Development.Shake.Progress:message from shake-0.15.5

Percentage Accurate: 99.4% → 99.7%
Time: 2.3s
Alternatives: 6
Speedup: 1.0×

Specification

?
\[\begin{array}{l} \\ \frac{x \cdot 100}{x + y} \end{array} \]
(FPCore (x y) :precision binary64 (/ (* x 100.0) (+ x y)))
double code(double x, double y) {
	return (x * 100.0) / (x + y);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 100.0d0) / (x + y)
end function

public static double code(double x, double y) {
	return (x * 100.0) / (x + y);
}

def code(x, y):
	return (x * 100.0) / (x + y)

function code(x, y)
	return Float64(Float64(x * 100.0) / Float64(x + y))
end

function tmp = code(x, y)
	tmp = (x * 100.0) / (x + y);
end

code[x_, y_] := N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x \cdot 100}{x + y}
\end{array}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.4% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{x \cdot 100}{x + y} \end{array} \]
(FPCore (x y) :precision binary64 (/ (* x 100.0) (+ x y)))
double code(double x, double y) {
	return (x * 100.0) / (x + y);
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * 100.0d0) / (x + y)
end function

public static double code(double x, double y) {
	return (x * 100.0) / (x + y);
}

def code(x, y):
	return (x * 100.0) / (x + y)

function code(x, y)
	return Float64(Float64(x * 100.0) / Float64(x + y))
end

function tmp = code(x, y)
	tmp = (x * 100.0) / (x + y);
end

code[x_, y_] := N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{x \cdot 100}{x + y}
\end{array}

Alternative 1: 99.7% accurate, 1.0× speedup?

\[\begin{array}{l} \\ x \cdot \frac{100}{y + x} \end{array} \]
(FPCore (x y) :precision binary64 (* x (/ 100.0 (+ y x))))
double code(double x, double y) {
	return x * (100.0 / (y + x));
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (100.0d0 / (y + x))
end function

public static double code(double x, double y) {
	return x * (100.0 / (y + x));
}

def code(x, y):
	return x * (100.0 / (y + x))

function code(x, y)
	return Float64(x * Float64(100.0 / Float64(y + x)))
end

function tmp = code(x, y)
	tmp = x * (100.0 / (y + x));
end

code[x_, y_] := N[(x * N[(100.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
x \cdot \frac{100}{y + x}
\end{array}
Derivation

  1. Initial program 99.4%

    \[\frac{x \cdot 100}{x + y} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \frac{\color{blue}{x \cdot 100}}{x + y} \]

    2. lift-+.f64N/A

      \[\leadsto \frac{x \cdot 100}{\color{blue}{x + y}} \]

    3. lift-/.f64N/A

      \[\leadsto \color{blue}{\frac{x \cdot 100}{x + y}} \]

    4. associate-/l*N/A

      \[\leadsto \color{blue}{x \cdot \frac{100}{x + y}} \]

    5. lower-*.f64N/A

      \[\leadsto \color{blue}{x \cdot \frac{100}{x + y}} \]

    6. lower-/.f64N/A

      \[\leadsto x \cdot \color{blue}{\frac{100}{x + y}} \]

    7. +-commutativeN/A

      \[\leadsto x \cdot \frac{100}{\color{blue}{y + x}} \]

    8. lower-+.f6499.7

      \[\leadsto x \cdot \frac{100}{\color{blue}{y + x}} \]

  4. Applied rewrites99.7%

    \[\leadsto \color{blue}{x \cdot \frac{100}{y + x}} \]
  5. Add Preprocessing

Alternative 2: 97.6% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot 100}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, -100, 100\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (if (<= (/ (* x 100.0) (+ x y)) 5e-12)
   (/ (* x 100.0) y)
   (fma (/ y x) -100.0 100.0)))
double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = (x * 100.0) / y;
	} else {
		tmp = fma((y / x), -100.0, 100.0);
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 100.0) / Float64(x + y)) <= 5e-12)
		tmp = Float64(Float64(x * 100.0) / y);
	else
		tmp = fma(Float64(y / x), -100.0, 100.0);
	end
	return tmp
end

code[x_, y_] := If[LessEqual[N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(x * 100.0), $MachinePrecision] / y), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * -100.0 + 100.0), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\
\;\;\;\;\frac{x \cdot 100}{y}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{x}, -100, 100\right)\\


\end{array}
\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y)) < 4.9999999999999997e-12

    1. Initial program 99.7%

      \[\frac{x \cdot 100}{x + y} \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \frac{x \cdot 100}{\color{blue}{y}} \]
    4. Step-by-step derivation

      1. Applied rewrites99.4%

        \[\leadsto \frac{x \cdot 100}{\color{blue}{y}} \]

      if 4.9999999999999997e-12 < (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y))

      1. Initial program 99.1%

        \[\frac{x \cdot 100}{x + y} \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

        \[\leadsto \color{blue}{100 + -100 \cdot \frac{y}{x}} \]
      4. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto -100 \cdot \frac{y}{x} + \color{blue}{100} \]

        2. *-commutativeN/A

          \[\leadsto \frac{y}{x} \cdot -100 + 100 \]

        3. lower-fma.f64N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, \color{blue}{-100}, 100\right) \]

        4. lower-/.f6499.2

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, -100, 100\right) \]

      5. Applied rewrites99.2%

        \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, -100, 100\right)} \]
    5. Recombined 2 regimes into one program.

    6. Final simplification99.3%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot 100}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, -100, 100\right)\\ \end{array} \]
    7. Add Preprocessing

    Alternative 3: 97.2% accurate, 0.5× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot 100}{y}\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \end{array} \]
    (FPCore (x y)
 :precision binary64
 (if (<= (/ (* x 100.0) (+ x y)) 5e-12) (/ (* x 100.0) y) 100.0))
    double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = (x * 100.0) / y;
	} else {
		tmp = 100.0;
	}
	return tmp;
}

    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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * 100.0d0) / (x + y)) <= 5d-12) then
        tmp = (x * 100.0d0) / y
    else
        tmp = 100.0d0
    end if
    code = tmp
end function

    public static double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = (x * 100.0) / y;
	} else {
		tmp = 100.0;
	}
	return tmp;
}

    def code(x, y):
	tmp = 0
	if ((x * 100.0) / (x + y)) <= 5e-12:
		tmp = (x * 100.0) / y
	else:
		tmp = 100.0
	return tmp

    function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 100.0) / Float64(x + y)) <= 5e-12)
		tmp = Float64(Float64(x * 100.0) / y);
	else
		tmp = 100.0;
	end
	return tmp
end

    function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * 100.0) / (x + y)) <= 5e-12)
		tmp = (x * 100.0) / y;
	else
		tmp = 100.0;
	end
	tmp_2 = tmp;
end

    code[x_, y_] := If[LessEqual[N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(x * 100.0), $MachinePrecision] / y), $MachinePrecision], 100.0]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\
\;\;\;\;\frac{x \cdot 100}{y}\\

\mathbf{else}:\\
\;\;\;\;100\\


\end{array}
\end{array}
    Derivation
    1. Split input into 2 regimes

    2. if (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y)) < 4.9999999999999997e-12

      1. Initial program 99.7%

        \[\frac{x \cdot 100}{x + y} \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \frac{x \cdot 100}{\color{blue}{y}} \]
      4. Step-by-step derivation

        1. Applied rewrites99.4%

          \[\leadsto \frac{x \cdot 100}{\color{blue}{y}} \]

        if 4.9999999999999997e-12 < (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y))

        1. Initial program 99.1%

          \[\frac{x \cdot 100}{x + y} \]
        2. Add Preprocessing

        3. Taylor expanded in x around inf

          \[\leadsto \color{blue}{100} \]
        4. Step-by-step derivation

          1. Applied rewrites98.2%

            \[\leadsto \color{blue}{100} \]
        5. Recombined 2 regimes into one program.

        6. Final simplification98.8%

          \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot 100}{y}\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \]
        7. Add Preprocessing

        Alternative 4: 97.2% accurate, 0.5× speedup?

        \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{100}{y}\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \end{array} \]
        (FPCore (x y)
 :precision binary64
 (if (<= (/ (* x 100.0) (+ x y)) 5e-12) (* x (/ 100.0 y)) 100.0))
        double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = x * (100.0 / y);
	} else {
		tmp = 100.0;
	}
	return tmp;
}

        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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * 100.0d0) / (x + y)) <= 5d-12) then
        tmp = x * (100.0d0 / y)
    else
        tmp = 100.0d0
    end if
    code = tmp
end function

        public static double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = x * (100.0 / y);
	} else {
		tmp = 100.0;
	}
	return tmp;
}

        def code(x, y):
	tmp = 0
	if ((x * 100.0) / (x + y)) <= 5e-12:
		tmp = x * (100.0 / y)
	else:
		tmp = 100.0
	return tmp

        function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 100.0) / Float64(x + y)) <= 5e-12)
		tmp = Float64(x * Float64(100.0 / y));
	else
		tmp = 100.0;
	end
	return tmp
end

        function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * 100.0) / (x + y)) <= 5e-12)
		tmp = x * (100.0 / y);
	else
		tmp = 100.0;
	end
	tmp_2 = tmp;
end

        code[x_, y_] := If[LessEqual[N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], 5e-12], N[(x * N[(100.0 / y), $MachinePrecision]), $MachinePrecision], 100.0]

        \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\
\;\;\;\;x \cdot \frac{100}{y}\\

\mathbf{else}:\\
\;\;\;\;100\\


\end{array}
\end{array}
        Derivation
        1. Split input into 2 regimes

        2. if (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y)) < 4.9999999999999997e-12

          1. Initial program 99.7%

            \[\frac{x \cdot 100}{x + y} \]
          2. Add Preprocessing
          3. Step-by-step derivation

            1. lift-*.f64N/A

              \[\leadsto \frac{\color{blue}{x \cdot 100}}{x + y} \]

            2. lift-+.f64N/A

              \[\leadsto \frac{x \cdot 100}{\color{blue}{x + y}} \]

            3. lift-/.f64N/A

              \[\leadsto \color{blue}{\frac{x \cdot 100}{x + y}} \]

            4. associate-/l*N/A

              \[\leadsto \color{blue}{x \cdot \frac{100}{x + y}} \]

            5. lower-*.f64N/A

              \[\leadsto \color{blue}{x \cdot \frac{100}{x + y}} \]

            6. lower-/.f64N/A

              \[\leadsto x \cdot \color{blue}{\frac{100}{x + y}} \]

            7. +-commutativeN/A

              \[\leadsto x \cdot \frac{100}{\color{blue}{y + x}} \]

            8. lower-+.f6499.6

              \[\leadsto x \cdot \frac{100}{\color{blue}{y + x}} \]

          4. Applied rewrites99.6%

            \[\leadsto \color{blue}{x \cdot \frac{100}{y + x}} \]

          5. Taylor expanded in x around 0

            \[\leadsto x \cdot \frac{100}{\color{blue}{y}} \]
          6. Step-by-step derivation

            1. Applied rewrites99.3%

              \[\leadsto x \cdot \frac{100}{\color{blue}{y}} \]

            if 4.9999999999999997e-12 < (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y))

            1. Initial program 99.1%

              \[\frac{x \cdot 100}{x + y} \]
            2. Add Preprocessing

            3. Taylor expanded in x around inf

              \[\leadsto \color{blue}{100} \]
            4. Step-by-step derivation

              1. Applied rewrites98.2%

                \[\leadsto \color{blue}{100} \]
            5. Recombined 2 regimes into one program.
            6. Add Preprocessing

            Alternative 5: 97.1% accurate, 0.5× speedup?

            \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{y} \cdot 100\\ \mathbf{else}:\\ \;\;\;\;100\\ \end{array} \end{array} \]
            (FPCore (x y)
 :precision binary64
 (if (<= (/ (* x 100.0) (+ x y)) 5e-12) (* (/ x y) 100.0) 100.0))
            double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = (x / y) * 100.0;
	} else {
		tmp = 100.0;
	}
	return tmp;
}

            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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (((x * 100.0d0) / (x + y)) <= 5d-12) then
        tmp = (x / y) * 100.0d0
    else
        tmp = 100.0d0
    end if
    code = tmp
end function

            public static double code(double x, double y) {
	double tmp;
	if (((x * 100.0) / (x + y)) <= 5e-12) {
		tmp = (x / y) * 100.0;
	} else {
		tmp = 100.0;
	}
	return tmp;
}

            def code(x, y):
	tmp = 0
	if ((x * 100.0) / (x + y)) <= 5e-12:
		tmp = (x / y) * 100.0
	else:
		tmp = 100.0
	return tmp

            function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x * 100.0) / Float64(x + y)) <= 5e-12)
		tmp = Float64(Float64(x / y) * 100.0);
	else
		tmp = 100.0;
	end
	return tmp
end

            function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x * 100.0) / (x + y)) <= 5e-12)
		tmp = (x / y) * 100.0;
	else
		tmp = 100.0;
	end
	tmp_2 = tmp;
end

            code[x_, y_] := If[LessEqual[N[(N[(x * 100.0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], 5e-12], N[(N[(x / y), $MachinePrecision] * 100.0), $MachinePrecision], 100.0]

            \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\frac{x \cdot 100}{x + y} \leq 5 \cdot 10^{-12}:\\
\;\;\;\;\frac{x}{y} \cdot 100\\

\mathbf{else}:\\
\;\;\;\;100\\


\end{array}
\end{array}
            Derivation
            1. Split input into 2 regimes

            2. if (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y)) < 4.9999999999999997e-12

              1. Initial program 99.7%

                \[\frac{x \cdot 100}{x + y} \]
              2. Add Preprocessing

              3. Taylor expanded in x around 0

                \[\leadsto \color{blue}{100 \cdot \frac{x}{y}} \]
              4. Step-by-step derivation

                1. *-commutativeN/A

                  \[\leadsto \frac{x}{y} \cdot \color{blue}{100} \]

                2. lower-*.f64N/A

                  \[\leadsto \frac{x}{y} \cdot \color{blue}{100} \]

                3. lower-/.f6499.1

                  \[\leadsto \frac{x}{y} \cdot 100 \]

              5. Applied rewrites99.1%

                \[\leadsto \color{blue}{\frac{x}{y} \cdot 100} \]

              if 4.9999999999999997e-12 < (/.f64 (*.f64 x #s(literal 100 binary64)) (+.f64 x y))

              1. Initial program 99.1%

                \[\frac{x \cdot 100}{x + y} \]
              2. Add Preprocessing

              3. Taylor expanded in x around inf

                \[\leadsto \color{blue}{100} \]
              4. Step-by-step derivation

                1. Applied rewrites98.2%

                  \[\leadsto \color{blue}{100} \]
              5. Recombined 2 regimes into one program.
              6. Add Preprocessing

              Alternative 6: 49.6% accurate, 20.0× speedup?

              \[\begin{array}{l} \\ 100 \end{array} \]
              (FPCore (x y) :precision binary64 100.0)
              double code(double x, double y) {
	return 100.0;
}

              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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 100.0d0
end function

              public static double code(double x, double y) {
	return 100.0;
}

              def code(x, y):
	return 100.0

              function code(x, y)
	return 100.0
end

              function tmp = code(x, y)
	tmp = 100.0;
end

              code[x_, y_] := 100.0

              \begin{array}{l}

\\
100
\end{array}
              Derivation

              1. Initial program 99.4%

                \[\frac{x \cdot 100}{x + y} \]
              2. Add Preprocessing

              3. Taylor expanded in x around inf

                \[\leadsto \color{blue}{100} \]
              4. Step-by-step derivation

                1. Applied rewrites52.6%

                  \[\leadsto \color{blue}{100} \]
                2. Add Preprocessing

                Developer Target 1: 99.7% accurate, 0.6× speedup?

                \[\begin{array}{l} \\ \frac{x}{1} \cdot \frac{100}{x + y} \end{array} \]
                (FPCore (x y) :precision binary64 (* (/ x 1.0) (/ 100.0 (+ x y))))
                double code(double x, double y) {
	return (x / 1.0) * (100.0 / (x + y));
}

                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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / 1.0d0) * (100.0d0 / (x + y))
end function

                public static double code(double x, double y) {
	return (x / 1.0) * (100.0 / (x + y));
}

                def code(x, y):
	return (x / 1.0) * (100.0 / (x + y))

                function code(x, y)
	return Float64(Float64(x / 1.0) * Float64(100.0 / Float64(x + y)))
end

                function tmp = code(x, y)
	tmp = (x / 1.0) * (100.0 / (x + y));
end

                code[x_, y_] := N[(N[(x / 1.0), $MachinePrecision] * N[(100.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

                \begin{array}{l}

\\
\frac{x}{1} \cdot \frac{100}{x + y}
\end{array}

                Reproduce

                ?
                herbie shell --seed 2025056 
(FPCore (x y)
  :name "Development.Shake.Progress:message from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (* (/ x 1) (/ 100 (+ x y))))

  (/ (* x 100.0) (+ x y)))