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

Percentage Accurate: 72.0% → 99.5%

Time: 21.6s

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: 72.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2900000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+70}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2900000000.0)
   (- (- 1.0 (log1p (- x))) (log (/ -1.0 y)))
   (if (<= y 9.8e+70)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2900000000.0) {
		tmp = (1.0 - log1p(-x)) - log((-1.0 / y));
	} else if (y <= 9.8e+70) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2900000000.0) {
		tmp = (1.0 - Math.log1p(-x)) - Math.log((-1.0 / y));
	} else if (y <= 9.8e+70) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2900000000.0:
		tmp = (1.0 - math.log1p(-x)) - math.log((-1.0 / y))
	elif y <= 9.8e+70:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2900000000.0)
		tmp = Float64(Float64(1.0 - log1p(Float64(-x))) - log(Float64(-1.0 / y)));
	elseif (y <= 9.8e+70)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2900000000.0], N[(N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.8e+70], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9e9

   1. Initial program 18.6%

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

      1. sub-neg 18.6%

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

      2. log1p-define 18.6%

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

      3. distribute-neg-frac2 18.6%

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

      4. neg-sub0 18.6%

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

      5. associate--r- 18.6%

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

      6. metadata-eval 18.6%

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

      7. +-commutative 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in y around -inf 99.5%

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

      1. associate--r+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. log1p-define 99.5%

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

      8. mul-1-neg 99.5%

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

   7. Simplified 99.5%

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

   if -2.9e9 < y < 9.80000000000000056e70

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 9.80000000000000056e70 < y

   1. Initial program 35.6%

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

      1. sub-neg 35.6%

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

      2. log1p-define 35.6%

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

      3. distribute-neg-frac2 35.6%

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

      4. neg-sub0 35.6%

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

      5. associate--r- 35.6%

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

      6. metadata-eval 35.6%

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

      7. +-commutative 35.6%

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

   3. Simplified 35.6%

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

   5. Taylor expanded in y around inf 98.3%

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

      1. log-rec 98.3%

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

      2. unsub-neg 98.3%

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

      3. sub-neg 98.3%

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

      4. metadata-eval 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2900000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{+70}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 5.00000000000000024e-5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 5.00000000000000024e-5 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.0%

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

      1. sub-neg 6.0%

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

      2. log1p-define 6.0%

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

      3. distribute-neg-frac2 6.0%

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

      4. neg-sub0 6.0%

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

      5. associate--r- 6.0%

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

      6. metadata-eval 6.0%

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

      7. +-commutative 6.0%

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

   3. Simplified 6.0%

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

      1. log1p-expm1-u 6.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)\right)} \]

   6. Applied egg-rr 6.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)\right)} \]

   7. Taylor expanded in y around 0 78.3%

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

      1. associate--l+ 78.3%

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

      2. exp-diff 78.3%

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

      3. exp-1-e 78.3%

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

      4. rem-exp-log 78.3%

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

      5. mul-1-neg 78.3%

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

      6. sub-neg 78.3%

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

      7. associate-*r* 78.3%

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

      8. fma-neg 78.3%

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

   9. Simplified 98.8%

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

4. Final simplification 99.6%

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

Alternative 3: 72.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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-define 71.0%

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

      3. distribute-neg-frac2 71.0%

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

      4. neg-sub0 71.0%

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

      5. associate--r- 71.0%

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

      6. metadata-eval 71.0%

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

      7. +-commutative 71.0%

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

   3. Simplified 71.0%

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

      1. clear-num 71.0%

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

      2. associate-/r/ 71.4%

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

   6. Applied egg-rr 71.4%

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

   7. Taylor expanded in x around inf 72.1%

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

      1. +-commutative 72.1%

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

      2. sub-neg 72.1%

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

      3. metadata-eval 72.1%

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

      4. +-commutative 72.1%

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

      5. mul-1-neg 72.1%

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

      6. unsub-neg 72.1%

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

      7. associate-/r* 71.2%

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

      8. sub-neg 71.2%

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

      9. metadata-eval 71.2%

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

      10. +-commutative 71.2%

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

   9. Simplified 71.2%

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

   10. Taylor expanded in x around inf 72.1%

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

       1. +-commutative 72.1%

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

       2. sub-neg 72.1%

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

       3. metadata-eval 72.1%

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

       4. +-commutative 72.1%

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

       5. mul-1-neg 72.1%

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

       6. *-commutative 72.1%

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

       7. sub-neg 72.1%

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

       8. metadata-eval 72.1%

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

       9. +-commutative 72.1%

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

       10. sub-neg 72.1%

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

       11. +-commutative 72.1%

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

       12. metadata-eval 72.1%

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

       13. sub-neg 72.1%

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

       14. *-commutative 72.1%

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

       15. associate-/r* 71.2%

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

       16. sub-neg 71.2%

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

       17. metadata-eval 71.2%

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

       18. +-commutative 71.2%

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

       19. div-sub 71.2%

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

   12. Simplified 71.2%

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

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

   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-define 71.0%

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

      3. distribute-neg-frac2 71.0%

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

      4. neg-sub0 71.0%

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

      5. associate--r- 71.0%

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

      6. metadata-eval 71.0%

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

      7. +-commutative 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around 0 41.3%

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

      1. sub-neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. neg-mul-1 41.3%

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

      4. distribute-neg-frac 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in y around -inf 21.9%

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

4. Final simplification 71.2%

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

Alternative 4: 72.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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-define 71.0%

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

      3. distribute-neg-frac2 71.0%

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

      4. neg-sub0 71.0%

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

      5. associate--r- 71.0%

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

      6. metadata-eval 71.0%

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

      7. +-commutative 71.0%

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

   3. Simplified 71.0%

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

      1. clear-num 71.0%

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

      2. associate-/r/ 71.4%

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

   6. Applied egg-rr 71.4%

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

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

   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-define 71.0%

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

      3. distribute-neg-frac2 71.0%

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

      4. neg-sub0 71.0%

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

      5. associate--r- 71.0%

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

      6. metadata-eval 71.0%

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

      7. +-commutative 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around 0 41.3%

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

      1. sub-neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. neg-mul-1 41.3%

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

      4. distribute-neg-frac 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in y around -inf 21.9%

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

4. Final simplification 71.4%

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

Alternative 5: 72.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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-define 71.0%

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

      3. distribute-neg-frac2 71.0%

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

      4. neg-sub0 71.0%

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

      5. associate--r- 71.0%

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

      6. metadata-eval 71.0%

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

      7. +-commutative 71.0%

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

   3. Simplified 71.0%

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

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

   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-define 71.0%

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

      3. distribute-neg-frac2 71.0%

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

      4. neg-sub0 71.0%

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

      5. associate--r- 71.0%

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

      6. metadata-eval 71.0%

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

      7. +-commutative 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in x around 0 41.3%

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

      1. sub-neg 41.3%

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

      2. metadata-eval 41.3%

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

      3. neg-mul-1 41.3%

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

      4. distribute-neg-frac 41.3%

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

   7. Simplified 41.3%

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

   8. Taylor expanded in y around -inf 21.9%

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

4. Final simplification 71.0%

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

Alternative 6: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+30}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.25e+30)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+30) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.25e+30) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.25e+30:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.25e+30)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.25e+30], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.25e30

   1. Initial program 16.6%

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

      1. sub-neg 16.6%

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

      2. log1p-define 16.6%

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

      3. distribute-neg-frac2 16.6%

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

      4. neg-sub0 16.6%

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

      5. associate--r- 16.6%

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

      6. metadata-eval 16.6%

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

      7. +-commutative 16.6%

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

   3. Simplified 16.6%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 69.6%

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

   if -1.25e30 < y

   1. Initial program 91.0%

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

      1. sub-neg 91.0%

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

      2. log1p-define 91.0%

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

      3. distribute-neg-frac2 91.0%

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

      4. neg-sub0 91.0%

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

      5. associate--r- 91.0%

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

      6. metadata-eval 91.0%

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

      7. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in x around inf 89.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+30}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.0) {
		tmp = 1.0 - log((-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 <= -5.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -5.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -5.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -5.0], N[(1.0 - N[Log[N[(-1.0 / y), $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 < -5

   1. Initial program 18.6%

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

      1. sub-neg 18.6%

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

      2. log1p-define 18.6%

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

      3. distribute-neg-frac2 18.6%

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

      4. neg-sub0 18.6%

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

      5. associate--r- 18.6%

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

      6. metadata-eval 18.6%

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

      7. +-commutative 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 68.5%

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

   if -5 < y

   1. Initial program 91.9%

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

      1. sub-neg 91.9%

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

      2. log1p-define 91.9%

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

      3. distribute-neg-frac2 91.9%

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

      4. neg-sub0 91.9%

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

      5. associate--r- 91.9%

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

      6. metadata-eval 91.9%

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

      7. +-commutative 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around 0 82.0%

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

      1. +-commutative 82.0%

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

      2. div-sub 82.0%

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

      3. *-commutative 82.0%

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

      4. mul-1-neg 82.0%

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

      5. sub-neg 82.0%

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

      6. *-inverses 82.0%

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

      7. metadata-eval 82.0%

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

      8. distribute-lft-neg-in 82.0%

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

      9. neg-mul-1 82.0%

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

      10. remove-double-neg 82.0%

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

      11. log1p-define 82.0%

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

      12. mul-1-neg 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 78.2%

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

Alternative 8: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -155:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -155.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -155.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -155.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -155.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -155.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -155.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -155

   1. Initial program 18.6%

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

      1. sub-neg 18.6%

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

      2. log1p-define 18.6%

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

      3. distribute-neg-frac2 18.6%

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

      4. neg-sub0 18.6%

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

      5. associate--r- 18.6%

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

      6. metadata-eval 18.6%

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

      7. +-commutative 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 68.5%

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

   if -155 < y

   1. Initial program 91.9%

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

      1. sub-neg 91.9%

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

      2. log1p-define 91.9%

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

      3. distribute-neg-frac2 91.9%

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

      4. neg-sub0 91.9%

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

      5. associate--r- 91.9%

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

      6. metadata-eval 91.9%

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

      7. +-commutative 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in y around 0 80.8%

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

      1. log1p-define 80.8%

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

      2. mul-1-neg 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -155:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% 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 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-define 71.0%

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

   3. distribute-neg-frac2 71.0%

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

   4. neg-sub0 71.0%

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

   5. associate--r- 71.0%

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

   6. metadata-eval 71.0%

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

   7. +-commutative 71.0%

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

3. Simplified 71.0%

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

5. Taylor expanded in y around 0 61.3%

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

   1. log1p-define 61.3%

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

   2. mul-1-neg 61.3%

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

7. Simplified 61.3%

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

8. Final simplification 61.3%

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

Alternative 10: 40.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 - y), $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-define 71.0%

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

   3. distribute-neg-frac2 71.0%

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

   4. neg-sub0 71.0%

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

   5. associate--r- 71.0%

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

   6. metadata-eval 71.0%

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

   7. +-commutative 71.0%

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

3. Simplified 71.0%

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

5. Taylor expanded in x around 0 41.3%

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

   1. sub-neg 41.3%

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

   2. metadata-eval 41.3%

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

   3. neg-mul-1 41.3%

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

   4. distribute-neg-frac 41.3%

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

7. Simplified 41.3%

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

8. Taylor expanded in y around 0 40.8%

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

9. Final simplification 40.8%

   \[\leadsto 1 - y \]
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 2024055 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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