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

Percentage Accurate: 71.1% → 99.3%

Time: 17.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x - y}{y + -1}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- x y) (+ y -1.0)))))
   (if (<= t_0 0.0) (- 1.0 (log (/ (+ x -1.0) y))) (- 1.0 (log t_0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x - y) / (y + -1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - log(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 + ((x - y) / (y + (-1.0d0)))
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 - log(((x + (-1.0d0)) / y))
    else
        tmp = 1.0d0 - log(t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + ((x - y) / (y + -1.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - Math.log(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + ((x - y) / (y + -1.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - math.log(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - log(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + ((x - y) / (y + -1.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 - log(((x + -1.0) / y));
	else
		tmp = 1.0 - log(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[t$95$0], $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. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

   5. Simplified 100.0%

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

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

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{\frac{y + -1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+21)
   (- 1.0 (log (/ (+ x -1.0) y)))
   (if (<= y 9.5e-26)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (log (/ 1.0 (/ (+ y -1.0) x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+21) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (y <= 9.5e-26) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((1.0 / ((y + -1.0) / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+21) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (y <= 9.5e-26) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((1.0 / ((y + -1.0) / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+21:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif y <= 9.5e-26:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((1.0 / ((y + -1.0) / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+21)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (y <= 9.5e-26)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(1.0 / Float64(Float64(y + -1.0) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+21], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-26], 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 / N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e21

   1. Initial program 16.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.8e21 < y < 9.4999999999999995e-26

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. log1p-define 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. neg-sub0 99.3%

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

      5. associate--r- 99.3%

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

      6. metadata-eval 99.3%

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

      7. +-commutative 99.3%

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

   3. Simplified 99.3%

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

   if 9.4999999999999995e-26 < y

   1. Initial program 63.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

      3. sub-neg 91.5%

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

      4. distribute-neg-in 91.5%

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

      5. metadata-eval 91.5%

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

      6. remove-double-neg 91.5%

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

      7. +-commutative 91.5%

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

   5. Simplified 91.5%

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

      1. clear-num 91.5%

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

      2. inv-pow 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. unpow-1 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{\frac{y + -1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{\frac{y + -1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+21)
   (- 1.0 (log (/ (+ x -1.0) y)))
   (if (<= y 2.8e-5)
     (- (- 1.0 y) (log1p (- x)))
     (- 1.0 (log (/ 1.0 (/ (+ y -1.0) x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+21) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (y <= 2.8e-5) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((1.0 / ((y + -1.0) / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+21) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (y <= 2.8e-5) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((1.0 / ((y + -1.0) / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+21:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif y <= 2.8e-5:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((1.0 / ((y + -1.0) / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+21)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (y <= 2.8e-5)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(1.0 / Float64(Float64(y + -1.0) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+21], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-5], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(1.0 / N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e21

   1. Initial program 16.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.8e21 < y < 2.79999999999999996e-5

   1. Initial program 99.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 97.4%

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

   5. Simplified 97.5%

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

   if 2.79999999999999996e-5 < y

   1. Initial program 59.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. distribute-neg-frac2 97.0%

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

      3. sub-neg 97.0%

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

      4. distribute-neg-in 97.0%

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

      5. metadata-eval 97.0%

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

      6. remove-double-neg 97.0%

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

      7. +-commutative 97.0%

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

   5. Simplified 97.0%

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

      1. clear-num 97.0%

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

      2. inv-pow 97.0%

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

   7. Applied egg-rr 97.0%

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

      1. unpow-1 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{\frac{y + -1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.8e+21)
   (- 1.0 (log (/ (+ x -1.0) y)))
   (if (<= y 2.8e-5)
     (- (- 1.0 y) (log1p (- x)))
     (- 1.0 (log (/ x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+21) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (y <= 2.8e-5) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.8e+21) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (y <= 2.8e-5) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.8e+21:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif y <= 2.8e-5:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.8e+21)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (y <= 2.8e-5)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.8e+21], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-5], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e21

   1. Initial program 16.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. neg-mul-1 100.0%

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

      5. mul-1-neg 100.0%

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

      6. remove-double-neg 100.0%

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

   5. Simplified 100.0%

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

   if -4.8e21 < y < 2.79999999999999996e-5

   1. Initial program 99.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 97.4%

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

   5. Simplified 97.5%

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

   if 2.79999999999999996e-5 < y

   1. Initial program 59.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. distribute-neg-frac2 97.0%

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

      3. sub-neg 97.0%

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

      4. distribute-neg-in 97.0%

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

      5. metadata-eval 97.0%

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

      6. remove-double-neg 97.0%

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

      7. +-commutative 97.0%

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

   5. Simplified 97.0%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -38:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -38.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 2.8e-5)
     (- (- 1.0 y) (log1p (- x)))
     (- 1.0 (log (/ x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -38.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 2.8e-5) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -38.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 2.8e-5) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -38.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 2.8e-5:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -38.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 2.8e-5)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -38.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-5], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -38

   1. Initial program 17.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 2.8%

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

      1. log1p-define 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in y around -inf 74.7%

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

   if -38 < y < 2.79999999999999996e-5

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.7%

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

   5. Simplified 98.7%

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

   if 2.79999999999999996e-5 < y

   1. Initial program 59.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.0%

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

      1. mul-1-neg 97.0%

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

      2. distribute-neg-frac2 97.0%

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

      3. sub-neg 97.0%

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

      4. distribute-neg-in 97.0%

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

      5. metadata-eval 97.0%

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

      6. remove-double-neg 97.0%

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

      7. +-commutative 97.0%

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

   5. Simplified 97.0%

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

Alternative 6: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -38:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x} \cdot e\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -38.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 9.5e-26) (- (- 1.0 y) (log1p (- x))) (log (* (/ y x) E)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -38.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 9.5e-26) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log(((y / x) * ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -38.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 9.5e-26) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log(((y / x) * Math.E));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -38.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 9.5e-26:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log(((y / x) * math.e))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -38.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 9.5e-26)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y / x) * exp(1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -38.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-26], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y / x), $MachinePrecision] * E), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -38

   1. Initial program 17.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 2.8%

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

      1. log1p-define 2.8%

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

   5. Simplified 2.8%

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

   6. Taylor expanded in y around -inf 74.7%

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

   if -38 < y < 9.4999999999999995e-26

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.6%

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

   5. Simplified 98.6%

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

   if 9.4999999999999995e-26 < y

   1. Initial program 63.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

      3. sub-neg 91.5%

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

      4. distribute-neg-in 91.5%

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

      5. metadata-eval 91.5%

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

      6. remove-double-neg 91.5%

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

      7. +-commutative 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in y around inf 85.1%

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

      1. add-log-exp 85.1%

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

      2. sub-neg 85.1%

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

      3. exp-sum 85.1%

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

      4. neg-log 85.1%

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

      5. clear-num 85.1%

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

      6. add-exp-log 85.1%

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

   8. Applied egg-rr 85.1%

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

      1. *-commutative 85.1%

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

      2. exp-1-e 85.1%

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

   10. Simplified 85.1%

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

Alternative 7: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+25} \lor \neg \left(y \leq 9.5 \cdot 10^{-26}\right):\\ \;\;\;\;\log \left(\frac{y}{x} \cdot e\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -6.5e+25) (not (<= y 9.5e-26)))
   (log (* (/ y x) E))
   (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+25) || !(y <= 9.5e-26)) {
		tmp = log(((y / x) * ((double) M_E)));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -6.5e+25) || !(y <= 9.5e-26)) {
		tmp = Math.log(((y / x) * Math.E));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -6.5e+25) or not (y <= 9.5e-26):
		tmp = math.log(((y / x) * math.e))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -6.5e+25) || !(y <= 9.5e-26))
		tmp = log(Float64(Float64(y / x) * exp(1)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -6.5e+25], N[Not[LessEqual[y, 9.5e-26]], $MachinePrecision]], N[Log[N[(N[(y / x), $MachinePrecision] * E), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.50000000000000005e25 or 9.4999999999999995e-26 < y

   1. Initial program 30.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 49.9%

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

      1. mul-1-neg 49.9%

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

      2. distribute-neg-frac2 49.9%

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

      3. sub-neg 49.9%

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

      4. distribute-neg-in 49.9%

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

      5. metadata-eval 49.9%

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

      6. remove-double-neg 49.9%

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

      7. +-commutative 49.9%

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

   5. Simplified 49.9%

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

   6. Taylor expanded in y around inf 48.0%

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

      1. add-log-exp 47.4%

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

      2. sub-neg 47.4%

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

      3. exp-sum 47.4%

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

      4. neg-log 47.0%

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

      5. clear-num 47.0%

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

      6. add-exp-log 47.0%

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

   8. Applied egg-rr 47.0%

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

      1. *-commutative 47.0%

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

      2. exp-1-e 47.0%

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

   10. Simplified 47.0%

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

   if -6.50000000000000005e25 < y < 9.4999999999999995e-26

   1. Initial program 98.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. mul-1-neg 96.6%

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

      3. log1p-define 96.6%

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

      4. mul-1-neg 96.6%

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

   5. Simplified 96.6%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{+25} \lor \neg \left(y \leq 9.5 \cdot 10^{-26}\right):\\ \;\;\;\;\log \left(\frac{y}{x} \cdot e\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+54}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y}{x} \cdot e\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.6e+54)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 9.5e-26) (- 1.0 (log1p (- x))) (log (* (/ y x) E)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+54) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 9.5e-26) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = log(((y / x) * ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.6e+54) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 9.5e-26) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = Math.log(((y / x) * Math.E));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.6e+54:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 9.5e-26:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = math.log(((y / x) * math.e))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.6e+54)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 9.5e-26)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(y / x) * exp(1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.6e+54], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-26], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y / x), $MachinePrecision] * E), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.60000000000000007e54

   1. Initial program 14.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 2.9%

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

      1. log1p-define 2.9%

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

   5. Simplified 2.9%

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

   6. Taylor expanded in y around -inf 76.1%

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

   if -2.60000000000000007e54 < y < 9.4999999999999995e-26

   1. Initial program 96.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 94.0%

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

      1. sub-neg 94.0%

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

      2. mul-1-neg 94.0%

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

      3. log1p-define 94.0%

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

      4. mul-1-neg 94.0%

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

   5. Simplified 94.0%

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

   if 9.4999999999999995e-26 < y

   1. Initial program 63.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 91.5%

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

      1. mul-1-neg 91.5%

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

      2. distribute-neg-frac2 91.5%

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

      3. sub-neg 91.5%

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

      4. distribute-neg-in 91.5%

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

      5. metadata-eval 91.5%

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

      6. remove-double-neg 91.5%

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

      7. +-commutative 91.5%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in y around inf 85.1%

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

      1. add-log-exp 85.1%

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

      2. sub-neg 85.1%

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

      3. exp-sum 85.1%

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

      4. neg-log 85.1%

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

      5. clear-num 85.1%

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

      6. add-exp-log 85.1%

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

   8. Applied egg-rr 85.1%

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

      1. *-commutative 85.1%

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

      2. exp-1-e 85.1%

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

   10. Simplified 85.1%

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

Alternative 9: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+14}:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.25e+14) (- 1.0 (log (- x))) (- 1.0 (log1p x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+14) {
		tmp = 1.0 - log(-x);
	} else {
		tmp = 1.0 - log1p(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.25e+14) {
		tmp = 1.0 - Math.log(-x);
	} else {
		tmp = 1.0 - Math.log1p(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.25e+14:
		tmp = 1.0 - math.log(-x)
	else:
		tmp = 1.0 - math.log1p(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.25e+14)
		tmp = Float64(1.0 - log(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.25e+14], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e14

   1. Initial program 86.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 71.4%

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

      1. mul-1-neg 71.4%

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

   8. Simplified 71.4%

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

   if -1.25e14 < x

   1. Initial program 64.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 58.3%

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

      1. sub-neg 58.3%

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

      2. mul-1-neg 58.3%

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

      3. log1p-define 58.3%

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

      4. mul-1-neg 58.3%

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

   5. Simplified 58.3%

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

      1. sub-neg 58.3%

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

      2. add-sqr-sqrt 27.2%

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

      3. sqrt-unprod 58.5%

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

      4. sqr-neg 58.5%

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

      5. sqrt-unprod 32.4%

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

      6. add-sqr-sqrt 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. sub-neg 56.8%

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

   9. Simplified 56.8%

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

Alternative 10: 61.7% 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 70.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 61.7%

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

   1. sub-neg 61.7%

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

   2. mul-1-neg 61.7%

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

   3. log1p-define 61.7%

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

   4. mul-1-neg 61.7%

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

5. Simplified 61.7%

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

Alternative 11: 42.9% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 61.7%

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

   1. sub-neg 61.7%

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

   2. mul-1-neg 61.7%

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

   3. log1p-define 61.7%

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

   4. mul-1-neg 61.7%

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

5. Simplified 61.7%

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

6. Taylor expanded in x around 0 42.9%

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

   1. +-commutative 42.9%

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

8. Simplified 42.9%

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

9. Final simplification 42.9%

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

Alternative 12: 42.7% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 70.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 61.7%

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

   1. sub-neg 61.7%

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

   2. mul-1-neg 61.7%

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

   3. log1p-define 61.7%

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

   4. mul-1-neg 61.7%

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

5. Simplified 61.7%

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

6. Taylor expanded in x around 0 42.7%

   \[\leadsto \color{blue}{1} \]
7. 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 2024107 
(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))))))