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

Average Error: 17.8 → 1.0

Time: 13.0s

Precision: binary64

Cost: 20868

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;1 - \log \left(\left(-1 - t_0\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{1 - x}{y}\right)\right) - \frac{1}{{y}^{2}}\right)\\ \end{array} \]

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 5e-12)
     (- 1.0 (log (+ (- -1.0 t_0) 2.0)))
     (-
      1.0
      (log
       (- (+ (/ x (pow y 2.0)) (- (/ (- 1.0 x) y))) (/ 1.0 (pow y 2.0))))))))

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 <= 5e-12) {
		tmp = 1.0 - log(((-1.0 - t_0) + 2.0));
	} else {
		tmp = 1.0 - log((((x / pow(y, 2.0)) + -((1.0 - x) / y)) - (1.0 / pow(y, 2.0))));
	}
	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 <= 5d-12) then
        tmp = 1.0d0 - log((((-1.0d0) - t_0) + 2.0d0))
    else
        tmp = 1.0d0 - log((((x / (y ** 2.0d0)) + -((1.0d0 - x) / y)) - (1.0d0 / (y ** 2.0d0))))
    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 <= 5e-12) {
		tmp = 1.0 - Math.log(((-1.0 - t_0) + 2.0));
	} else {
		tmp = 1.0 - Math.log((((x / Math.pow(y, 2.0)) + -((1.0 - x) / y)) - (1.0 / Math.pow(y, 2.0))));
	}
	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 <= 5e-12:
		tmp = 1.0 - math.log(((-1.0 - t_0) + 2.0))
	else:
		tmp = 1.0 - math.log((((x / math.pow(y, 2.0)) + -((1.0 - x) / y)) - (1.0 / math.pow(y, 2.0))))
	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 <= 5e-12)
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 - t_0) + 2.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x / (y ^ 2.0)) + Float64(-Float64(Float64(1.0 - x) / y))) - Float64(1.0 / (y ^ 2.0)))));
	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 <= 5e-12)
		tmp = 1.0 - log(((-1.0 - t_0) + 2.0));
	else
		tmp = 1.0 - log((((x / (y ^ 2.0)) + -((1.0 - x) / y)) - (1.0 / (y ^ 2.0))));
	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, 5e-12], N[(1.0 - N[Log[N[(N[(-1.0 - t$95$0), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] + (-N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision])), $MachinePrecision] - N[(1.0 / N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	17.8
Target	0.1
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 4.9999999999999997e-12

   1. Initial program 0.0

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

   2. Applied egg-rr 0.0

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

   if 4.9999999999999997e-12 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 58.6

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

   2. Taylor expanded in y around inf 3.1

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

   3. Simplified 3.1

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

      [Start] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + -1 \cdot \frac{1 + -1 \cdot x}{y}\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-2 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \color{blue}{\frac{1 + -1 \cdot x}{y} \cdot -1}\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-9 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \color{blue}{\left(-\frac{1 + -1 \cdot x}{y}\right)}\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-17 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{\color{blue}{-1 \cdot x - -1}}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-2 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{\color{blue}{x \cdot -1} - -1}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-9 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{\color{blue}{\left(-x\right)} - -1}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-12 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{\color{blue}{\left(0 - x\right)} - -1}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      rational.json-simplify-42 [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{\color{blue}{\left(0 - -1\right) - x}}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]
      metadata-eval [=>] 3.1	\[ 1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{\color{blue}{1} - x}{y}\right)\right) - \frac{1}{{y}^{2}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;1 - \log \left(\left(-1 - \frac{x - y}{1 - y}\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{{y}^{2}} + \left(-\frac{1 - x}{y}\right)\right) - \frac{1}{{y}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	14468

\[\begin{array}{l} \mathbf{if}\;y \leq -490000:\\ \;\;\;\;1 - \left(\left(-\frac{\frac{x}{1 - x} - \frac{1}{1 - x}}{y}\right) + \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2000000000000:\\ \;\;\;\;1 - \log \left(\left(-1 - \frac{x - y}{1 - y}\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\frac{1 - x}{y}\right)\\ \end{array} \]

Alternative 2
Error	1.1
Cost	7748

\[\begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;1 - \log \left(\left(-1 - t_0\right) + 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\frac{1 - x}{y}\right)\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7620

\[\begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;1 - \log \left(1 - t_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\frac{1 - x}{y}\right)\\ \end{array} \]

Alternative 4
Error	7.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -40:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.135:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\frac{x}{1 - y}\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	7176

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \log \left(-\frac{1 - x}{y}\right)\\ \mathbf{elif}\;y \leq 0.023:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\frac{x}{1 - y}\right)\\ \end{array} \]

Alternative 6
Error	7.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -75:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(-\left(-\frac{x}{y}\right)\right)\\ \end{array} \]

Alternative 7
Error	13.4
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -23.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 8
Error	23.6
Cost	6788

\[\begin{array}{l} \mathbf{if}\;x \leq -0.85:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(-x\right)\\ \end{array} \]

Alternative 9
Error	23.4
Cost	6720

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

Alternative 10
Error	36.2
Cost	256

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

Alternative 11
Error	37.7
Cost	192

\[1 - y \]

Reproduce

herbie shell --seed 2023077 
(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))))))