Logistic regression 2

Percentage Accurate: 99.2% → 99.3%
Time: 5.5s
Alternatives: 11
Speedup: 1.6×

Specification

?
\[\begin{array}{l} \\ \log \left(1 + e^{x}\right) - x \cdot y \end{array} \]
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
double code(double x, double y) {
	return log((1.0 + exp(x))) - (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 = log((1.0d0 + exp(x))) - (x * y)
end function

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\log \left(1 + e^{x}\right) - x \cdot 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 11 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: 99.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \log \left(1 + e^{x}\right) - x \cdot y \end{array} \]
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
double code(double x, double y) {
	return log((1.0 + exp(x))) - (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 = log((1.0d0 + exp(x))) - (x * y)
end function

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\log \left(1 + e^{x}\right) - x \cdot y
\end{array}

Alternative 1: 99.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right) \end{array} \]
(FPCore (x y) :precision binary64 (fma (- y) x (log1p (exp x))))
double code(double x, double y) {
	return fma(-y, x, log1p(exp(x)));
}

function code(x, y)
	return fma(Float64(-y), x, log1p(exp(x)))
end

code[x_, y_] := N[((-y) * x + N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right)
\end{array}
Derivation

  1. Initial program 98.8%

    \[\log \left(1 + e^{x}\right) - x \cdot y \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift--.f64N/A

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

    2. lift-log.f64N/A

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

    3. lift-+.f64N/A

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

    4. lift-exp.f64N/A

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

    5. lift-*.f64N/A

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

    6. fp-cancel-sub-sign-invN/A

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

    7. mul-1-negN/A

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

    8. associate-*r*N/A

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

    9. +-commutativeN/A

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

    10. *-commutativeN/A

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

    11. associate-*r*N/A

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

    12. lower-fma.f64N/A

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

    13. mul-1-negN/A

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

    14. lower-neg.f64N/A

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

    15. lower-log1p.f64N/A

      \[\leadsto \mathsf{fma}\left(-y, x, \color{blue}{\mathsf{log1p}\left(e^{x}\right)}\right) \]

    16. lift-exp.f6499.2

      \[\leadsto \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(\color{blue}{e^{x}}\right)\right) \]

  4. Applied rewrites99.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right)} \]
  5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-35} \lor \neg \left(t\_0 \leq 200\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))))
   (if (or (<= t_0 5e-35) (not (<= t_0 200.0)))
     (* (- x) y)
     (fma 0.5 x (log 2.0)))))
double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0)) {
		tmp = -x * y;
	} else {
		tmp = fma(0.5, x, log(2.0));
	}
	return tmp;
}

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(0.5, x, log(2.0));
	end
	return tmp
end

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-35], N[Not[LessEqual[t$95$0, 200.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[(0.5 * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\
\mathbf{if}\;t\_0 \leq 5 \cdot 10^{-35} \lor \neg \left(t\_0 \leq 200\right):\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\


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

  2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 4.99999999999999964e-35 or 200 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

    1. Initial program 97.5%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f6497.3

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites97.3%

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

    if 4.99999999999999964e-35 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 200

    1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
    4. Step-by-step derivation

      1. lower-log1p.f64N/A

        \[\leadsto \mathsf{log1p}\left(e^{x}\right) \]

      2. lift-exp.f6497.5

        \[\leadsto \mathsf{log1p}\left(e^{x}\right) \]

    5. Applied rewrites97.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

    6. Taylor expanded in x around 0

      \[\leadsto \log 2 + \color{blue}{\frac{1}{2} \cdot x} \]
    7. Step-by-step derivation

      1. +-commutativeN/A

        \[\leadsto \frac{1}{2} \cdot x + \log 2 \]

      2. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \log 2\right) \]

      3. lift-log.f6495.4

        \[\leadsto \mathsf{fma}\left(0.5, x, \log 2\right) \]

    8. Applied rewrites95.4%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x}, \log 2\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification96.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5 \cdot 10^{-35} \lor \neg \left(\log \left(1 + e^{x}\right) - x \cdot y \leq 200\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, \log 2\right)\\ \end{array} \]
  5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-35} \lor \neg \left(t\_0 \leq 200\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x - -1\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))))
   (if (or (<= t_0 5e-35) (not (<= t_0 200.0)))
     (* (- x) y)
     (log1p (- x -1.0)))))
double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0)) {
		tmp = -x * y;
	} else {
		tmp = log1p((x - -1.0));
	}
	return tmp;
}

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0)) {
		tmp = -x * y;
	} else {
		tmp = Math.log1p((x - -1.0));
	}
	return tmp;
}

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	tmp = 0
	if (t_0 <= 5e-35) or not (t_0 <= 200.0):
		tmp = -x * y
	else:
		tmp = math.log1p((x - -1.0))
	return tmp

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = log1p(Float64(x - -1.0));
	end
	return tmp
end

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-35], N[Not[LessEqual[t$95$0, 200.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[Log[1 + N[(x - -1.0), $MachinePrecision]], $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\
\mathbf{if}\;t\_0 \leq 5 \cdot 10^{-35} \lor \neg \left(t\_0 \leq 200\right):\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{log1p}\left(x - -1\right)\\


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

  2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 4.99999999999999964e-35 or 200 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

    1. Initial program 97.5%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f6497.3

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites97.3%

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

    if 4.99999999999999964e-35 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 200

    1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
    4. Step-by-step derivation

      1. lower-log1p.f64N/A

        \[\leadsto \mathsf{log1p}\left(e^{x}\right) \]

      2. lift-exp.f6497.5

        \[\leadsto \mathsf{log1p}\left(e^{x}\right) \]

    5. Applied rewrites97.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

    6. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(1 + x\right) \]
    7. Step-by-step derivation

      1. +-commutativeN/A

        \[\leadsto \mathsf{log1p}\left(x + 1\right) \]

      2. metadata-evalN/A

        \[\leadsto \mathsf{log1p}\left(x + 1 \cdot 1\right) \]

      3. fp-cancel-sign-sub-invN/A

        \[\leadsto \mathsf{log1p}\left(x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right) \]

      4. metadata-evalN/A

        \[\leadsto \mathsf{log1p}\left(x - -1 \cdot 1\right) \]

      5. metadata-evalN/A

        \[\leadsto \mathsf{log1p}\left(x - -1\right) \]

      6. lower--.f6495.2

        \[\leadsto \mathsf{log1p}\left(x - -1\right) \]

    8. Applied rewrites95.2%

      \[\leadsto \mathsf{log1p}\left(x - -1\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification96.2%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5 \cdot 10^{-35} \lor \neg \left(\log \left(1 + e^{x}\right) - x \cdot y \leq 200\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x - -1\right)\\ \end{array} \]
  5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.4× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-35} \lor \neg \left(t\_0 \leq 200\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))))
   (if (or (<= t_0 5e-35) (not (<= t_0 200.0))) (* (- x) y) (log 2.0))))
double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0)) {
		tmp = -x * y;
	} else {
		tmp = log(2.0);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = log((1.0d0 + exp(x))) - (x * y)
    if ((t_0 <= 5d-35) .or. (.not. (t_0 <= 200.0d0))) then
        tmp = -x * y
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double tmp;
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0)) {
		tmp = -x * y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	tmp = 0
	if (t_0 <= 5e-35) or not (t_0 <= 200.0):
		tmp = -x * y
	else:
		tmp = math.log(2.0)
	return tmp

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	tmp = 0.0
	if ((t_0 <= 5e-35) || !(t_0 <= 200.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = log(2.0);
	end
	return tmp
end

function tmp_2 = code(x, y)
	t_0 = log((1.0 + exp(x))) - (x * y);
	tmp = 0.0;
	if ((t_0 <= 5e-35) || ~((t_0 <= 200.0)))
		tmp = -x * y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e-35], N[Not[LessEqual[t$95$0, 200.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[Log[2.0], $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\
\mathbf{if}\;t\_0 \leq 5 \cdot 10^{-35} \lor \neg \left(t\_0 \leq 200\right):\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\log 2\\


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

  2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 4.99999999999999964e-35 or 200 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

    1. Initial program 97.5%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f6497.3

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites97.3%

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

    if 4.99999999999999964e-35 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 200

    1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} \]
    4. Step-by-step derivation

      1. lower-log.f6494.0

        \[\leadsto \log 2 \]

    5. Applied rewrites94.0%

      \[\leadsto \color{blue}{\log 2} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification95.5%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5 \cdot 10^{-35} \lor \neg \left(\log \left(1 + e^{x}\right) - x \cdot y \leq 200\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
  5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \log 2\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6)
   (* (- x) y)
   (fma
    (fma (fma (* x x) -0.005208333333333333 0.125) x (- 0.5 y))
    x
    (log 2.0))))
double code(double x, double y) {
	double tmp;
	if (x <= -2.6) {
		tmp = -x * y;
	} else {
		tmp = fma(fma(fma((x * x), -0.005208333333333333, 0.125), x, (0.5 - y)), x, log(2.0));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(fma(fma(Float64(x * x), -0.005208333333333333, 0.125), x, Float64(0.5 - y)), x, log(2.0));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[x, -2.6], N[((-x) * y), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * x + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -2.6:\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \log 2\right)\\


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

  2. if x < -2.60000000000000009

    1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f6498.8

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites98.8%

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

    if -2.60000000000000009 < x

    1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)} \]
    4. Step-by-step derivation

      1. +-commutativeN/A

        \[\leadsto x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right) + \color{blue}{\log 2} \]

      2. *-commutativeN/A

        \[\leadsto \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right) \cdot x + \log \color{blue}{2} \]

      3. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y, \color{blue}{x}, \log 2\right) \]

      4. +-commutativeN/A

        \[\leadsto \mathsf{fma}\left(\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \frac{1}{2}\right) - y, x, \log 2\right) \]

      5. associate--l+N/A

        \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) + \left(\frac{1}{2} - y\right), x, \log 2\right) \]

      6. *-commutativeN/A

        \[\leadsto \mathsf{fma}\left(\left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) \cdot x + \left(\frac{1}{2} - y\right), x, \log 2\right) \]

      7. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      8. +-commutativeN/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{192} \cdot {x}^{2} + \frac{1}{8}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      9. *-commutativeN/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \frac{-1}{192} + \frac{1}{8}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      10. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-1}{192}, \frac{1}{8}\right), x, \frac{1}{2} - y\right), x, \log 2\right) \]

      11. unpow2N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{192}, \frac{1}{8}\right), x, \frac{1}{2} - y\right), x, \log 2\right) \]

      12. lower-*.f64N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{192}, \frac{1}{8}\right), x, \frac{1}{2} - y\right), x, \log 2\right) \]

      13. lower--.f64N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-1}{192}, \frac{1}{8}\right), x, \frac{1}{2} - y\right), x, \log 2\right) \]

      14. lower-log.f6498.2

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \log 2\right) \]

    5. Applied rewrites98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.005208333333333333, 0.125\right), x, 0.5 - y\right), x, \log 2\right)} \]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 6: 98.4% accurate, 1.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+19) (* (- x) y) (fma (fma 0.125 x (- 0.5 y)) x (log 2.0))))
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+19) {
		tmp = -x * y;
	} else {
		tmp = fma(fma(0.125, x, (0.5 - y)), x, log(2.0));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+19)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(fma(0.125, x, Float64(0.5 - y)), x, log(2.0));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[x, -1.35e+19], N[((-x) * y), $MachinePrecision], N[(N[(0.125 * x + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1.35 \cdot 10^{+19}:\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)\\


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

  2. if x < -1.35e19

    1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f64100.0

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites100.0%

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

    if -1.35e19 < x

    1. Initial program 98.3%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)} \]
    4. Step-by-step derivation

      1. +-commutativeN/A

        \[\leadsto x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) + \color{blue}{\log 2} \]

      2. *-commutativeN/A

        \[\leadsto \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot x + \log \color{blue}{2} \]

      3. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y, \color{blue}{x}, \log 2\right) \]

      4. +-commutativeN/A

        \[\leadsto \mathsf{fma}\left(\left(\frac{1}{8} \cdot x + \frac{1}{2}\right) - y, x, \log 2\right) \]

      5. associate--l+N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{8} \cdot x + \left(\frac{1}{2} - y\right), x, \log 2\right) \]

      6. lower-fma.f64N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      7. lower--.f64N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, x, \frac{1}{2} - y\right), x, \log 2\right) \]

      8. lower-log.f6497.5

        \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right) \]

    5. Applied rewrites97.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, x, 0.5 - y\right), x, \log 2\right)} \]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 7: 99.0% accurate, 1.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35) (* (- x) y) (fma (- 0.5 y) x (log 2.0))))
double code(double x, double y) {
	double tmp;
	if (x <= -1.35) {
		tmp = -x * y;
	} else {
		tmp = fma((0.5 - y), x, log(2.0));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(Float64(0.5 - y), x, log(2.0));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[x, -1.35], N[((-x) * y), $MachinePrecision], N[(N[(0.5 - y), $MachinePrecision] * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1.35:\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.5 - y, x, \log 2\right)\\


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

  2. if x < -1.3500000000000001

    1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f6498.8

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites98.8%

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

    if -1.3500000000000001 < x

    1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around 0

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

      1. +-commutativeN/A

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

      2. *-commutativeN/A

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

      3. lower-fma.f64N/A

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

      4. lower--.f64N/A

        \[\leadsto \mathsf{fma}\left(\frac{1}{2} - y, x, \log 2\right) \]

      5. lower-log.f6497.4

        \[\leadsto \mathsf{fma}\left(0.5 - y, x, \log 2\right) \]

    5. Applied rewrites97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - y, x, \log 2\right)} \]
  3. Recombined 2 regimes into one program.
  4. Add Preprocessing

Alternative 8: 97.8% accurate, 1.8× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(1\right)\right)\\ \end{array} \end{array} \]
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+19) (* (- x) y) (fma (- y) x (log1p 1.0))))
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+19) {
		tmp = -x * y;
	} else {
		tmp = fma(-y, x, log1p(1.0));
	}
	return tmp;
}

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+19)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(Float64(-y), x, log1p(1.0));
	end
	return tmp
end

code[x_, y_] := If[LessEqual[x, -1.35e+19], N[((-x) * y), $MachinePrecision], N[((-y) * x + N[Log[1 + 1.0], $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1.35 \cdot 10^{+19}:\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(1\right)\right)\\


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

  2. if x < -1.35e19

    1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing

    3. Taylor expanded in x around inf

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

      1. associate-*r*N/A

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

      2. mul-1-negN/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

      3. lower-*.f64N/A

        \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

      4. lower-neg.f64100.0

        \[\leadsto \left(-x\right) \cdot y \]

    5. Applied rewrites100.0%

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

    if -1.35e19 < x

    1. Initial program 98.3%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
    2. Add Preprocessing
    3. Step-by-step derivation

      1. lift--.f64N/A

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

      2. lift-log.f64N/A

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

      3. lift-+.f64N/A

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

      4. lift-exp.f64N/A

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

      5. lift-*.f64N/A

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

      6. fp-cancel-sub-sign-invN/A

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

      7. mul-1-negN/A

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

      8. associate-*r*N/A

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

      9. +-commutativeN/A

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

      10. *-commutativeN/A

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

      11. associate-*r*N/A

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

      12. lower-fma.f64N/A

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

      13. mul-1-negN/A

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

      14. lower-neg.f64N/A

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

      15. lower-log1p.f64N/A

        \[\leadsto \mathsf{fma}\left(-y, x, \color{blue}{\mathsf{log1p}\left(e^{x}\right)}\right) \]

      16. lift-exp.f6498.9

        \[\leadsto \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(\color{blue}{e^{x}}\right)\right) \]

    4. Applied rewrites98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right)} \]

    5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(\color{blue}{1}\right)\right) \]
    6. Step-by-step derivation

      1. Applied rewrites95.8%

        \[\leadsto \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(\color{blue}{1}\right)\right) \]
    7. Recombined 2 regimes into one program.
    8. Add Preprocessing

    Alternative 9: 97.8% accurate, 1.8× speedup?

    \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]
    (FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+19) (* (- x) y) (- (log 2.0) (* x y))))
    double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+19) {
		tmp = -x * y;
	} else {
		tmp = log(2.0) - (x * y);
	}
	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 <= (-1.35d+19)) then
        tmp = -x * y
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

    public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+19) {
		tmp = -x * y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

    def code(x, y):
	tmp = 0
	if x <= -1.35e+19:
		tmp = -x * y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

    function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+19)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

    function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e+19)
		tmp = -x * y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

    code[x_, y_] := If[LessEqual[x, -1.35e+19], N[((-x) * y), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

    \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -1.35 \cdot 10^{+19}:\\
\;\;\;\;\left(-x\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\log 2 - x \cdot y\\


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

    2. if x < -1.35e19

      1. Initial program 100.0%

        \[\log \left(1 + e^{x}\right) - x \cdot y \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

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

        1. associate-*r*N/A

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

        2. mul-1-negN/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

        3. lower-*.f64N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

        4. lower-neg.f64100.0

          \[\leadsto \left(-x\right) \cdot y \]

      5. Applied rewrites100.0%

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

      if -1.35e19 < x

      1. Initial program 98.3%

        \[\log \left(1 + e^{x}\right) - x \cdot y \]
      2. Add Preprocessing

      3. Taylor expanded in x around 0

        \[\leadsto \log \color{blue}{2} - x \cdot y \]
      4. Step-by-step derivation

        1. Applied rewrites95.8%

          \[\leadsto \log \color{blue}{2} - x \cdot y \]
      5. Recombined 2 regimes into one program.
      6. Add Preprocessing

      Alternative 10: 50.0% accurate, 26.5× speedup?

      \[\begin{array}{l} \\ \left(-x\right) \cdot y \end{array} \]
      (FPCore (x y) :precision binary64 (* (- x) y))
      double code(double x, double y) {
	return -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 * y
end function

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

      def code(x, y):
	return -x * y

      function code(x, y)
	return Float64(Float64(-x) * y)
end

      function tmp = code(x, y)
	tmp = -x * y;
end

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

      \begin{array}{l}

\\
\left(-x\right) \cdot y
\end{array}
      Derivation

      1. Initial program 98.8%

        \[\log \left(1 + e^{x}\right) - x \cdot y \]
      2. Add Preprocessing

      3. Taylor expanded in x around inf

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

        1. associate-*r*N/A

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

        2. mul-1-negN/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot y \]

        3. lower-*.f64N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) \cdot \color{blue}{y} \]

        4. lower-neg.f6447.2

          \[\leadsto \left(-x\right) \cdot y \]

      5. Applied rewrites47.2%

        \[\leadsto \color{blue}{\left(-x\right) \cdot y} \]
      6. Add Preprocessing

      Alternative 11: 3.5% accurate, 35.3× speedup?

      \[\begin{array}{l} \\ 0.5 \cdot x \end{array} \]
      (FPCore (x y) :precision binary64 (* 0.5 x))
      double code(double x, double y) {
	return 0.5 * 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 = 0.5d0 * x
end function

      public static double code(double x, double y) {
	return 0.5 * x;
}

      def code(x, y):
	return 0.5 * x

      function code(x, y)
	return Float64(0.5 * x)
end

      function tmp = code(x, y)
	tmp = 0.5 * x;
end

      code[x_, y_] := N[(0.5 * x), $MachinePrecision]

      \begin{array}{l}

\\
0.5 \cdot x
\end{array}
      Derivation

      1. Initial program 98.8%

        \[\log \left(1 + e^{x}\right) - x \cdot y \]
      2. Add Preprocessing

      3. Taylor expanded in y around 0

        \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
      4. Step-by-step derivation

        1. lower-log1p.f64N/A

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \]

        2. lift-exp.f6453.9

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \]

      5. Applied rewrites53.9%

        \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

      6. Taylor expanded in x around 0

        \[\leadsto \log 2 + \color{blue}{\frac{1}{2} \cdot x} \]
      7. Step-by-step derivation

        1. +-commutativeN/A

          \[\leadsto \frac{1}{2} \cdot x + \log 2 \]

        2. lower-fma.f64N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, x, \log 2\right) \]

        3. lift-log.f6452.4

          \[\leadsto \mathsf{fma}\left(0.5, x, \log 2\right) \]

      8. Applied rewrites52.4%

        \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{x}, \log 2\right) \]

      9. Taylor expanded in x around inf

        \[\leadsto \frac{1}{2} \cdot x \]
      10. Step-by-step derivation

        1. lower-*.f643.2

          \[\leadsto 0.5 \cdot x \]

      11. Applied rewrites3.2%

        \[\leadsto 0.5 \cdot x \]
      12. Add Preprocessing

      Developer Target 1: 99.9% accurate, 0.9× speedup?

      \[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]
      (FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))
      double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	}
	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 <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

      public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

      def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

      function code(x, y)
	tmp = 0.0
	if (x <= 0.0)
		tmp = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y));
	else
		tmp = Float64(log(Float64(1.0 + exp(Float64(-x)))) - Float64(Float64(-x) * Float64(1.0 - y)));
	end
	return tmp
end

      function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = log((1.0 + exp(x))) - (x * y);
	else
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	end
	tmp_2 = tmp;
end

      code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

      \begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 0:\\
\;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\

\mathbf{else}:\\
\;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\


\end{array}
\end{array}

      Reproduce

      ?
      herbie shell --seed 2025051 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

  (- (log (+ 1.0 (exp x))) (* x y)))