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

Percentage Accurate: 84.2% → 99.6%

Time: 11.9s

Alternatives: 7

Speedup: 15.6×

• could not determine a ground truth

  1. x = -1.0367072061932578e+230
  2. y = -1.9447299206958087e+40
  3. z = -1.8345683015434623e-165

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

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

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

Python

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

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
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]

Initial Program: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

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

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

Python

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

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
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]

Alternative 1: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -115.0)
   (+ x (/ 1.0 (* y (exp z))))
   (if (<= y 1.25e-11) (+ x (/ 1.0 y)) (+ x (/ (exp (- z)) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -115.0) {
		tmp = x + (1.0 / (y * exp(z)));
	} else if (y <= 1.25e-11) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (exp(-z) / 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
    real(8) :: tmp
    if (y <= (-115.0d0)) then
        tmp = x + (1.0d0 / (y * exp(z)))
    else if (y <= 1.25d-11) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (exp(-z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -115.0) {
		tmp = x + (1.0 / (y * Math.exp(z)));
	} else if (y <= 1.25e-11) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (Math.exp(-z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -115.0:
		tmp = x + (1.0 / (y * math.exp(z)))
	elif y <= 1.25e-11:
		tmp = x + (1.0 / y)
	else:
		tmp = x + (math.exp(-z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -115.0)
		tmp = Float64(x + Float64(1.0 / Float64(y * exp(z))));
	elseif (y <= 1.25e-11)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -115.0)
		tmp = x + (1.0 / (y * exp(z)));
	elseif (y <= 1.25e-11)
		tmp = x + (1.0 / y);
	else
		tmp = x + (exp(-z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -115.0], N[(x + N[(1.0 / N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-11], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -115

   1. Initial program 86.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto x + \frac{e^{\color{blue}{-z}}}{y} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{y}{e^{\mathsf{neg}\left(z\right)}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{y}{e^{\mathsf{neg}\left(z\right)}}}} \]

      4. div-inv N/A

         \[\leadsto x + \frac{1}{\color{blue}{y \cdot \frac{1}{e^{\mathsf{neg}\left(z\right)}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{1}{\color{blue}{y \cdot \frac{1}{e^{\mathsf{neg}\left(z\right)}}}} \]

      6. lift-exp.f64 N/A

         \[\leadsto x + \frac{1}{y \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(z\right)}}}} \]

      7. rec-exp N/A

         \[\leadsto x + \frac{1}{y \cdot \color{blue}{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto x + \frac{1}{y \cdot \color{blue}{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)}}} \]

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

      \[\leadsto x + \frac{1}{y \cdot \color{blue}{e^{\mathsf{neg}\left(-1 \cdot z\right)}}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \frac{1}{y \cdot e^{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}} \]

      2. remove-double-neg N/A

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

      3. lower-exp.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -115 < y < 1.25000000000000005e-11

   1. Initial program 82.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.7%

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

   if 1.25000000000000005e-11 < y

   1. Initial program 88.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{e^{-z}}{y}\\ \mathbf{if}\;y \leq -115:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ (exp (- z)) y))))
   (if (<= y -115.0) t_0 (if (<= y 1.25e-11) (+ x (/ 1.0 y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (exp(-z) / y);
	double tmp;
	if (y <= -115.0) {
		tmp = t_0;
	} else if (y <= 1.25e-11) {
		tmp = x + (1.0 / 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (exp(-z) / y)
    if (y <= (-115.0d0)) then
        tmp = t_0
    else if (y <= 1.25d-11) then
        tmp = x + (1.0d0 / y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x + (Math.exp(-z) / y);
	double tmp;
	if (y <= -115.0) {
		tmp = t_0;
	} else if (y <= 1.25e-11) {
		tmp = x + (1.0 / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x + (math.exp(-z) / y)
	tmp = 0
	if y <= -115.0:
		tmp = t_0
	elif y <= 1.25e-11:
		tmp = x + (1.0 / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(exp(Float64(-z)) / y))
	tmp = 0.0
	if (y <= -115.0)
		tmp = t_0;
	elseif (y <= 1.25e-11)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x + (exp(-z) / y);
	tmp = 0.0;
	if (y <= -115.0)
		tmp = t_0;
	elseif (y <= 1.25e-11)
		tmp = x + (1.0 / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -115.0], t$95$0, If[LessEqual[y, 1.25e-11], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -115 or 1.25000000000000005e-11 < y

   1. Initial program 87.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -115 < y < 1.25000000000000005e-11

   1. Initial program 82.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.7%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{y \cdot y}\\ \mathbf{if}\;y \leq -115:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, \left(-0.16666666666666666\right) - \left(\frac{0.5}{y} + t\_0\right), 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5 + \mathsf{fma}\left(z, \left(0.16666666666666666 + t\_0\right) + \frac{-0.5}{y}, \frac{-0.5}{y}\right), 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (* y y))))
   (if (<= y -115.0)
     (+
      x
      (/
       (fma
        z
        (fma
         z
         (+
          (/ 0.5 y)
          (fma z (- (- 0.16666666666666666) (+ (/ 0.5 y) t_0)) 0.5))
         -1.0)
        1.0)
       y))
     (if (<= y 1.25e-11)
       (+ x (/ 1.0 y))
       (+
        x
        (/
         1.0
         (*
          y
          (fma
           z
           (fma
            z
            (+
             0.5
             (fma z (+ (+ 0.16666666666666666 t_0) (/ -0.5 y)) (/ -0.5 y)))
            1.0)
           1.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.3333333333333333 / (y * y);
	double tmp;
	if (y <= -115.0) {
		tmp = x + (fma(z, fma(z, ((0.5 / y) + fma(z, (-0.16666666666666666 - ((0.5 / y) + t_0)), 0.5)), -1.0), 1.0) / y);
	} else if (y <= 1.25e-11) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (1.0 / (y * fma(z, fma(z, (0.5 + fma(z, ((0.16666666666666666 + t_0) + (-0.5 / y)), (-0.5 / y))), 1.0), 1.0)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(0.3333333333333333 / Float64(y * y))
	tmp = 0.0
	if (y <= -115.0)
		tmp = Float64(x + Float64(fma(z, fma(z, Float64(Float64(0.5 / y) + fma(z, Float64(Float64(-0.16666666666666666) - Float64(Float64(0.5 / y) + t_0)), 0.5)), -1.0), 1.0) / y));
	elseif (y <= 1.25e-11)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(1.0 / Float64(y * fma(z, fma(z, Float64(0.5 + fma(z, Float64(Float64(0.16666666666666666 + t_0) + Float64(-0.5 / y)), Float64(-0.5 / y))), 1.0), 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.3333333333333333 / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -115.0], N[(x + N[(N[(z * N[(z * N[(N[(0.5 / y), $MachinePrecision] + N[(z * N[((-0.16666666666666666) - N[(N[(0.5 / y), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e-11], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(y * N[(z * N[(z * N[(0.5 + N[(z * N[(N[(0.16666666666666666 + t$95$0), $MachinePrecision] + N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -115

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)}}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right) + 1}}{y} \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1, 1\right)}}{y} \]

   5. Applied rewrites 77.7%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, -\left(0.16666666666666666 + \left(\frac{0.5}{y} + \frac{0.3333333333333333}{y \cdot y}\right)\right), 0.5\right), -1\right), 1\right)}}{y} \]

   if -115 < y < 1.25000000000000005e-11

   1. Initial program 82.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.7%

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

   if 1.25000000000000005e-11 < y

   1. Initial program 88.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \frac{e^{\color{blue}{-1 \cdot z}}}{y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto x + \frac{e^{\color{blue}{-z}}}{y} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{y}{e^{\mathsf{neg}\left(z\right)}}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\frac{y}{e^{\mathsf{neg}\left(z\right)}}}} \]

      4. div-inv N/A

         \[\leadsto x + \frac{1}{\color{blue}{y \cdot \frac{1}{e^{\mathsf{neg}\left(z\right)}}}} \]

      5. lower-*.f64 N/A

         \[\leadsto x + \frac{1}{\color{blue}{y \cdot \frac{1}{e^{\mathsf{neg}\left(z\right)}}}} \]

      6. lift-exp.f64 N/A

         \[\leadsto x + \frac{1}{y \cdot \frac{1}{\color{blue}{e^{\mathsf{neg}\left(z\right)}}}} \]

      7. rec-exp N/A

         \[\leadsto x + \frac{1}{y \cdot \color{blue}{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)}}} \]

      8. lower-exp.f64 N/A

         \[\leadsto x + \frac{1}{y \cdot \color{blue}{e^{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)}}} \]

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in z around 0

      \[\leadsto x + \frac{1}{y \cdot \color{blue}{\left(1 + z \cdot \left(1 + z \cdot \left(\left(\frac{1}{2} + z \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{y}\right)\right) - \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{1}{y \cdot \color{blue}{\left(z \cdot \left(1 + z \cdot \left(\left(\frac{1}{2} + z \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{y}\right)\right) - \frac{1}{2} \cdot \frac{1}{y}\right)\right) + 1\right)}} \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \frac{1}{y \cdot \color{blue}{\mathsf{fma}\left(z, 1 + z \cdot \left(\left(\frac{1}{2} + z \cdot \left(\left(\frac{1}{6} + \frac{1}{3} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{2} \cdot \frac{1}{y}\right)\right) - \frac{1}{2} \cdot \frac{1}{y}\right), 1\right)}} \]

   10. Applied rewrites 89.7%

       \[\leadsto x + \frac{1}{y \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5 + \mathsf{fma}\left(z, \left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right) + \frac{-0.5}{y}, \frac{-0.5}{y}\right), 1\right), 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, \left(-0.16666666666666666\right) - \left(\frac{0.5}{y} + \frac{0.3333333333333333}{y \cdot y}\right), 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5 + \mathsf{fma}\left(z, \left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right) + \frac{-0.5}{y}, \frac{-0.5}{y}\right), 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, \left(-0.16666666666666666\right) - \left(\frac{0.5}{y} + \frac{0.3333333333333333}{y \cdot y}\right), 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -115.0)
   (+
    x
    (/
     (fma
      z
      (fma
       z
       (+
        (/ 0.5 y)
        (fma
         z
         (-
          (- 0.16666666666666666)
          (+ (/ 0.5 y) (/ 0.3333333333333333 (* y y))))
         0.5))
       -1.0)
      1.0)
     y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -115.0) {
		tmp = x + (fma(z, fma(z, ((0.5 / y) + fma(z, (-0.16666666666666666 - ((0.5 / y) + (0.3333333333333333 / (y * y)))), 0.5)), -1.0), 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -115.0)
		tmp = Float64(x + Float64(fma(z, fma(z, Float64(Float64(0.5 / y) + fma(z, Float64(Float64(-0.16666666666666666) - Float64(Float64(0.5 / y) + Float64(0.3333333333333333 / Float64(y * y)))), 0.5)), -1.0), 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -115.0], N[(x + N[(N[(z * N[(z * N[(N[(0.5 / y), $MachinePrecision] + N[(z * N[((-0.16666666666666666) - N[(N[(0.5 / y), $MachinePrecision] + N[(0.3333333333333333 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -115

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{\color{blue}{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)}}{y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{\color{blue}{z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right) + 1}}{y} \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1, 1\right)}}{y} \]

   5. Applied rewrites 77.7%

      \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, -\left(0.16666666666666666 + \left(\frac{0.5}{y} + \frac{0.3333333333333333}{y \cdot y}\right)\right), 0.5\right), -1\right), 1\right)}}{y} \]

   if -115 < y

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.1%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \mathsf{fma}\left(z, \left(-0.16666666666666666\right) - \left(\frac{0.5}{y} + \frac{0.3333333333333333}{y \cdot y}\right), 0.5\right), -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -115:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -115.0) (+ x (/ (fma z (fma z 0.5 -1.0) 1.0) y)) (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -115.0) {
		tmp = x + (fma(z, fma(z, 0.5, -1.0), 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -115.0)
		tmp = Float64(x + Float64(fma(z, fma(z, 0.5, -1.0), 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -115.0], N[(x + N[(N[(z * N[(z * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -115

   1. Initial program 86.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}\right) + \frac{1}{y}\right)} \]
   4. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) - \frac{1}{y}, \frac{1}{y}\right)} \]

      2. sub-neg N/A

         \[\leadsto x + \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \left(\mathsf{neg}\left(\frac{1}{y}\right)\right)}, \frac{1}{y}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto x + \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \color{blue}{\frac{\mathsf{neg}\left(1\right)}{y}}, \frac{1}{y}\right) \]

      4. metadata-eval N/A

         \[\leadsto x + \mathsf{fma}\left(z, z \cdot \left(\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \frac{\color{blue}{-1}}{y}, \frac{1}{y}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \frac{-1}{y}\right)}, \frac{1}{y}\right) \]

      6. lower-+.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot \frac{1}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      7. associate-*r/ N/A

         \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{1}{2} \cdot 1}{y}} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      8. metadata-eval N/A

         \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\color{blue}{\frac{1}{2}}}{y} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{\frac{1}{2}}{y}} + \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      10. associate-*r/ N/A

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{1}{2}}{y} + \color{blue}{\frac{\frac{1}{2} \cdot 1}{{y}^{2}}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      11. metadata-eval N/A

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{1}{2}}{y} + \frac{\color{blue}{\frac{1}{2}}}{{y}^{2}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{1}{2}}{y} + \color{blue}{\frac{\frac{1}{2}}{{y}^{2}}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      13. unpow2 N/A

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{1}{2}}{y} + \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{1}{2}}{y} + \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \frac{-1}{y}\right), \frac{1}{y}\right) \]

      15. lower-/.f64 N/A

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{\frac{1}{2}}{y} + \frac{\frac{1}{2}}{y \cdot y}, \color{blue}{\frac{-1}{y}}\right), \frac{1}{y}\right) \]

      16. lower-/.f64 68.5

          \[\leadsto x + \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \frac{0.5}{y \cdot y}, \frac{-1}{y}\right), \color{blue}{\frac{1}{y}}\right) \]

   5. Applied rewrites 68.5%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.5}{y} + \frac{0.5}{y \cdot y}, \frac{-1}{y}\right), \frac{1}{y}\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto x + \frac{1 + z \cdot \left(\frac{1}{2} \cdot z - 1\right)}{\color{blue}{y}} \]

   8. Applied rewrites 77.7%

      \[\leadsto x + \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.5, -1\right), 1\right)}{\color{blue}{y}} \]

   if -115 < y

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
   4. Step-by-step derivation

   5. Applied rewrites 93.1%

      \[\leadsto x + \frac{\color{blue}{1}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.4% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ x + \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (/ 1.0 y)))

C

double code(double x, double y, double z) {
	return x + (1.0 / y);
}

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 + (1.0d0 / y)
end function

Java

public static double code(double x, double y, double z) {
	return x + (1.0 / y);
}

Python

def code(x, y, z):
	return x + (1.0 / y)

Julia

function code(x, y, z)
	return Float64(x + Float64(1.0 / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (1.0 / y);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in y around 0

   \[\leadsto x + \frac{\color{blue}{1}}{y} \]
4. Step-by-step derivation

5. Applied rewrites 86.7%

   \[\leadsto x + \frac{\color{blue}{1}}{y} \]
6. Add Preprocessing

Alternative 7: 39.7% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 1.0 y))

C

double code(double x, double y, double z) {
	return 1.0 / y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 / y
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 / y;
}

Python

def code(x, y, z):
	return 1.0 / y

Julia

function code(x, y, z)
	return Float64(1.0 / y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 / y;
end

Wolfram

code[x_, y_, z_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{1}{y}} \]
4. Step-by-step derivation

   1. lower-/.f64 41.2

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

5. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{1}{y}} \]
6. Add Preprocessing

Developer Target 1: 90.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< (/ y (+ z y)) 7.11541576e-315)
   (+ x (/ (exp (/ -1.0 z)) y))
   (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (exp((-1.0 / z)) / y);
	} else {
		tmp = x + (exp(log(pow((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
    real(8) :: tmp
    if ((y / (z + y)) < 7.11541576d-315) then
        tmp = x + (exp(((-1.0d0) / z)) / y)
    else
        tmp = x + (exp(log(((y / (y + z)) ** y))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y / (z + y)) < 7.11541576e-315) {
		tmp = x + (Math.exp((-1.0 / z)) / y);
	} else {
		tmp = x + (Math.exp(Math.log(Math.pow((y / (y + z)), y))) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y / (z + y)) < 7.11541576e-315:
		tmp = x + (math.exp((-1.0 / z)) / y)
	else:
		tmp = x + (math.exp(math.log(math.pow((y / (y + z)), y))) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(y / Float64(z + y)) < 7.11541576e-315)
		tmp = Float64(x + Float64(exp(Float64(-1.0 / z)) / y));
	else
		tmp = Float64(x + Float64(exp(log((Float64(y / Float64(y + z)) ^ y))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y / (z + y)) < 7.11541576e-315)
		tmp = x + (exp((-1.0 / z)) / y);
	else
		tmp = x + (exp(log(((y / (y + z)) ^ y))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision], 7.11541576e-315], N[(x + N[(N[Exp[N[(-1.0 / z), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Exp[N[Log[N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ y (+ z y)) 17788539399477/2500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (exp (/ -1 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y))))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))