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

Average Error: 6.4 → 1.4

Time: 7.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 2.6512083876927243 \cdot 10^{-77}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\left|\frac{\sqrt{y}}{\sqrt{y + z}}\right|\right)}^{y} \cdot {\left(\sqrt{\frac{y}{y + z}}\right)}^{y}}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.6512083876927243e-77)
   (+ x (/ 1.0 y))
   (+
    x
    (/
     (*
      (pow (fabs (/ (sqrt y) (sqrt (+ y z)))) y)
      (pow (sqrt (/ y (+ y z))) y))
     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 <= 2.6512083876927243e-77) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((pow(fabs((sqrt(y) / sqrt((y + z)))), y) * pow(sqrt((y / (y + z))), y)) / 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 <= 2.6512083876927243d-77) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (((abs((sqrt(y) / sqrt((y + z)))) ** y) * (sqrt((y / (y + z))) ** y)) / 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 <= 2.6512083876927243e-77) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((Math.pow(Math.abs((Math.sqrt(y) / Math.sqrt((y + z)))), y) * Math.pow(Math.sqrt((y / (y + z))), y)) / 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 <= 2.6512083876927243e-77:
		tmp = x + (1.0 / y)
	else:
		tmp = x + ((math.pow(math.fabs((math.sqrt(y) / math.sqrt((y + z)))), y) * math.pow(math.sqrt((y / (y + z))), y)) / 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 <= 2.6512083876927243e-77)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64((abs(Float64(sqrt(y) / sqrt(Float64(y + z)))) ^ y) * (sqrt(Float64(y / Float64(y + z))) ^ y)) / 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 <= 2.6512083876927243e-77)
		tmp = x + (1.0 / y);
	else
		tmp = x + (((abs((sqrt(y) / sqrt((y + z)))) ^ y) * (sqrt((y / (y + z))) ^ y)) / 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, 2.6512083876927243e-77], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[Power[N[Abs[N[(N[Sqrt[y], $MachinePrecision] / N[Sqrt[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y], $MachinePrecision] * N[Power[N[Sqrt[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.4
Target	1.2
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if y < 2.6512083876927243e-77

   1. Initial program 9.0

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

   2. Simplified 9.0

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

   3. Taylor expanded in y around 0 1.1

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

   if 2.6512083876927243e-77 < y

   1. Initial program 1.9

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

   2. Simplified 1.9

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

   3. Applied add-sqr-sqrt_binary64 1.9

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

   4. Applied unpow-prod-down_binary64 1.9

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

   5. Applied add-sqr-sqrt_binary64 3.6

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

   6. Applied add-sqr-sqrt_binary64 1.9

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

   7. Applied times-frac_binary64 1.9

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

   8. Applied rem-sqrt-square_binary64 1.9

      \[\leadsto x + \frac{{\color{blue}{\left(\left|\frac{\sqrt{y}}{\sqrt{y + z}}\right|\right)}}^{y} \cdot {\left(\sqrt{\frac{y}{y + z}}\right)}^{y}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6512083876927243 \cdot 10^{-77}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\left|\frac{\sqrt{y}}{\sqrt{y + z}}\right|\right)}^{y} \cdot {\left(\sqrt{\frac{y}{y + z}}\right)}^{y}}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))