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

Average Error: 11.1 → 1.5

Time: 9.2s

Precision: binary64

Cost: 13188

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+133}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e+133)
   (pow (* x (exp y)) -1.0)
   (if (<= x 1.95e-13) (/ 1.0 x) (/ (exp (- y)) x))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e+133) {
		tmp = pow((x * exp(y)), -1.0);
	} else if (x <= 1.95e-13) {
		tmp = 1.0 / x;
	} else {
		tmp = exp(-y) / x;
	}
	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) :: tmp
    if (x <= (-9.2d+133)) then
        tmp = (x * exp(y)) ** (-1.0d0)
    else if (x <= 1.95d-13) then
        tmp = 1.0d0 / x
    else
        tmp = exp(-y) / x
    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 tmp;
	if (x <= -9.2e+133) {
		tmp = Math.pow((x * Math.exp(y)), -1.0);
	} else if (x <= 1.95e-13) {
		tmp = 1.0 / x;
	} else {
		tmp = Math.exp(-y) / x;
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if x <= -9.2e+133:
		tmp = math.pow((x * math.exp(y)), -1.0)
	elif x <= 1.95e-13:
		tmp = 1.0 / x
	else:
		tmp = math.exp(-y) / x
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e+133)
		tmp = Float64(x * exp(y)) ^ -1.0;
	elseif (x <= 1.95e-13)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(exp(Float64(-y)) / x);
	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)
	tmp = 0.0;
	if (x <= -9.2e+133)
		tmp = (x * exp(y)) ^ -1.0;
	elseif (x <= 1.95e-13)
		tmp = 1.0 / x;
	else
		tmp = exp(-y) / x;
	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_] := If[LessEqual[x, -9.2e+133], N[Power[N[(x * N[Exp[y], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[x, 1.95e-13], N[(1.0 / x), $MachinePrecision], N[(N[Exp[(-y)], $MachinePrecision] / x), $MachinePrecision]]]

Target

Original	11.1
Target	8.2
Herbie	1.5

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

Derivation

1. Split input into 3 regimes

2. if x < -9.1999999999999996e133

   1. Initial program 14.9

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

   2. Simplified 14.9

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      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 0.0

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

   4. Simplified 0.0

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

   5. Applied egg-rr 0.0

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

   if -9.1999999999999996e133 < x < 1.95000000000000002e-13

   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): 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 0 2.0

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if 1.95000000000000002e-13 < x

   1. Initial program 9.1

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

   2. Simplified 9.1

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}} \]
      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 1.2

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

   4. Simplified 1.2

      \[\leadsto \color{blue}{\frac{e^{-y}}{x}} \]
      Proof
      (/.f64 (exp.f64 (neg.f64 y)) x): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y))) x): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+133}:\\ \;\;\;\;{\left(x \cdot e^{y}\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-y}}{x}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	6920

\[\begin{array}{l} t_0 := \frac{e^{-y}}{x}\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	7.6
Cost	324

\[\begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+39}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Error	53.0
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022308 
(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))