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

Average Error: 5.8 → 2.3

Time: 6.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y}{y + z}\\ t_1 := x + \frac{e^{y \cdot \log t_0}}{y}\\ \mathbf{if}\;t_1 \leq -3.773824059460863 \cdot 10^{-188}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;t_1 \leq 2.9769590526251184 \cdot 10^{-166}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{t_0}^{y}}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (+ y z))) (t_1 (+ x (/ (exp (* y (log t_0))) y))))
   (if (<= t_1 -3.773824059460863e-188)
     (+ x (/ 1.0 y))
     (if (<= t_1 2.9769590526251184e-166)
       (+ x (/ (exp (- z)) y))
       (+ x (/ (pow t_0 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 t_0 = y / (y + z);
	double t_1 = x + (exp((y * log(t_0))) / y);
	double tmp;
	if (t_1 <= -3.773824059460863e-188) {
		tmp = x + (1.0 / y);
	} else if (t_1 <= 2.9769590526251184e-166) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (pow(t_0, 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y / (y + z)
    t_1 = x + (exp((y * log(t_0))) / y)
    if (t_1 <= (-3.773824059460863d-188)) then
        tmp = x + (1.0d0 / y)
    else if (t_1 <= 2.9769590526251184d-166) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + ((t_0 ** 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 t_0 = y / (y + z);
	double t_1 = x + (Math.exp((y * Math.log(t_0))) / y);
	double tmp;
	if (t_1 <= -3.773824059460863e-188) {
		tmp = x + (1.0 / y);
	} else if (t_1 <= 2.9769590526251184e-166) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (Math.pow(t_0, 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):
	t_0 = y / (y + z)
	t_1 = x + (math.exp((y * math.log(t_0))) / y)
	tmp = 0
	if t_1 <= -3.773824059460863e-188:
		tmp = x + (1.0 / y)
	elif t_1 <= 2.9769590526251184e-166:
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (math.pow(t_0, 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)
	t_0 = Float64(y / Float64(y + z))
	t_1 = Float64(x + Float64(exp(Float64(y * log(t_0))) / y))
	tmp = 0.0
	if (t_1 <= -3.773824059460863e-188)
		tmp = Float64(x + Float64(1.0 / y));
	elseif (t_1 <= 2.9769590526251184e-166)
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64((t_0 ^ 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)
	t_0 = y / (y + z);
	t_1 = x + (exp((y * log(t_0))) / y);
	tmp = 0.0;
	if (t_1 <= -3.773824059460863e-188)
		tmp = x + (1.0 / y);
	elseif (t_1 <= 2.9769590526251184e-166)
		tmp = x + (exp(-z) / y);
	else
		tmp = x + ((t_0 ^ 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_] := Block[{t$95$0 = N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x + N[(N[Exp[N[(y * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3.773824059460863e-188], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.9769590526251184e-166], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Power[t$95$0, y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.0
Herbie	2.3

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < -3.7738240594608627e-188

   1. Initial program 7.4

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

   2. Simplified 7.4

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

   3. Taylor expanded in y around 0 1.3

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

   if -3.7738240594608627e-188 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < 2.9769590526251184e-166

   1. Initial program 15.9

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

   2. Simplified 15.9

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

   3. Taylor expanded in y around inf 9.6

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

   if 2.9769590526251184e-166 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y))

   1. Initial program 2.3

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

   2. Simplified 2.3

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

   3. Applied egg-rr 2.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{{\left(\frac{y}{y + z}\right)}^{\left(y + y\right)}}}{1}, \frac{\sqrt[3]{{\left(\frac{y}{y + z}\right)}^{y}}}{y}, x\right)} \]

   4. Applied egg-rr 2.3

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

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -3.773824059460863 \cdot 10^{-188}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 2.9769590526251184 \cdot 10^{-166}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(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)))