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

Average Error: 5.8 → 0.9

Time: 5.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2.9865184356591876 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{e^{z}}}{-y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.9865184356591876e-19)
   (+ x (/ 1.0 y))
   (+ x (/ (/ -1.0 (exp z)) (- y)))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.9865184356591876e-19) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((-1.0 / exp(z)) / -y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.9865184356591876d-19) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (((-1.0d0) / exp(z)) / -y)
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.9865184356591876e-19) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((-1.0 / Math.exp(z)) / -y);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= 2.9865184356591876e-19:
		tmp = x + (1.0 / y)
	else:
		tmp = x + ((-1.0 / math.exp(z)) / -y)
	return tmp

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.9865184356591876e-19)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(-1.0 / exp(z)) / Float64(-y)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.9865184356591876e-19)
		tmp = x + (1.0 / y);
	else
		tmp = x + ((-1.0 / exp(z)) / -y);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := If[LessEqual[y, 2.9865184356591876e-19], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(-1.0 / N[Exp[z], $MachinePrecision]), $MachinePrecision] / (-y)), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.1
Herbie	0.9

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < 2.9865184356591876e-19

   1. Initial program 7.5

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

   2. Simplified 7.5

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

   3. Taylor expanded in y around 0 1.0

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

   if 2.9865184356591876e-19 < y

   1. Initial program 1.8

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

   2. Simplified 1.8

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

   3. Taylor expanded in y around inf 0.6

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y} \]

   4. Applied frac-2neg_binary64 0.6

      \[\leadsto x + \color{blue}{\frac{-e^{-z}}{-y}} \]

   5. Simplified 0.6

      \[\leadsto x + \frac{\color{blue}{\frac{-1}{e^{z}}}}{-y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9865184356591876 \cdot 10^{-19}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{-1}{e^{z}}}{-y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.11541576e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

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