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

Average Error: 18.7 → 0.2

Time: 19.6s

Precision: binary64

Cost: 28164

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t_0 \leq 0.1:\\ \;\;\;\;\left(1 - \left(\log \left(1 - t_0\right) + -1\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(-1 \cdot \left(\frac{1 - x}{{y}^{4}} + \left(\frac{1 - x}{{y}^{2}} + \frac{1 - x}{{y}^{3}}\right)\right) - \frac{1}{y}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 0.1)
     (+ (- 1.0 (+ (log (- 1.0 t_0)) -1.0)) -1.0)
     (-
      1.0
      (log
       (+
        (/ x y)
        (-
         (*
          -1.0
          (+
           (/ (- 1.0 x) (pow y 4.0))
           (+ (/ (- 1.0 x) (pow y 2.0)) (/ (- 1.0 x) (pow y 3.0)))))
         (/ 1.0 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 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = (1.0 - (log((1.0 - t_0)) + -1.0)) + -1.0;
	} else {
		tmp = 1.0 - log(((x / y) + ((-1.0 * (((1.0 - x) / pow(y, 4.0)) + (((1.0 - x) / pow(y, 2.0)) + ((1.0 - x) / pow(y, 3.0))))) - (1.0 / y))));
	}
	return tmp;
}

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

↓

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 = (x - y) / (1.0d0 - y)
    if (t_0 <= 0.1d0) then
        tmp = (1.0d0 - (log((1.0d0 - t_0)) + (-1.0d0))) + (-1.0d0)
    else
        tmp = 1.0d0 - log(((x / y) + (((-1.0d0) * (((1.0d0 - x) / (y ** 4.0d0)) + (((1.0d0 - x) / (y ** 2.0d0)) + ((1.0d0 - x) / (y ** 3.0d0))))) - (1.0d0 / y))))
    end if
    code = tmp
end function

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 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = (1.0 - (Math.log((1.0 - t_0)) + -1.0)) + -1.0;
	} else {
		tmp = 1.0 - Math.log(((x / y) + ((-1.0 * (((1.0 - x) / Math.pow(y, 4.0)) + (((1.0 - x) / Math.pow(y, 2.0)) + ((1.0 - x) / Math.pow(y, 3.0))))) - (1.0 / 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 = (x - y) / (1.0 - y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = (1.0 - (math.log((1.0 - t_0)) + -1.0)) + -1.0
	else:
		tmp = 1.0 - math.log(((x / y) + ((-1.0 * (((1.0 - x) / math.pow(y, 4.0)) + (((1.0 - x) / math.pow(y, 2.0)) + ((1.0 - x) / math.pow(y, 3.0))))) - (1.0 / 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 = Float64(Float64(x - y) / Float64(1.0 - y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(Float64(1.0 - Float64(log(Float64(1.0 - t_0)) + -1.0)) + -1.0);
	else
		tmp = Float64(1.0 - log(Float64(Float64(x / y) + Float64(Float64(-1.0 * Float64(Float64(Float64(1.0 - x) / (y ^ 4.0)) + Float64(Float64(Float64(1.0 - x) / (y ^ 2.0)) + Float64(Float64(1.0 - x) / (y ^ 3.0))))) - Float64(1.0 / y)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	t_0 = (x - y) / (1.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = (1.0 - (log((1.0 - t_0)) + -1.0)) + -1.0;
	else
		tmp = 1.0 - log(((x / y) + ((-1.0 * (((1.0 - x) / (y ^ 4.0)) + (((1.0 - x) / (y ^ 2.0)) + ((1.0 - x) / (y ^ 3.0))))) - (1.0 / y))));
	end
	tmp_2 = 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[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], N[(N[(1.0 - N[(N[Log[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x / y), $MachinePrecision] + N[(N[(-1.0 * N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.7
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 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.10000000000000001

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   if 0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 61.0

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

   2. Applied egg-rr 61.1

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

   3. Simplified 61.1

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

      [Start] 61.1	\[ 1 - \log \left(1 - \frac{1}{x - y} \cdot \frac{1 - y}{\left(1 - y\right) \cdot \frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-35 [=>] 61.1	\[ 1 - \log \left(1 - \color{blue}{\frac{1 + 1}{\left(x - y\right) + \left(x - y\right)}} \cdot \frac{1 - y}{\left(1 - y\right) \cdot \frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      metadata-eval [=>] 61.1	\[ 1 - \log \left(1 - \frac{\color{blue}{2}}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{1 - y}{\left(1 - y\right) \cdot \frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-46 [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \color{blue}{\frac{\frac{1 - y}{1 - y}}{\frac{\frac{1 - y}{x - y}}{x - y}}}\right) \]
      rational.json-simplify-50 [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\color{blue}{\frac{-\left(1 - y\right)}{y - 1}}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-12 [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\color{blue}{0 - \left(1 - y\right)}}{y - 1}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-45 [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\color{blue}{y - \left(1 - 0\right)}}{y - 1}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      metadata-eval [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{y - \color{blue}{1}}{y - 1}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-15 [<=] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\color{blue}{y + -1}}{y - 1}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-50 [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\color{blue}{\frac{-\left(y + -1\right)}{1 - y}}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-8 [=>] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\color{blue}{\left(y + -1\right) \cdot -1}}{1 - y}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      metadata-eval [<=] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\left(y + -1\right) \cdot \color{blue}{\frac{-2}{2}}}{1 - y}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-49 [<=] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\color{blue}{\frac{-2 \cdot \left(y + -1\right)}{2}}}{1 - y}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-2 [<=] 60.2	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \frac{\frac{\frac{\color{blue}{\left(y + -1\right) \cdot -2}}{2}}{1 - y}}{\frac{\frac{1 - y}{x - y}}{x - y}}\right) \]
      rational.json-simplify-46 [<=] 61.1	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \color{blue}{\frac{\frac{\left(y + -1\right) \cdot -2}{2}}{\left(1 - y\right) \cdot \frac{\frac{1 - y}{x - y}}{x - y}}}\right) \]
      rational.json-simplify-44 [=>] 61.1	\[ 1 - \log \left(1 - \frac{2}{\left(x - y\right) + \left(x - y\right)} \cdot \color{blue}{\frac{\frac{\left(y + -1\right) \cdot -2}{\left(1 - y\right) \cdot \frac{\frac{1 - y}{x - y}}{x - y}}}{2}}\right) \]

   4. Taylor expanded in y around inf 0.5

      \[\leadsto 1 - \log \color{blue}{\left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{4}} + \left(\frac{x}{y} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}}\right)\right)\right) - \frac{1}{y}\right)} \]

   5. Simplified 0.5

      \[\leadsto 1 - \log \color{blue}{\left(\frac{x}{y} + \left(-1 \cdot \left(\frac{1 - x}{{y}^{4}} + \left(\frac{1 - x}{{y}^{2}} + \frac{1 - x}{{y}^{3}}\right)\right) - \frac{1}{y}\right)\right)} \]
      Proof

      [Start] 0.5	\[ 1 - \log \left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{4}} + \left(\frac{x}{y} + \left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}}\right)\right)\right) - \frac{1}{y}\right) \]
      rational.json-simplify-41 [=>] 0.5	\[ 1 - \log \left(\color{blue}{\left(\frac{x}{y} + \left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}}\right) + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{4}}\right)\right)} - \frac{1}{y}\right) \]
      rational.json-simplify-1 [=>] 0.5	\[ 1 - \log \left(\color{blue}{\left(\left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}}\right) + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{4}}\right) + \frac{x}{y}\right)} - \frac{1}{y}\right) \]
      rational.json-simplify-48 [=>] 0.5	\[ 1 - \log \color{blue}{\left(\frac{x}{y} + \left(\left(\left(-1 \cdot \frac{1 + -1 \cdot x}{{y}^{3}} + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{2}}\right) + -1 \cdot \frac{1 + -1 \cdot x}{{y}^{4}}\right) - \frac{1}{y}\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.1:\\ \;\;\;\;\left(1 - \left(\log \left(1 - \frac{x - y}{1 - y}\right) + -1\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(-1 \cdot \left(\frac{1 - x}{{y}^{4}} + \left(\frac{1 - x}{{y}^{2}} + \frac{1 - x}{{y}^{3}}\right)\right) - \frac{1}{y}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	14468

\[\begin{array}{l} \mathbf{if}\;y \leq -1900000:\\ \;\;\;\;1 - \left(\left(-\frac{\frac{x}{1 - x} - \frac{1}{1 - x}}{y}\right) + \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 30000000000000:\\ \;\;\;\;\left(1 - \left(\log \left(1 - \frac{x - y}{1 - y}\right) + -1\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7624

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{if}\;y \leq -880000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+14}:\\ \;\;\;\;\left(1 - \left(\log \left(1 - \frac{x - y}{1 - y}\right) + -1\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7368

\[\begin{array}{l} t_0 := 1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{if}\;y \leq -880000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	7.6
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -108:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(2 - \log \left(1 - x\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	7.5
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -32:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;\left(2 - \log \left(1 - x\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]

Alternative 6
Error	1.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-9}:\\ \;\;\;\;\left(2 - \log \left(1 - x\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]

Alternative 7
Error	7.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -700:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 8
Error	13.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -92:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 9
Error	24.7
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -0.82:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x - -1\\ \end{array} \]

Alternative 10
Error	24.5
Cost	6720

\[1 - \log \left(1 - x\right) \]

Alternative 11
Error	36.7
Cost	192

\[x - -1 \]

Alternative 12
Error	36.9
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023074 
(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))))))