum 74.1%

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

      4. sub-neg 74.1%

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

      5. metadata-eval 74.1%

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

      6. distribute-lft-in 74.1%

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

      7. metadata-eval 74.1%

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

      8. +-commutative 74.1%

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

      9. rem-exp-log 74.2%

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

      10. mul-1-neg 74.2%

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

      11. sub-neg 74.2%

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

      12. rem-exp-log 99.2%

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

   9. Simplified 99.2%

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

   if -1.69999999999999996 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. div-sub 99.0%

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

      3. mul-1-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. *-inverses 99.0%

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

      6. *-rgt-identity 99.0%

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

      7. log1p-def 99.0%

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

      8. mul-1-neg 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\log \left(\frac{e}{\left(1 - x\right) \cdot \frac{-1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -82:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -82.0)
   (- 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 <= -82.0) {
		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 <= -82.0) {
		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 <= -82.0:
		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 <= -82.0)
		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, -82.0], 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 < -82

   1. Initial program 19.0%

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

      1. sub-neg 19.0%

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

      2. log1p-def 19.0%

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

      3. distribute-neg-frac 19.0%

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

      4. sub-neg 19.0%

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

      5. distribute-neg-in 19.0%

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

      6. remove-double-neg 19.0%

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

      7. +-commutative 19.0%

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

      8. sub-neg 19.0%

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

   3. Simplified 19.0%

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

   5. Taylor expanded in y around inf 18.5%

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

   6. Taylor expanded in x around 0 68.8%

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

      1. distribute-neg-frac 68.8%

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

      2. metadata-eval 68.8%

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

   8. Simplified 68.8%

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

   if -82 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. div-sub 99.0%

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

      3. mul-1-neg 99.0%

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

      4. sub-neg 99.0%

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

      5. *-inverses 99.0%

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

      6. *-rgt-identity 99.0%

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

      7. log1p-def 99.0%

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

      8. mul-1-neg 99.0%

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

   7. Simplified 99.0%

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

   if 1 < y

   1. Initial program 65.3%

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

      1. sub-neg 65.3%

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

      2. log1p-def 65.3%

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

      3. distribute-neg-frac 65.3%

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

      4. sub-neg 65.3%

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

      5. distribute-neg-in 65.3%

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

      6. remove-double-neg 65.3%

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

      7. +-commutative 65.3%

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

      8. sub-neg 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 56.3%

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

      1. neg-mul-1 56.3%

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

      2. distribute-neg-frac 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in y around inf 55.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -82:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2400000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\log \left(\frac{e}{1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2400000.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (log (/ E (- 1.0 x))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2400000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = log((((double) M_E) / (1.0 - x)));
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2400000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = Math.log((Math.E / (1.0 - x)));
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2400000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = math.log((math.e / (1.0 - x)))
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2400000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = log(Float64(exp(1) / Float64(1.0 - x)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2400000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[Log[N[(E / N[(1.0 - 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 < -2.4e6

   1. Initial program 18.0%

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

      1. sub-neg 18.0%

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

      2. log1p-def 18.0%

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

      3. distribute-neg-frac 18.0%

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

      4. sub-neg 18.0%

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

      5. distribute-neg-in 18.0%

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

      6. remove-double-neg 18.0%

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

      7. +-commutative 18.0%

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

      8. sub-neg 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in y around inf 18.0%

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

   6. Taylor expanded in x around 0 69.7%

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

      1. distribute-neg-frac 69.7%

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

      2. metadata-eval 69.7%

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

   8. Simplified 69.7%

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

   if -2.4e6 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 97.7%

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

      1. exp-diff 97.7%

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

      2. e-exp-1 97.7%

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

      3. rem-exp-log 97.7%

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

      4. mul-1-neg 97.7%

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

      5. sub-neg 97.7%

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

   9. Simplified 97.7%

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

   if 1 < y

   1. Initial program 65.3%

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

      1. sub-neg 65.3%

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

      2. log1p-def 65.3%

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

      3. distribute-neg-frac 65.3%

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

      4. sub-neg 65.3%

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

      5. distribute-neg-in 65.3%

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

      6. remove-double-neg 65.3%

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

      7. +-commutative 65.3%

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

      8. sub-neg 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 56.3%

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

      1. neg-mul-1 56.3%

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

      2. distribute-neg-frac 56.3%

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

   7. Simplified 56.3%

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

   8. Taylor expanded in y around inf 55.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2400000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\log \left(\frac{e}{1 - x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2650000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2650000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2650000000.0) {
		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 <= -2650000000.0) {
		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 <= -2650000000.0:
		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 <= -2650000000.0)
		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, -2650000000.0], 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 < -2.65e9

   1. Initial program 18.0%

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

      1. sub-neg 18.0%

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

      2. log1p-def 18.0%

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

      3. distribute-neg-frac 18.0%

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

      4. sub-neg 18.0%

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

      5. distribute-neg-in 18.0%

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

      6. remove-double-neg 18.0%

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

      7. +-commutative 18.0%

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

      8. sub-neg 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in y around inf 18.0%

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

   6. Taylor expanded in x around 0 69.7%

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

      1. distribute-neg-frac 69.7%

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

      2. metadata-eval 69.7%

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

   8. Simplified 69.7%

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

   if -2.65e9 < y

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. log1p-def 95.2%

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

      3. distribute-neg-frac 95.2%

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

      4. sub-neg 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. remove-double-neg 95.2%

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

      7. +-commutative 95.2%

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

      8. sub-neg 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around inf 92.6%

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

      1. neg-mul-1 92.6%

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

      2. distribute-neg-frac 92.6%

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

   7. Simplified 92.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2650000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2400000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{1 - x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2400000.0) (- 1.0 (log (/ -1.0 y))) (log (/ E (- 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2400000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = log((((double) M_E) / (1.0 - x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2400000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = Math.log((Math.E / (1.0 - x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2400000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = math.log((math.e / (1.0 - x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2400000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = log(Float64(exp(1) / Float64(1.0 - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2400000.0)
		tmp = 1.0 - log((-1.0 / y));
	else
		tmp = log((2.71828182845904523536 / (1.0 - x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2400000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e6

   1. Initial program 18.0%

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

      1. sub-neg 18.0%

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

      2. log1p-def 18.0%

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

      3. distribute-neg-frac 18.0%

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

      4. sub-neg 18.0%

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

      5. distribute-neg-in 18.0%

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

      6. remove-double-neg 18.0%

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

      7. +-commutative 18.0%

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

      8. sub-neg 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in y around inf 18.0%

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

   6. Taylor expanded in x around 0 69.7%

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

      1. distribute-neg-frac 69.7%

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

      2. metadata-eval 69.7%

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

   8. Simplified 69.7%

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

   if -2.4e6 < y

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. log1p-def 95.2%

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

      3. distribute-neg-frac 95.2%

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

      4. sub-neg 95.2%

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

      5. distribute-neg-in 95.2%

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

      6. remove-double-neg 95.2%

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

      7. +-commutative 95.2%

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

      8. sub-neg 95.2%

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

   3. Simplified 95.2%

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

      1. add-log-exp 95.2%

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around 0 84.3%

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

      1. exp-diff 84.3%

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

      2. e-exp-1 84.3%

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

      3. rem-exp-log 84.3%

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

      4. mul-1-neg 84.3%

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

      5. sub-neg 84.3%

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

   9. Simplified 84.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2400000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{1 - x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. log1p-def 72.9%

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

   3. distribute-neg-frac 72.9%

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

   4. sub-neg 72.9%

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

   5. distribute-neg-in 72.9%

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

   6. remove-double-neg 72.9%

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

   7. +-commutative 72.9%

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

   8. sub-neg 72.9%

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

3. Simplified 72.9%

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

   1. add-log-exp 72.9%

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

6. Applied egg-rr 72.9%

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

7. Taylor expanded in y around 0 63.5%

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

   1. exp-diff 63.5%

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

   2. e-exp-1 63.5%

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

   3. rem-exp-log 63.5%

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

   4. mul-1-neg 63.5%

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

   5. sub-neg 63.5%

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

9. Simplified 63.5%

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

10. Final simplification 63.5%

    \[\leadsto \log \left(\frac{e}{1 - x}\right) \]
11. Add Preprocessing

Alternative 9: 63.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.9%

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

   1. sub-neg 72.9%

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

   2. log1p-def 72.9%

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

   3. distribute-neg-frac 72.9%

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

   4. sub-neg 72.9%

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

   5. distribute-neg-in 72.9%

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

   6. remove-double-neg 72.9%

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

   7. +-commutative 72.9%

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

   8. sub-neg 72.9%

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

3. Simplified 72.9%

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

5. Taylor expanded in y around 0 63.5%

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

   1. log1p-def 63.5%

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

   2. mul-1-neg 63.5%

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

7. Simplified 63.5%

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

8. Final simplification 63.5%

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

Alternative 10: 43.7% 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 72.9%

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

   1. sub-neg 72.9%

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

   2. log1p-def 72.9%

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

   3. distribute-neg-frac 72.9%

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

   4. sub-neg 72.9%

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

   5. distribute-neg-in 72.9%

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

   6. remove-double-neg 72.9%

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

   7. +-commutative 72.9%

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

   8. sub-neg 72.9%

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

3. Simplified 72.9%

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

5. Taylor expanded in y around 0 63.5%

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

   1. log1p-def 63.5%

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

   2. mul-1-neg 63.5%

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

7. Simplified 63.5%

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

8. Taylor expanded in x around 0 46.1%

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

9. Final simplification 46.1%

   \[\leadsto 1 \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(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))))))