Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Percentage Accurate: 77.7% → 95.8%

Time: 8.4s

Alternatives: 7

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * log((x / y))) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left({\log x}^{2} - {\log y}^{2}\right) \cdot x}{\log \left(y \cdot x\right)} - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (if (<= x 2.65e+164)
     (- (/ (* (- (pow (log x) 2.0) (pow (log y) 2.0)) x) (log (* y x))) z)
     (* (- (log x) (log y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else if (x <= 2.65e+164) {
		tmp = (((pow(log(x), 2.0) - pow(log(y), 2.0)) * x) / log((y * x))) - z;
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5d-310)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else if (x <= 2.65d+164) then
        tmp = ((((log(x) ** 2.0d0) - (log(y) ** 2.0d0)) * x) / log((y * x))) - z
    else
        tmp = (log(x) - log(y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else if (x <= 2.65e+164) {
		tmp = (((Math.pow(Math.log(x), 2.0) - Math.pow(Math.log(y), 2.0)) * x) / Math.log((y * x))) - z;
	} else {
		tmp = (Math.log(x) - Math.log(y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5e-310:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	elif x <= 2.65e+164:
		tmp = (((math.pow(math.log(x), 2.0) - math.pow(math.log(y), 2.0)) * x) / math.log((y * x))) - z
	else:
		tmp = (math.log(x) - math.log(y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	elseif (x <= 2.65e+164)
		tmp = Float64(Float64(Float64(Float64((log(x) ^ 2.0) - (log(y) ^ 2.0)) * x) / log(Float64(y * x))) - z);
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	elseif (x <= 2.65e+164)
		tmp = ((((log(x) ^ 2.0) - (log(y) ^ 2.0)) * x) / log((y * x))) - z;
	else
		tmp = (log(x) - log(y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 2.65e+164], N[(N[(N[(N[(N[Power[N[Log[x], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[y], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 70.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -4.999999999999985e-310 < x < 2.6500000000000002e164

   1. Initial program 87.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. log-div N/A

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

      6. flip-- N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. pow2 N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-log.f64 N/A

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

      17. +-commutative N/A

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

      18. sum-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if 2.6500000000000002e164 < x

   1. Initial program 69.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 94.9

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

   5. Applied rewrites 94.9%

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

Alternative 2: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0 - z\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{+291}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (- (* x t_0) z)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e+291)))
     (- z)
     (fma t_0 x (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = (x * t_0) - z;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e+291)) {
		tmp = -z;
	} else {
		tmp = fma(t_0, x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(Float64(x * t_0) - z)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e+291))
		tmp = Float64(-z);
	else
		tmp = fma(t_0, x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e+291]], $MachinePrecision]], (-z), N[(t$95$0 * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -inf.0 or 1.9999999999999999e291 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 9.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 53.8

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 53.8%

      \[\leadsto \color{blue}{-z} \]

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 1.9999999999999999e291

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.9%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) - z \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) - z \leq 2 \cdot 10^{+291}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0 - z\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (- (* x t_0) z)))
   (if (<= t_1 (- INFINITY))
     (* (- (log (- x)) (log (- y))) x)
     (if (<= t_1 2e+291) (fma t_0 x (- z)) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = (x * t_0) - z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (t_1 <= 2e+291) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(Float64(x * t_0) - z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (t_1 <= 2e+291)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, 2e+291], N[(t$95$0 * x + (-z)), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -inf.0

   1. Initial program 5.6%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.5%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 1.9999999999999999e291

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.9%

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

   if 1.9999999999999999e291 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 62.5

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 62.5%

      \[\leadsto \color{blue}{-z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-227}:\\ \;\;\;\;\mathsf{fma}\left({\left(\sqrt{\log x - \log y}\right)}^{2}, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (if (<= y 1.85e-227)
     (fma (pow (sqrt (- (log x) (log y))) 2.0) x (- z))
     (- (* x (log (/ x y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else if (y <= 1.85e-227) {
		tmp = fma(pow(sqrt((log(x) - log(y))), 2.0), x, -z);
	} else {
		tmp = (x * log((x / y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	elseif (y <= 1.85e-227)
		tmp = fma((sqrt(Float64(log(x) - log(y))) ^ 2.0), x, Float64(-z));
	else
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.85e-227], N[(N[Power[N[Sqrt[N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * x + (-z)), $MachinePrecision], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 70.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -4.999999999999985e-310 < y < 1.84999999999999989e-227

   1. Initial program 50.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 50.3%

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

   7. Applied rewrites 50.0%

      \[\leadsto \mathsf{fma}\left({\left(\sqrt{\log \left(\frac{x}{y}\right)}\right)}^{2}, x, -z\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.5%

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

   if 1.84999999999999989e-227 < y

   1. Initial program 89.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-280}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-203}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+169}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e-280)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (if (<= x 1.55e-203)
     (- z)
     (if (<= x 1.3e+169)
       (- (* x (log (/ x y))) z)
       (* (- (log x) (log y)) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-280) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else if (x <= 1.55e-203) {
		tmp = -z;
	} else if (x <= 1.3e+169) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3d-280)) then
        tmp = (x * (log(-x) - log(-y))) - z
    else if (x <= 1.55d-203) then
        tmp = -z
    else if (x <= 1.3d+169) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (log(x) - log(y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e-280) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else if (x <= 1.55e-203) {
		tmp = -z;
	} else if (x <= 1.3e+169) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (Math.log(x) - Math.log(y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e-280:
		tmp = (x * (math.log(-x) - math.log(-y))) - z
	elif x <= 1.55e-203:
		tmp = -z
	elif x <= 1.3e+169:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (math.log(x) - math.log(y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e-280)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	elseif (x <= 1.55e-203)
		tmp = Float64(-z);
	elseif (x <= 1.3e+169)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e-280)
		tmp = (x * (log(-x) - log(-y))) - z;
	elseif (x <= 1.55e-203)
		tmp = -z;
	elseif (x <= 1.3e+169)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (log(x) - log(y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e-280], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.55e-203], (-z), If[LessEqual[x, 1.3e+169], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.99999999999999987e-280

   1. Initial program 71.3%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -2.99999999999999987e-280 < x < 1.54999999999999989e-203

   1. Initial program 74.2%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 100.0

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{-z} \]

   if 1.54999999999999989e-203 < x < 1.3e169

   1. Initial program 90.1%

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

   if 1.3e169 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 94.7

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

   5. Applied rewrites 94.7%

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

Alternative 6: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-107} \lor \neg \left(z \leq 4.5 \cdot 10^{-57}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.4e-107) (not (<= z 4.5e-57))) (- z) (* (log (/ x y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.4e-107) || !(z <= 4.5e-57)) {
		tmp = -z;
	} else {
		tmp = log((x / y)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-4.4d-107)) .or. (.not. (z <= 4.5d-57))) then
        tmp = -z
    else
        tmp = log((x / y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.4e-107) || !(z <= 4.5e-57)) {
		tmp = -z;
	} else {
		tmp = Math.log((x / y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.4e-107) or not (z <= 4.5e-57):
		tmp = -z
	else:
		tmp = math.log((x / y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.4e-107) || !(z <= 4.5e-57))
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(x / y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.4e-107) || ~((z <= 4.5e-57)))
		tmp = -z;
	else
		tmp = log((x / y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.4e-107], N[Not[LessEqual[z, 4.5e-57]], $MachinePrecision]], (-z), N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.40000000000000025e-107 or 4.49999999999999973e-57 < z

   1. Initial program 79.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 73.2

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 73.2%

      \[\leadsto \color{blue}{-z} \]

   if -4.40000000000000025e-107 < z < 4.49999999999999973e-57

   1. Initial program 73.0%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 61.7

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

   5. Applied rewrites 61.7%

      \[\leadsto \color{blue}{\log \left(\frac{x}{y}\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-107} \lor \neg \left(z \leq 4.5 \cdot 10^{-57}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.2% accurate, 40.0× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 77.2%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 56.0

      \[\leadsto \color{blue}{-z} \]

5. Applied rewrites 56.0%

   \[\leadsto \color{blue}{-z} \]
6. Add Preprocessing

Developer Target 1: 88.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< y 7.595077799083773e-308)
   (- (* x (log (/ x y))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * log((x / y))) - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y < 7.595077799083773d-308) then
        tmp = (x * log((x / y))) - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y < 7.595077799083773e-308) {
		tmp = (x * Math.log((x / y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y < 7.595077799083773e-308:
		tmp = (x * math.log((x / y))) - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y < 7.595077799083773e-308)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y < 7.595077799083773e-308)
		tmp = (x * log((x / y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[y, 7.595077799083773e-308], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024352 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y 7595077799083773/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z)))

  (- (* x (log (/ x y))) z))