Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 11.3 → 0.6

Time: 9.8s

Precision: binary64

Cost: 6920

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

↓

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -0.00126:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 112000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (exp (* x (log (/ x (+ x y))))) x))

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (exp (- y)) x)))
   (if (<= x -0.00126) t_0 (if (<= x 112000000.0) (/ 1.0 x) t_0))))

C

double code(double x, double y) {
	return exp((x * log((x / (x + y))))) / x;
}

↓

double code(double x, double y) {
	double t_0 = exp(-y) / x;
	double tmp;
	if (x <= -0.00126) {
		tmp = t_0;
	} else if (x <= 112000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = exp((x * log((x / (x + y))))) / x
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 = exp(-y) / x
    if (x <= (-0.00126d0)) then
        tmp = t_0
    else if (x <= 112000000.0d0) then
        tmp = 1.0d0 / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	return Math.exp((x * Math.log((x / (x + y))))) / x;
}

↓

public static double code(double x, double y) {
	double t_0 = Math.exp(-y) / x;
	double tmp;
	if (x <= -0.00126) {
		tmp = t_0;
	} else if (x <= 112000000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	return math.exp((x * math.log((x / (x + y))))) / x

↓

def code(x, y):
	t_0 = math.exp(-y) / x
	tmp = 0
	if x <= -0.00126:
		tmp = t_0
	elif x <= 112000000.0:
		tmp = 1.0 / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	return Float64(exp(Float64(x * log(Float64(x / Float64(x + y))))) / x)
end

↓

function code(x, y)
	t_0 = Float64(exp(Float64(-y)) / x)
	tmp = 0.0
	if (x <= -0.00126)
		tmp = t_0;
	elseif (x <= 112000000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp = code(x, y)
	tmp = exp((x * log((x / (x + y))))) / x;
end

↓

function tmp_2 = code(x, y)
	t_0 = exp(-y) / x;
	tmp = 0.0;
	if (x <= -0.00126)
		tmp = t_0;
	elseif (x <= 112000000.0)
		tmp = 1.0 / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := N[(N[Exp[N[(x * N[Log[N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision]

↓

code[x_, y_] := Block[{t$95$0 = N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -0.00126], t$95$0, If[LessEqual[x, 112000000.0], N[(1.0 / x), $MachinePrecision], t$95$0]]]

Target

Original	11.3
Target	8.2
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;y < -3.7311844206647956 \cdot 10^{+94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y < 2.817959242728288 \cdot 10^{+37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y < 2.347387415166998 \cdot 10^{+178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -0.00126000000000000005 or 1.12e8 < x

   1. Initial program 11.2

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 11.2

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 x (+.f64 x y)) x) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 x (+.f64 x y))) x))) x): 1 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around inf 0.2

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x} \]

   4. Simplified 0.2

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x} \]
      Proof
      (exp.f64 (neg.f64 y)): 0 points increase in error, 0 points decrease in error
      (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))): 0 points increase in error, 0 points decrease in error

   if -0.00126000000000000005 < x < 1.12e8

   1. Initial program 11.3

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x} \]

   2. Simplified 11.3

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      Proof
      (/.f64 (pow.f64 (/.f64 x (+.f64 x y)) x) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 x (+.f64 x y))) x))) x): 1 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 x (log.f64 (/.f64 x (+.f64 x y)))))) x): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00126:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 112000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	5.2
Cost	712

\[\begin{array}{l} t_0 := \frac{1}{x + x \cdot y}\\ \mathbf{if}\;x \leq -0.00126:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 112000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 10000000000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{x}\right) + -1\\ \end{array} \]

Alternative 3
Error	10.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot y}\\ \end{array} \]

Alternative 4
Error	9.7
Cost	192

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

Reproduce

herbie shell --seed 2022316 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1.0 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1.0 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))