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

Percentage Accurate: 72.0% → 100.0%

Time: 8.6s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9995:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{-1}{1 - y}, x - y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x - \frac{\left(1 - \frac{x - 1}{y}\right) - x}{y}\right) - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- y 1.0)) 0.9995)
   (- 1.0 (log (fma (/ -1.0 (- 1.0 y)) (- x y) 1.0)))
   (- 1.0 (log (/ (- (- x (/ (- (- 1.0 (/ (- x 1.0) y)) x) y)) 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / (y - 1.0)) <= 0.9995) {
		tmp = 1.0 - log(fma((-1.0 / (1.0 - y)), (x - y), 1.0));
	} else {
		tmp = 1.0 - log((((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(y - 1.0)) <= 0.9995)
		tmp = Float64(1.0 - log(fma(Float64(-1.0 / Float64(1.0 - y)), Float64(x - y), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x - Float64(Float64(Float64(1.0 - Float64(Float64(x - 1.0) / y)) - x) / y)) - 1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], 0.9995], N[(1.0 - N[Log[N[(N[(-1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x - N[(N[(N[(1.0 - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99950000000000006

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. inv-pow N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left(\color{blue}{{\left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)}^{-1}}, x - y, 1\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left(\color{blue}{{\left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)}^{-1}}, x - y, 1\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(0 - \left(1 - y\right)\right)}}^{-1}, x - y, 1\right)\right) \]

      13. lift--.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 - y\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      14. sub-neg N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - 1\right)}}^{-1}, x - y, 1\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} - 1\right)}^{-1}, x - y, 1\right)\right) \]

      18. remove-double-neg N/A

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

      19. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left({\left(y - 1\right)}^{-1}, x - y, 1\right)\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 0.99950000000000006 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9995:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{-1}{1 - y}, x - y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x - \frac{\left(1 - \frac{x - 1}{y}\right) - x}{y}\right) - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - 1}\\ \mathbf{if}\;t\_0 \leq -100:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \mathbf{elif}\;t\_0 \leq 0.6:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y 1.0))))
   (if (<= t_0 -100.0)
     (- 1.0 (log (/ x (- y 1.0))))
     (if (<= t_0 0.6)
       (- 1.0 (+ (log1p (- x)) y))
       (- 1.0 (log (/ (- x 1.0) y)))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = 1.0 - log((x / (y - 1.0)));
	} else if (t_0 <= 0.6) {
		tmp = 1.0 - (log1p(-x) + y);
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= -100.0) {
		tmp = 1.0 - Math.log((x / (y - 1.0)));
	} else if (t_0 <= 0.6) {
		tmp = 1.0 - (Math.log1p(-x) + y);
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (y - 1.0)
	tmp = 0
	if t_0 <= -100.0:
		tmp = 1.0 - math.log((x / (y - 1.0)))
	elif t_0 <= 0.6:
		tmp = 1.0 - (math.log1p(-x) + y)
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(y - 1.0))
	tmp = 0.0
	if (t_0 <= -100.0)
		tmp = Float64(1.0 - log(Float64(x / Float64(y - 1.0))));
	elseif (t_0 <= 0.6)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + y));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -100.0], N[(1.0 - N[Log[N[(x / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.6], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < -100

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -100 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.599999999999999978

   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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 99.0%

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

   if 0.599999999999999978 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.5%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 97.5

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

   5. Applied rewrites 97.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq -100:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \mathbf{elif}\;\frac{y - x}{y - 1} \leq 0.6:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9995:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{-1}{1 - y}, x - y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(\frac{x - 1}{y} + x\right) - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- y 1.0)) 0.9995)
   (- 1.0 (log (fma (/ -1.0 (- 1.0 y)) (- x y) 1.0)))
   (- 1.0 (log (/ (- (+ (/ (- x 1.0) y) x) 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / (y - 1.0)) <= 0.9995) {
		tmp = 1.0 - log(fma((-1.0 / (1.0 - y)), (x - y), 1.0));
	} else {
		tmp = 1.0 - log((((((x - 1.0) / y) + x) - 1.0) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(y - 1.0)) <= 0.9995)
		tmp = Float64(1.0 - log(fma(Float64(-1.0 / Float64(1.0 - y)), Float64(x - y), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(Float64(Float64(x - 1.0) / y) + x) - 1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], 0.9995], N[(1.0 - N[Log[N[(N[(-1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99950000000000006

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. inv-pow N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left(\color{blue}{{\left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)}^{-1}}, x - y, 1\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left(\color{blue}{{\left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)}^{-1}}, x - y, 1\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(0 - \left(1 - y\right)\right)}}^{-1}, x - y, 1\right)\right) \]

      13. lift--.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 - y\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      14. sub-neg N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - 1\right)}}^{-1}, x - y, 1\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} - 1\right)}^{-1}, x - y, 1\right)\right) \]

      18. remove-double-neg N/A

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

      19. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left({\left(y - 1\right)}^{-1}, x - y, 1\right)\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if 0.99950000000000006 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.2%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9995:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{-1}{1 - y}, x - y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(\frac{x - 1}{y} + x\right) - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9999976487049771:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{-1}{1 - y}, x - y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- y 1.0)) 0.9999976487049771)
   (- 1.0 (log (fma (/ -1.0 (- 1.0 y)) (- x y) 1.0)))
   (- 1.0 (log (/ (- x 1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / (y - 1.0)) <= 0.9999976487049771) {
		tmp = 1.0 - log(fma((-1.0 / (1.0 - y)), (x - y), 1.0));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(y - 1.0)) <= 0.9999976487049771)
		tmp = Float64(1.0 - log(fma(Float64(-1.0 / Float64(1.0 - y)), Float64(x - y), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], 0.9999976487049771], N[(1.0 - N[Log[N[(N[(-1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99999764870497709

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. distribute-lft-neg-in N/A

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. inv-pow N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left(\color{blue}{{\left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)}^{-1}}, x - y, 1\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left(\color{blue}{{\left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)}^{-1}}, x - y, 1\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(0 - \left(1 - y\right)\right)}}^{-1}, x - y, 1\right)\right) \]

      13. lift--.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 - y\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      14. sub-neg N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - 1\right)}}^{-1}, x - y, 1\right)\right) \]

      17. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)} - 1\right)}^{-1}, x - y, 1\right)\right) \]

      18. remove-double-neg N/A

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

      19. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left({\left(y - 1\right)}^{-1}, x - y, 1\right)\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lift--.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 0.99999764870497709 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   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

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9999976487049771:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{-1}{1 - y}, x - y, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - 1}\\ \mathbf{if}\;t\_0 \leq 0.9999976487049771:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y 1.0))))
   (if (<= t_0 0.9999976487049771)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (- x 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= 0.9999976487049771) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	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 = (y - x) / (y - 1.0d0)
    if (t_0 <= 0.9999976487049771d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log(((x - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= 0.9999976487049771) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (y - 1.0)
	tmp = 0
	if t_0 <= 0.9999976487049771:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(y - 1.0))
	tmp = 0.0
	if (t_0 <= 0.9999976487049771)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / (y - 1.0);
	tmp = 0.0;
	if (t_0 <= 0.9999976487049771)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log(((x - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.9999976487049771], N[(1.0 - N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99999764870497709

   1. Initial program 99.7%

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

   if 0.99999764870497709 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   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

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

      1. mul-1-neg N/A

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

      2. distribute-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.9999976487049771:\\ \;\;\;\;1 - \log \left(1 - \frac{y - x}{y - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y - 1}\right)\\ \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ x (- y 1.0))))))
   (if (<= x -1.35) t_0 (if (<= x 1.0) (- 1.0 (log1p (- x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((x / (y - 1.0)));
	double tmp;
	if (x <= -1.35) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((x / (y - 1.0)));
	double tmp;
	if (x <= -1.35) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((x / (y - 1.0)))
	tmp = 0
	if x <= -1.35:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(x / Float64(y - 1.0))))
	tmp = 0.0
	if (x <= -1.35)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(x / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35], t$95$0, If[LessEqual[x, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001 or 1 < x

   1. Initial program 78.2%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1.3500000000000001 < x < 1

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 69.9

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

   5. Applied rewrites 69.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ x y)))))
   (if (<= y -100.0) t_0 (if (<= y 1.0) (- 1.0 (+ (log1p (- x)) y)) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((x / y));
	double tmp;
	if (y <= -100.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - (log1p(-x) + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -100.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - (Math.log1p(-x) + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -100.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - (math.log1p(-x) + y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -100.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -100.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -100 or 1 < y

   1. Initial program 34.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 51.7%

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

   if -100 < y < 1

   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

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 98.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -100:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 60.7

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

5. Applied rewrites 60.7%

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

Alternative 9: 43.5% accurate, 20.7× speedup

Math

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

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 - Float64(-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 72.2%

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

3. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 60.7

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

5. Applied rewrites 60.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 46.3%

   \[\leadsto 1 - \left(-x\right) \]
9. Add Preprocessing

Developer Target 1: 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 2024255 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

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