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

Average Error: 5.8 → 1.4

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ t_1 := x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-103}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ 1.0 y)))
        (t_1 (+ x (/ (exp (* y (log (/ y (+ y z))))) y))))
   (if (<= t_1 -1e-103) t_0 (if (<= t_1 4e-48) (+ x (/ (exp (- z)) y)) t_0))))

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 t_0 = x + (1.0 / y);
	double t_1 = x + (exp((y * log((y / (y + z))))) / y);
	double tmp;
	if (t_1 <= -1e-103) {
		tmp = t_0;
	} else if (t_1 <= 4e-48) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + (1.0d0 / y)
    t_1 = x + (exp((y * log((y / (y + z))))) / y)
    if (t_1 <= (-1d-103)) then
        tmp = t_0
    else if (t_1 <= 4d-48) then
        tmp = x + (exp(-z) / y)
    else
        tmp = t_0
    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 t_0 = x + (1.0 / y);
	double t_1 = x + (Math.exp((y * Math.log((y / (y + z))))) / y);
	double tmp;
	if (t_1 <= -1e-103) {
		tmp = t_0;
	} else if (t_1 <= 4e-48) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = x + (1.0 / y)
	t_1 = x + (math.exp((y * math.log((y / (y + z))))) / y)
	tmp = 0
	if t_1 <= -1e-103:
		tmp = t_0
	elif t_1 <= 4e-48:
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = t_0
	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)
	t_0 = Float64(x + Float64(1.0 / y))
	t_1 = Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(y + z))))) / y))
	tmp = 0.0
	if (t_1 <= -1e-103)
		tmp = t_0;
	elseif (t_1 <= 4e-48)
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = t_0;
	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)
	t_0 = x + (1.0 / y);
	t_1 = x + (exp((y * log((y / (y + z))))) / y);
	tmp = 0.0;
	if (t_1 <= -1e-103)
		tmp = t_0;
	elseif (t_1 <= 4e-48)
		tmp = x + (exp(-z) / y);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-103], t$95$0, If[LessEqual[t$95$1, 4e-48], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.0
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 (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < -9.99999999999999958e-104 or 3.9999999999999999e-48 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y))

   1. Initial program 4.8

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

   2. Simplified 4.8

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

   3. Taylor expanded in y around 0 0.5

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

   if -9.99999999999999958e-104 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < 3.9999999999999999e-48

   1. Initial program 10.2

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

   2. Simplified 10.2

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

   3. Taylor expanded in y around inf 5.5

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

   4. Simplified 5.5

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -1 \cdot 10^{-103}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 4 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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)))