Average Error: 6.2 → 1.6
Time: 10.5s
Precision: binary64
Cost: 33928
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
↓
\[\begin{array}{l} t_0 := x + \frac{1}{y}\\ t_1 := x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{if}\;t_1 \leq -4 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_1 \leq 10^{-80}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
(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 -4e-140) t_0 (if (<= t_1 1e-80) (+ x (/ (exp (- z)) y)) t_0))))
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 <= -4e-140) {
		tmp = t_0;
	} else if (t_1 <= 1e-80) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 <= (-4d-140)) then
        tmp = t_0
    else if (t_1 <= 1d-80) then
        tmp = x + (exp(-z) / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

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 <= -4e-140) {
		tmp = t_0;
	} else if (t_1 <= 1e-80) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 <= -4e-140:
		tmp = t_0
	elif t_1 <= 1e-80:
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = t_0
	return tmp

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 <= -4e-140)
		tmp = t_0;
	elseif (t_1 <= 1e-80)
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = t_0;
	end
	return tmp
end

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 <= -4e-140)
		tmp = t_0;
	elseif (t_1 <= 1e-80)
		tmp = x + (exp(-z) / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

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, -4e-140], t$95$0, If[LessEqual[t$95$1, 1e-80], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

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 -4 \cdot 10^{-140}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;t_1 \leq 10^{-80}:\\
\;\;\;\;x + \frac{e^{-z}}{y}\\

\mathbf{else}:\\
\;\;\;\;t_0\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.2
Target1.0
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < -3.9999999999999999e-140 or 9.99999999999999961e-81 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y))

    1. Initial program 5.4

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

    2. Simplified5.4

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      Proof
      (+.f64 x (/.f64 (pow.f64 (/.f64 y (+.f64 y z)) y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (pow.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 z y))) y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (/.f64 y (+.f64 z y)) (/.f64 y 2)) (pow.f64 (/.f64 y (+.f64 z y)) (/.f64 y 2)))) y)): 0 points increase in error, 1 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (/.f64 y (+.f64 z y)) y)) y)): 1 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 y (+.f64 z y))) y))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y)))))) y)): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in z around 0 0.9

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

    if -3.9999999999999999e-140 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < 9.99999999999999961e-81

    1. Initial program 11.3

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

    2. Simplified11.3

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      Proof
      (+.f64 x (/.f64 (pow.f64 (/.f64 y (+.f64 y z)) y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (pow.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 z y))) y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (/.f64 y (+.f64 z y)) (/.f64 y 2)) (pow.f64 (/.f64 y (+.f64 z y)) (/.f64 y 2)))) y)): 0 points increase in error, 1 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (/.f64 y (+.f64 z y)) y)) y)): 1 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 y (+.f64 z y))) y))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y)))))) y)): 0 points increase in error, 0 points decrease in error

    3. Taylor expanded in y around inf 6.1

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

    4. Simplified6.1

      \[\leadsto x + \color{blue}{\frac{e^{-z}}{y}} \]
      Proof
      (/.f64 (exp.f64 (neg.f64 z)) y): 0 points increase in error, 0 points decrease in error
      (/.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))) y): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification1.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -4 \cdot 10^{-140}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{elif}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 10^{-80}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error15.7
Cost720
\[\begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-40}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 2
Error2.5
Cost320
\[x + \frac{1}{y} \]
Alternative 3
Error28.2
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022317 
(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)))