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

Percentage Accurate: 83.9% → 98.9%

Time: 7.9s

Alternatives: 9

Speedup: 15.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 83.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+26} \lor \neg \left(y \leq 0.18\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8e+26) (not (<= y 0.18)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e+26) || !(y <= 0.18)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-8d+26)) .or. (.not. (y <= 0.18d0))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e+26) || !(y <= 0.18)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8e+26) or not (y <= 0.18):
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8e+26) || !(y <= 0.18))
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8e+26) || ~((y <= 0.18)))
		tmp = x + (exp(-z) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8e+26], N[Not[LessEqual[y, 0.18]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e26 or 0.17999999999999999 < y

   1. Initial program 81.6%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 -8.00000000000000038e26 < y < 0.17999999999999999

   1. Initial program 83.7%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 99.1%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+26} \lor \neg \left(y \leq 0.18\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right) + \frac{0.5}{y}, -z, \frac{0.5}{y}\right) - -0.5\right) \cdot z - 1, z, 1\right)}{x}}{y}, x, x\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+46)
   (fma
    (/
     (/
      (fma
       (-
        (*
         (-
          (fma
           (+ (+ 0.16666666666666666 (/ 0.3333333333333333 (* y y))) (/ 0.5 y))
           (- z)
           (/ 0.5 y))
          -0.5)
         z)
        1.0)
       z
       1.0)
      x)
     y)
    x
    x)
   (if (<= y 1.55e+226)
     (+ x (/ 1.0 y))
     (+ x (/ (/ (fma (fma (- (* 0.5 z) 1.0) z 1.0) y (* (* z z) 0.5)) y) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+46) {
		tmp = fma(((fma((((fma(((0.16666666666666666 + (0.3333333333333333 / (y * y))) + (0.5 / y)), -z, (0.5 / y)) - -0.5) * z) - 1.0), z, 1.0) / x) / y), x, x);
	} else if (y <= 1.55e+226) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((fma(fma(((0.5 * z) - 1.0), z, 1.0), y, ((z * z) * 0.5)) / y) / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+46)
		tmp = fma(Float64(Float64(fma(Float64(Float64(Float64(fma(Float64(Float64(0.16666666666666666 + Float64(0.3333333333333333 / Float64(y * y))) + Float64(0.5 / y)), Float64(-z), Float64(0.5 / y)) - -0.5) * z) - 1.0), z, 1.0) / x) / y), x, x);
	elseif (y <= 1.55e+226)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(fma(fma(Float64(Float64(0.5 * z) - 1.0), z, 1.0), y, Float64(Float64(z * z) * 0.5)) / y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+46], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(0.16666666666666666 + N[(0.3333333333333333 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] * (-z) + N[(0.5 / y), $MachinePrecision]), $MachinePrecision] - -0.5), $MachinePrecision] * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / x), $MachinePrecision] / y), $MachinePrecision] * x + x), $MachinePrecision], If[LessEqual[y, 1.55e+226], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[(N[(N[(0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000007e46

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 80.0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1 + z \cdot \left(z \cdot \left(\frac{1}{2} + \left(-1 \cdot \left(z \cdot \left(\frac{1}{6} + \left(\frac{1}{3} \cdot \frac{1}{{y}^{2}} + \frac{1}{2} \cdot \frac{1}{y}\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{y}\right)\right) - 1\right)}{x}}{y}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.0%

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

   if -1.30000000000000007e46 < y < 1.54999999999999988e226

   1. Initial program 86.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 94.9%

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

   if 1.54999999999999988e226 < y

   1. Initial program 65.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\left(0.16666666666666666 + \frac{0.3333333333333333}{y \cdot y}\right) + \frac{0.5}{y}, -z, \frac{0.5}{y}\right) - -0.5\right) \cdot z - 1, z, 1\right)}{x}}{y}, x, x\right)\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\frac{0.5 \cdot \left(z \cdot y\right)}{y} - 1, z, 1\right)}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+46)
   (+ x (/ (fma (- (/ (* 0.5 (* z y)) y) 1.0) z 1.0) y))
   (if (<= y 1.55e+226)
     (+ x (/ 1.0 y))
     (+ x (/ (/ (fma (fma (- (* 0.5 z) 1.0) z 1.0) y (* (* z z) 0.5)) y) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+46) {
		tmp = x + (fma((((0.5 * (z * y)) / y) - 1.0), z, 1.0) / y);
	} else if (y <= 1.55e+226) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + ((fma(fma(((0.5 * z) - 1.0), z, 1.0), y, ((z * z) * 0.5)) / y) / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+46)
		tmp = Float64(x + Float64(fma(Float64(Float64(Float64(0.5 * Float64(z * y)) / y) - 1.0), z, 1.0) / y));
	elseif (y <= 1.55e+226)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(Float64(fma(fma(Float64(Float64(0.5 * z) - 1.0), z, 1.0), y, Float64(Float64(z * z) * 0.5)) / y) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+46], N[(x + N[(N[(N[(N[(N[(0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+226], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(N[(N[(N[(0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] * y + N[(N[(z * z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000007e46

   1. Initial program 80.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot z + \frac{1}{2} \cdot \left(y \cdot z\right)}{y} - 1, z, 1\right)}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in y around inf

      \[\leadsto x + \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(y \cdot z\right)}{y} - 1, z, 1\right)}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.2%

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

   if -1.30000000000000007e46 < y < 1.54999999999999988e226

   1. Initial program 86.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 94.9%

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

   if 1.54999999999999988e226 < y

   1. Initial program 65.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\frac{0.5 \cdot \left(z \cdot y\right)}{y} - 1, z, 1\right)}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+226}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right), y, \left(z \cdot z\right) \cdot 0.5\right)}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\frac{0.5 \cdot \left(z \cdot y\right)}{y} - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+46)
   (+ x (/ (fma (- (/ (* 0.5 (* z y)) y) 1.0) z 1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+46) {
		tmp = x + (fma((((0.5 * (z * y)) / y) - 1.0), z, 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+46)
		tmp = Float64(x + Float64(fma(Float64(Float64(Float64(0.5 * Float64(z * y)) / y) - 1.0), z, 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+46], N[(x + N[(N[(N[(N[(N[(0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000007e46

   1. Initial program 80.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot z + \frac{1}{2} \cdot \left(y \cdot z\right)}{y} - 1, z, 1\right)}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.2%

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

   9. Taylor expanded in y around inf

      \[\leadsto x + \frac{\mathsf{fma}\left(\frac{\frac{1}{2} \cdot \left(y \cdot z\right)}{y} - 1, z, 1\right)}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.2%

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

   if -1.30000000000000007e46 < y

   1. Initial program 83.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 90.6%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(\frac{0.5 \cdot \left(z \cdot y\right)}{y} - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+46)
   (+ (/ (fma (fma (fma -0.16666666666666666 z 0.5) z -1.0) z 1.0) y) x)
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+46) {
		tmp = (fma(fma(fma(-0.16666666666666666, z, 0.5), z, -1.0), z, 1.0) / y) + x;
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+46)
		tmp = Float64(Float64(fma(fma(fma(-0.16666666666666666, z, 0.5), z, -1.0), z, 1.0) / y) + x);
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+46], N[(N[(N[(N[(N[(-0.16666666666666666 * z + 0.5), $MachinePrecision] * z + -1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000007e46

   1. Initial program 80.0%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \frac{\mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot z\right) \cdot z - 1, z, 1\right)}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.8%

      \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right) \cdot z - 1, z, 1\right)}{y} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 80.8

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

   10. Applied rewrites 80.8%

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

   if -1.30000000000000007e46 < y

   1. Initial program 83.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 90.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, z, 0.5\right), z, -1\right), z, 1\right)}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+46)
   (+ x (/ (fma (- (* 0.5 z) 1.0) z 1.0) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e+46) {
		tmp = x + (fma(((0.5 * z) - 1.0), z, 1.0) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e+46)
		tmp = Float64(x + Float64(fma(Float64(Float64(0.5 * z) - 1.0), z, 1.0) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e+46], N[(x + N[(N[(N[(N[(0.5 * z), $MachinePrecision] - 1.0), $MachinePrecision] * z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000007e46

   1. Initial program 80.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto x + \frac{\mathsf{fma}\left(\frac{1}{2} \cdot z - 1, z, 1\right)}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   if -1.30000000000000007e46 < y

   1. Initial program 83.3%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
   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 90.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+46}:\\ \;\;\;\;x + \frac{\mathsf{fma}\left(0.5 \cdot z - 1, z, 1\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 82.5%

   \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
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 85.1%

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

6. Final simplification 85.1%

   \[\leadsto x + \frac{1}{y} \]
7. Add Preprocessing

Alternative 8: 39.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 82.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 40.4

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

5. Applied rewrites 40.4%

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

Alternative 9: 2.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 82.5%

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

3. Taylor expanded in y around 0

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

   1. lower-/.f64 40.4

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

5. Applied rewrites 40.4%

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

7. Applied rewrites 12.1%

   \[\leadsto {\left(y \cdot y\right)}^{\color{blue}{-0.5}} \]

8. Taylor expanded in y around -inf

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

10. Applied rewrites 2.1%

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

Developer Target 1: 91.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025017 
(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)))