Average Error: 6.1 → 1.4
Time: 12.1s
Precision: binary64
Cost: 33929
\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
↓
\[\begin{array}{l} t_0 := x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-100} \lor \neg \left(t_0 \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]
(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (* y (log (/ y (+ y z))))) y))))
   (if (or (<= t_0 -5e-100) (not (<= t_0 2e-49)))
     (+ x (/ 1.0 y))
     (+ x (/ (exp (- z)) y)))))
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 + (exp((y * log((y / (y + z))))) / y);
	double tmp;
	if ((t_0 <= -5e-100) || !(t_0 <= 2e-49)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (exp(-z) / y);
	}
	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) :: tmp
    t_0 = x + (exp((y * log((y / (y + z))))) / y)
    if ((t_0 <= (-5d-100)) .or. (.not. (t_0 <= 2d-49))) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (exp(-z) / y)
    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 + (Math.exp((y * Math.log((y / (y + z))))) / y);
	double tmp;
	if ((t_0 <= -5e-100) || !(t_0 <= 2e-49)) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (Math.exp(-z) / y);
	}
	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 + (math.exp((y * math.log((y / (y + z))))) / y)
	tmp = 0
	if (t_0 <= -5e-100) or not (t_0 <= 2e-49):
		tmp = x + (1.0 / y)
	else:
		tmp = x + (math.exp(-z) / y)
	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(exp(Float64(y * log(Float64(y / Float64(y + z))))) / y))
	tmp = 0.0
	if ((t_0 <= -5e-100) || !(t_0 <= 2e-49))
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	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 + (exp((y * log((y / (y + z))))) / y);
	tmp = 0.0;
	if ((t_0 <= -5e-100) || ~((t_0 <= 2e-49)))
		tmp = x + (1.0 / y);
	else
		tmp = x + (exp(-z) / y);
	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[(N[Exp[N[(y * N[Log[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-100], N[Not[LessEqual[t$95$0, 2e-49]], $MachinePrecision]], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y}
↓
\begin{array}{l}
t_0 := x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y}\\
\mathbf{if}\;t_0 \leq -5 \cdot 10^{-100} \lor \neg \left(t_0 \leq 2 \cdot 10^{-49}\right):\\
\;\;\;\;x + \frac{1}{y}\\

\mathbf{else}:\\
\;\;\;\;x + \frac{e^{-z}}{y}\\


\end{array}

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.1
Target1.1
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < -5.0000000000000001e-100 or 1.99999999999999987e-49 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y))

    1. Initial program 5.1

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

    2. Simplified0.2

      \[\leadsto \color{blue}{x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}} \]
      Proof
      (+.f64 x (/.f64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (+.f64 y z)))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 z y))))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (exp.f64 y) (/.f64 (log.f64 (/.f64 y (+.f64 z y))) 2)) (pow.f64 (exp.f64 y) (/.f64 (log.f64 (/.f64 y (+.f64 z y))) 2)))) y)): 1 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (+.f64 z y))))) y)): 0 points increase in error, 1 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.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 0.5

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

    if -5.0000000000000001e-100 < (+.f64 x (/.f64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y))))) y)) < 1.99999999999999987e-49

    1. Initial program 10.8

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

    2. Simplified10.8

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      Proof
      (+.f64 x (/.f64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (+.f64 y z)))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 z y))))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (exp.f64 y) (/.f64 (log.f64 (/.f64 y (+.f64 z y))) 2)) (pow.f64 (exp.f64 y) (/.f64 (log.f64 (/.f64 y (+.f64 z y))) 2)))) y)): 1 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (+.f64 z y))))) y)): 0 points increase in error, 1 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.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 5.7

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

    4. Simplified5.7

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y} \]
      Proof
      (+.f64 x (/.f64 (exp.f64 (neg.f64 z)) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.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 simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq -5 \cdot 10^{-100} \lor \neg \left(x + \frac{e^{y \cdot \log \left(\frac{y}{y + z}\right)}}{y} \leq 2 \cdot 10^{-49}\right):\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error1.1
Cost19840
\[x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y} \]
Alternative 2
Error15.3
Cost720
\[\begin{array}{l} \mathbf{if}\;y \leq -10:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-55}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 3
Error2.3
Cost320
\[x + \frac{1}{y} \]
Alternative 4
Error28.2
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2022343 
(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)))