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

Percentage Accurate: 73.0% → 95.0%

Time: 12.1s

Alternatives: 14

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: 73.0% 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: 95.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + -1} - \frac{x}{x + -1}\\ t_1 := \log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\\ t_2 := -1 - t\_1\\ t_3 := {t\_1}^{2}\\ \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999999999985:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{t\_0 \cdot \left(1 - t\_3\right)}{{\left(1 + t\_1\right)}^{2}} + -2 \cdot \frac{t\_1 \cdot t\_0}{t\_2}}{y} + \frac{-1}{t\_2}\right) + \frac{t\_3}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 (+ x -1.0)) (/ x (+ x -1.0))))
        (t_1 (+ (log (- 1.0 x)) (log (/ -1.0 y))))
        (t_2 (- -1.0 t_1))
        (t_3 (pow t_1 2.0)))
   (if (<= (/ (- x y) (- 1.0 y)) 0.9999999999985)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+
      (+
       (/
        (+
         (/ (* t_0 (- 1.0 t_3)) (pow (+ 1.0 t_1) 2.0))
         (* -2.0 (/ (* t_1 t_0) t_2)))
        y)
       (/ -1.0 t_2))
      (/ t_3 t_2)))))

C

double code(double x, double y) {
	double t_0 = (1.0 / (x + -1.0)) - (x / (x + -1.0));
	double t_1 = log((1.0 - x)) + log((-1.0 / y));
	double t_2 = -1.0 - t_1;
	double t_3 = pow(t_1, 2.0);
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999999999985) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (((((t_0 * (1.0 - t_3)) / pow((1.0 + t_1), 2.0)) + (-2.0 * ((t_1 * t_0) / t_2))) / y) + (-1.0 / t_2)) + (t_3 / t_2);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (1.0 / (x + -1.0)) - (x / (x + -1.0));
	double t_1 = Math.log((1.0 - x)) + Math.log((-1.0 / y));
	double t_2 = -1.0 - t_1;
	double t_3 = Math.pow(t_1, 2.0);
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999999999985) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (((((t_0 * (1.0 - t_3)) / Math.pow((1.0 + t_1), 2.0)) + (-2.0 * ((t_1 * t_0) / t_2))) / y) + (-1.0 / t_2)) + (t_3 / t_2);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (1.0 / (x + -1.0)) - (x / (x + -1.0))
	t_1 = math.log((1.0 - x)) + math.log((-1.0 / y))
	t_2 = -1.0 - t_1
	t_3 = math.pow(t_1, 2.0)
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9999999999985:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = (((((t_0 * (1.0 - t_3)) / math.pow((1.0 + t_1), 2.0)) + (-2.0 * ((t_1 * t_0) / t_2))) / y) + (-1.0 / t_2)) + (t_3 / t_2)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(1.0 / Float64(x + -1.0)) - Float64(x / Float64(x + -1.0)))
	t_1 = Float64(log(Float64(1.0 - x)) + log(Float64(-1.0 / y)))
	t_2 = Float64(-1.0 - t_1)
	t_3 = t_1 ^ 2.0
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9999999999985)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(t_0 * Float64(1.0 - t_3)) / (Float64(1.0 + t_1) ^ 2.0)) + Float64(-2.0 * Float64(Float64(t_1 * t_0) / t_2))) / y) + Float64(-1.0 / t_2)) + Float64(t_3 / t_2));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 - t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.9999999999985], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(t$95$0 * N[(1.0 - t$95$3), $MachinePrecision]), $MachinePrecision] / N[Power[N[(1.0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(N[(t$95$1 * t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(-1.0 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999999999998499978

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. log1p-define 99.5%

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

      3. distribute-neg-frac2 99.5%

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

      4. neg-sub0 99.5%

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

      5. associate--r- 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   3. Simplified 99.5%

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

   if 0.999999999998499978 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 4.3%

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

      1. sub-neg 4.3%

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

      2. log1p-define 4.3%

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

      3. distribute-neg-frac2 4.3%

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

      4. neg-sub0 4.3%

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

      5. associate--r- 4.3%

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

      6. metadata-eval 4.3%

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

      7. +-commutative 4.3%

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

   3. Simplified 4.3%

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

      1. flip-- 1.3%

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

      2. metadata-eval 1.3%

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

      3. pow2 1.3%

         \[\leadsto \frac{1 - \color{blue}{{\left(\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}^{2}}}{1 + \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]

      4. +-commutative 1.3%

         \[\leadsto \frac{1 - {\left(\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}^{2}}{\color{blue}{\mathsf{log1p}\left(\frac{x - y}{y + -1}\right) + 1}} \]

   6. Applied egg-rr 1.3%

      \[\leadsto \color{blue}{\frac{1 - {\left(\mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\right)}^{2}}{\mathsf{log1p}\left(\frac{x - y}{y + -1}\right) + 1}} \]

   7. Taylor expanded in y around -inf 83.9%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999999999985:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\left(\frac{1}{x + -1} - \frac{x}{x + -1}\right) \cdot \left(1 - {\left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)}^{2}\right)}{{\left(1 + \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\right)}^{2}} + -2 \cdot \frac{\left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right) \cdot \left(\frac{1}{x + -1} - \frac{x}{x + -1}\right)}{-1 - \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)}}{y} + \frac{-1}{-1 - \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)}\right) + \frac{{\left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)}^{2}}{-1 - \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(1.0 - N[(1.0 / y), $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x * N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(y / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-define 3.1%

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

      3. distribute-neg-frac2 3.1%

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

      4. neg-sub0 3.1%

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

      5. associate--r- 3.1%

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

      6. metadata-eval 3.1%

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

      7. +-commutative 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in y around -inf 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in x around 0 69.7%

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

   if 0.0 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 98.9%

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

      1. sub-neg 98.9%

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

      2. log1p-define 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. neg-sub0 98.9%

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

      5. associate--r- 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in x around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. sub-neg 98.5%

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

      3. metadata-eval 98.5%

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

      4. mul-1-neg 98.5%

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

      5. unsub-neg 98.5%

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

      6. sub-neg 98.5%

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

      7. metadata-eval 98.5%

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

      8. *-commutative 98.5%

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

      9. associate-/r* 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 91.8%

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

Alternative 3: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - x \cdot \left(-1 - x \cdot 0.5\right)\right) + \frac{-1}{y}\right) - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 1.0)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- (+ (- 1.0 (* x (- -1.0 (* x 0.5)))) (/ -1.0 y)) (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 - (x * (-1.0 - (x * 0.5)))) + (-1.0 / y)) - 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 - (x * (-1.0 - (x * 0.5)))) + (-1.0 / y)) - 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 - (x * (-1.0 - (x * 0.5)))) + (-1.0 / y)) - 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(Float64(Float64(1.0 - Float64(x * Float64(-1.0 - Float64(x * 0.5)))) + Float64(-1.0 / y)) - 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[(N[(N[(1.0 - N[(x * N[(-1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 1

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. log1p-define 75.7%

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

      3. distribute-neg-frac2 75.7%

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

      4. neg-sub0 75.7%

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

      5. associate--r- 75.7%

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

      6. metadata-eval 75.7%

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

      7. +-commutative 75.7%

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

   3. Simplified 75.7%

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

   if 1 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 75.7%

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

      1. sub-neg 75.7%

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

      2. log1p-define 75.7%

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

      3. distribute-neg-frac2 75.7%

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

      4. neg-sub0 75.7%

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

      5. associate--r- 75.7%

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

      6. metadata-eval 75.7%

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

      7. +-commutative 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in y around -inf 28.3%

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

   7. Simplified 28.3%

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

   8. Taylor expanded in x around 0 18.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(1 - x \cdot \left(-1 - x \cdot 0.5\right)\right) + \frac{-1}{y}\right) - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (- 1.0 (log1p (* x (- (/ 1.0 (+ y -1.0)) (/ y (* x (+ y -1.0))))))))

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in x around -inf 76.3%

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

   1. associate-*r* 76.3%

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

   2. mul-1-neg 76.3%

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

   3. sub-neg 76.3%

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

   4. sub-neg 76.3%

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

   5. metadata-eval 76.3%

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

   6. sub-neg 76.3%

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

   7. metadata-eval 76.3%

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

   8. distribute-neg-frac 76.3%

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

   9. metadata-eval 76.3%

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

7. Simplified 76.3%

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

8. Final simplification 76.3%

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

Alternative 5: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 76.9%

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

      1. sub-neg 76.9%

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

      2. log1p-define 76.9%

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

      3. distribute-neg-frac2 76.9%

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

      4. neg-sub0 76.9%

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

      5. associate--r- 76.9%

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

      6. metadata-eval 76.9%

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

      7. +-commutative 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in y around 0 70.5%

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

      1. log1p-define 70.5%

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

      2. mul-1-neg 70.5%

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

   7. Simplified 70.5%

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

   if 1 < y

   1. Initial program 66.7%

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

      1. sub-neg 66.7%

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

      2. log1p-define 66.7%

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

      3. distribute-neg-frac2 66.7%

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

      4. neg-sub0 66.7%

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

      5. associate--r- 66.7%

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

      6. metadata-eval 66.7%

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

      7. +-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. clear-num 66.5%

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

      2. associate-/r/ 67.0%

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

   6. Applied egg-rr 67.0%

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

   7. Taylor expanded in y around inf 67.0%

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

   8. Taylor expanded in y around inf 66.7%

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

4. Final simplification 70.1%

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

Alternative 6: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5) (- 1.0 (log1p (/ x y))) (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = 1.0 - log1p((x / y));
	} else {
		tmp = (1.0 - y) - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = 1.0 - Math.log1p((x / y));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.5:
		tmp = 1.0 - math.log1p((x / y))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5)
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5

   1. Initial program 25.5%

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

      1. sub-neg 25.5%

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

      2. log1p-define 25.5%

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

      3. distribute-neg-frac2 25.5%

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

      4. neg-sub0 25.5%

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

      5. associate--r- 25.5%

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

      6. metadata-eval 25.5%

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

      7. +-commutative 25.5%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in x around inf 32.0%

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

   6. Taylor expanded in y around inf 31.0%

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

   if -2.5 < y

   1. Initial program 94.6%

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

      1. sub-neg 94.6%

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

      2. log1p-define 94.6%

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

      3. distribute-neg-frac2 94.6%

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

      4. neg-sub0 94.6%

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

      5. associate--r- 94.6%

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

      6. metadata-eval 94.6%

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

      7. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in y around 0 81.7%

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

   7. Simplified 81.8%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (- 1.0 (log1p (* (- x y) (/ 1.0 (+ y -1.0))))))

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

   1. clear-num 75.7%

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

   2. associate-/r/ 75.9%

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

6. Applied egg-rr 75.9%

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

7. Final simplification 75.9%

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

Alternative 8: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0033:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0033) (- 1.0 (log1p (- x))) (- 1.0 (log1p (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0033) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0033) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0033:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0033)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0033], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0033

   1. Initial program 76.8%

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

      1. sub-neg 76.8%

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

      2. log1p-define 76.8%

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

      3. distribute-neg-frac2 76.8%

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

      4. neg-sub0 76.8%

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

      5. associate--r- 76.8%

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

      6. metadata-eval 76.8%

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

      7. +-commutative 76.8%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in y around 0 70.7%

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

      1. log1p-define 70.7%

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

      2. mul-1-neg 70.7%

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

   7. Simplified 70.7%

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

   if 0.0033 < y

   1. Initial program 67.7%

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

      1. sub-neg 67.7%

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

      2. log1p-define 67.7%

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

      3. distribute-neg-frac2 67.7%

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

      4. neg-sub0 67.7%

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

      5. associate--r- 67.7%

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

      6. metadata-eval 67.7%

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

      7. +-commutative 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in x around inf 62.3%

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

   6. Taylor expanded in y around inf 62.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0033:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Final simplification 75.7%

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

Alternative 10: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in x around inf 74.8%

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

6. Final simplification 74.8%

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

Alternative 11: 63.0% 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 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in y around 0 62.3%

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

   1. log1p-define 62.3%

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

   2. mul-1-neg 62.3%

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

7. Simplified 62.3%

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

8. Final simplification 62.3%

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

Alternative 12: 45.1% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in x around inf 74.8%

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

6. Taylor expanded in x around 0 46.5%

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

   1. mul-1-neg 46.5%

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

   2. sub-neg 46.5%

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

   3. metadata-eval 46.5%

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

   4. unsub-neg 46.5%

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

   5. +-commutative 46.5%

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

8. Simplified 46.5%

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

9. Final simplification 46.5%

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

Alternative 13: 43.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in y around 0 62.3%

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

   1. log1p-define 62.3%

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

   2. mul-1-neg 62.3%

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

7. Simplified 62.3%

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

8. Taylor expanded in x around 0 45.1%

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

9. Final simplification 45.1%

   \[\leadsto x + 1 \]
10. Add Preprocessing

Alternative 14: 43.4% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 75.7%

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

   1. sub-neg 75.7%

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

   2. log1p-define 75.7%

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

   3. distribute-neg-frac2 75.7%

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

   4. neg-sub0 75.7%

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

   5. associate--r- 75.7%

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

   6. metadata-eval 75.7%

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

   7. +-commutative 75.7%

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

3. Simplified 75.7%

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

5. Taylor expanded in y around 0 62.3%

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

   1. log1p-define 62.3%

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

   2. mul-1-neg 62.3%

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

7. Simplified 62.3%

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

8. Taylor expanded in x around 0 44.6%

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

9. Final simplification 44.6%

   \[\leadsto 1 \]
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 2024066 
(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))))))