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

Average Error: 11.1 → 0.4

Time: 9.4s

Precision: binary64

Cost: 6920

Math

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

↓

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 40:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 -3.6e-6) t_0 (if (<= x 40.0) (+ 1.0 (+ (/ 1.0 x) -1.0)) 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 <= -3.6e-6) {
		tmp = t_0;
	} else if (x <= 40.0) {
		tmp = 1.0 + ((1.0 / x) + -1.0);
	} 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 <= (-3.6d-6)) then
        tmp = t_0
    else if (x <= 40.0d0) then
        tmp = 1.0d0 + ((1.0d0 / x) + (-1.0d0))
    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 <= -3.6e-6) {
		tmp = t_0;
	} else if (x <= 40.0) {
		tmp = 1.0 + ((1.0 / x) + -1.0);
	} 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 <= -3.6e-6:
		tmp = t_0
	elif x <= 40.0:
		tmp = 1.0 + ((1.0 / x) + -1.0)
	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 <= -3.6e-6)
		tmp = t_0;
	elseif (x <= 40.0)
		tmp = Float64(1.0 + Float64(Float64(1.0 / x) + -1.0));
	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 <= -3.6e-6)
		tmp = t_0;
	elseif (x <= 40.0)
		tmp = 1.0 + ((1.0 / x) + -1.0);
	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, -3.6e-6], t$95$0, If[LessEqual[x, 40.0], N[(1.0 + N[(N[(1.0 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	11.1
Target	7.8
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999984e-6 or 40 < x

   1. Initial program 10.6

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

   2. Simplified 10.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      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): 0 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 -3.59999999999999984e-6 < x < 40

   1. Initial program 11.6

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

   2. Simplified 11.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      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): 0 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 y around 0 25.8

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

   4. Simplified 25.8

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

   5. Applied egg-rr 25.8

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

   6. Taylor expanded in y around 0 0.5

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

   7. Applied egg-rr 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \mathbf{elif}\;x \leq 40:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 100:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{x} + -1\right)\\ \end{array} \]

Alternative 2
Error	9.5
Cost	192

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

Reproduce

herbie shell --seed 2022317 
(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))