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

Average Error: 18.4 → 0.2

Time: 1.4min

Precision: binary64

Cost: 26956

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(x - 1\right)\\ t_1 := \log y - -1\\ \mathbf{if}\;y \leq -630000:\\ \;\;\;\;\left(\left(1 + \frac{\frac{x - 1}{1 - x}}{y}\right) - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{elif}\;t_1 \ne 0:\\ \;\;\;\;t_1 \cdot \left(1 - \frac{t_0}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(t_0 - \log y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (- x 1.0))) (t_1 (- (log y) -1.0)))
   (if (<= y -630000.0)
     (-
      (- (+ 1.0 (/ (/ (- x 1.0) (- 1.0 x)) y)) (log1p (- x)))
      (log (/ -1.0 y)))
     (if (<= y 2.7e+31)
       (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
       (if (!= t_1 0.0)
         (* t_1 (- 1.0 (/ t_0 t_1)))
         (- 1.0 (- t_0 (log y))))))))

C

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

↓

double code(double x, double y) {
	double t_0 = log((x - 1.0));
	double t_1 = log(y) - -1.0;
	double tmp;
	if (y <= -630000.0) {
		tmp = ((1.0 + (((x - 1.0) / (1.0 - x)) / y)) - log1p(-x)) - log((-1.0 / y));
	} else if (y <= 2.7e+31) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else if (t_1 != 0.0) {
		tmp = t_1 * (1.0 - (t_0 / t_1));
	} else {
		tmp = 1.0 - (t_0 - log(y));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y) {
	double t_0 = Math.log((x - 1.0));
	double t_1 = Math.log(y) - -1.0;
	double tmp;
	if (y <= -630000.0) {
		tmp = ((1.0 + (((x - 1.0) / (1.0 - x)) / y)) - Math.log1p(-x)) - Math.log((-1.0 / y));
	} else if (y <= 2.7e+31) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else if (t_1 != 0.0) {
		tmp = t_1 * (1.0 - (t_0 / t_1));
	} else {
		tmp = 1.0 - (t_0 - Math.log(y));
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	t_0 = math.log((x - 1.0))
	t_1 = math.log(y) - -1.0
	tmp = 0
	if y <= -630000.0:
		tmp = ((1.0 + (((x - 1.0) / (1.0 - x)) / y)) - math.log1p(-x)) - math.log((-1.0 / y))
	elif y <= 2.7e+31:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	elif t_1 != 0.0:
		tmp = t_1 * (1.0 - (t_0 / t_1))
	else:
		tmp = 1.0 - (t_0 - math.log(y))
	return tmp

Julia

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

↓

function code(x, y)
	t_0 = log(Float64(x - 1.0))
	t_1 = Float64(log(y) - -1.0)
	tmp = 0.0
	if (y <= -630000.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(Float64(x - 1.0) / Float64(1.0 - x)) / y)) - log1p(Float64(-x))) - log(Float64(-1.0 / y)));
	elseif (y <= 2.7e+31)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	elseif (t_1 != 0.0)
		tmp = Float64(t_1 * Float64(1.0 - Float64(t_0 / t_1)));
	else
		tmp = Float64(1.0 - Float64(t_0 - log(y)));
	end
	return tmp
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]

↓

code[x_, y_] := Block[{t$95$0 = N[Log[N[(x - 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[y, -630000.0], N[(N[(N[(1.0 + N[(N[(N[(x - 1.0), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+31], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Unequal[t$95$1, 0.0], N[(t$95$1 * N[(1.0 - N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(t$95$0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Target

Original	18.4
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if y < -6.3e5

   1. Initial program 52.1

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

   2. Simplified 52.1

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

   3. Taylor expanded in y around -inf 0.3

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

   4. Simplified 0.3

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

   if -6.3e5 < y < 2.69999999999999986e31

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if 2.69999999999999986e31 < y

   1. Initial program 34.0

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

   2. Simplified 34.0

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

   3. Taylor expanded in y around inf 1.0

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

   4. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;1 - \log \left({y}^{-1}\right) \ne 0:\\ \;\;\;\;\left(1 - \log \left({y}^{-1}\right)\right) \cdot \left(1 + \frac{-\log \left(-1 + x\right)}{1 - \log \left({y}^{-1}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left({y}^{-1}\right) + \log \left(-1 + x\right)\right)\\ } \end{array}} \]

   5. Simplified 1.1

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\log y - -1 \ne 0:\\ \;\;\;\;\left(\log y - -1\right) \cdot \left(1 - \frac{\log \left(x - 1\right)}{\log y - -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x - 1\right) - \log y\right)\\ } \end{array}} \]
      Proof
3. Recombined 3 regimes into one program.

Alternatives

Alternative 1
Error	0.2
Cost	14084

\[\begin{array}{l} \mathbf{if}\;y \leq -540000:\\ \;\;\;\;\left(\left(1 + \frac{\frac{x - 1}{1 - x}}{y}\right) - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+27}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x - 1\right) - \log y\right)\\ \end{array} \]

Alternative 2
Error	0.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;y \leq -7800000000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+27}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x - 1\right) - \log y\right)\\ \end{array} \]

Alternative 3
Error	18.4
Cost	7492

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

Alternative 4
Error	17.3
Cost	6848

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

Alternative 5
Error	23.7
Cost	6656

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

Alternative 6
Error	35.4
Cost	712

\[\begin{array}{l} t_0 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -7.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 + \left(y \cdot x + x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	35.4
Cost	448

\[1 - \frac{x}{y - 1} \]

Alternative 8
Error	36.5
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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