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

Percentage Accurate: 77.0% → 99.5%

Time: 4.4s

Alternatives: 7

Speedup: 0.6×

Specification

Math

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

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: 77.0% accurate, 1.0× speedup

Math

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

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = (x * (log(-x) - log(-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 <= (-2d-310)) then
        tmp = (x * (log(-x) - log(-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 <= -2e-310) {
		tmp = (x * (Math.log(-x) - Math.log(-y))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2e-310:
		tmp = (x * (math.log(-x) - math.log(-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 <= -2e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-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 <= -2e-310)
		tmp = (x * (log(-x) - log(-y))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2e-310], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.999999999999994e-310

   1. Initial program 77.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. 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-unsound--.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-unsound-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-unsound-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 49.2%

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

   3. Applied rewrites 49.2%

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

   if -1.999999999999994e-310 < y

   1. Initial program 77.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. 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. log-div N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-log.f64 N/A

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

      6. lower-unsound-log.f64 50.3%

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

   3. Applied rewrites 50.3%

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

Alternative 2: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-121}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e+152)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -4.5e-121)
     (- (* x (log (/ x y))) z)
     (if (<= x 1.1e-307) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+152) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -4.5e-121) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= 1.1e-307) {
		tmp = -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 (x <= (-4.5d+152)) then
        tmp = x * (log(-x) - log(-y))
    else if (x <= (-4.5d-121)) then
        tmp = (x * log((x / y))) - z
    else if (x <= 1.1d-307) then
        tmp = -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 (x <= -4.5e+152) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (x <= -4.5e-121) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= 1.1e-307) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4.5e+152:
		tmp = x * (math.log(-x) - math.log(-y))
	elif x <= -4.5e-121:
		tmp = (x * math.log((x / y))) - z
	elif x <= 1.1e-307:
		tmp = -z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.5e+152)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -4.5e-121)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= 1.1e-307)
		tmp = Float64(-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 (x <= -4.5e+152)
		tmp = x * (log(-x) - log(-y));
	elseif (x <= -4.5e-121)
		tmp = (x * log((x / y))) - z;
	elseif (x <= 1.1e-307)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.5e+152], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-121], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.1e-307], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.5000000000000001e152

   1. Initial program 77.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 40.0%

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

   4. Applied rewrites 40.0%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-unsound-log.f64 N/A

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

      9. lower-neg.f64 25.4%

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

   6. Applied rewrites 25.4%

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

   if -4.5000000000000001e152 < x < -4.5000000000000003e-121

   1. Initial program 77.0%

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

   if -4.5000000000000003e-121 < x < 1.1e-307

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. neg-log N/A

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

      14. lower-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lower-/.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto -\color{blue}{z} \]
   5. Step-by-step derivation

   6. Applied rewrites 49.3%

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

   if 1.1e-307 < x

   1. Initial program 77.0%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   2. 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. log-div N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-log.f64 N/A

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

      6. lower-unsound-log.f64 50.3%

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

   3. Applied rewrites 50.3%

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

Alternative 3: 86.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log (/ x y))) z)))
   (if (<= t_0 (- INFINITY))
     (* x (- (log (- x)) (log (- y))))
     (if (<= t_0 2e+297) t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log((x / y))) - z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = x * (log(-x) - log(-y));
	} else if (t_0 <= 2e+297) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log((x / y))) - z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = x * (Math.log(-x) - Math.log(-y));
	} else if (t_0 <= 2e+297) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log((x / y))) - z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = x * (math.log(-x) - math.log(-y))
	elif t_0 <= 2e+297:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(Float64(x / y))) - z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (t_0 <= 2e+297)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log((x / y))) - z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = x * (log(-x) - log(-y));
	elseif (t_0 <= 2e+297)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+297], t$95$0, (-z)]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 77.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 40.0%

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

   4. Applied rewrites 40.0%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower-unsound--.f64 N/A

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

      6. lower-unsound-log.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-unsound-log.f64 N/A

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

      9. lower-neg.f64 25.4%

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

   6. Applied rewrites 25.4%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 2e297

   1. Initial program 77.0%

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

   if 2e297 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. neg-log N/A

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

      14. lower-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lower-/.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto -\color{blue}{z} \]
   5. Step-by-step derivation

   6. Applied rewrites 49.3%

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

Alternative 4: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+297}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log (/ x y))) z)))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+297) t_0 (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x * log((x / y))) - z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+297) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * Math.log((x / y))) - z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+297) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log((x / y))) - z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+297:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * log(Float64(x / y))) - z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+297)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * log((x / y))) - z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+297)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+297], t$95$0, (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -inf.0 or 2e297 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. neg-log N/A

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

      14. lower-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lower-/.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto -\color{blue}{z} \]
   5. Step-by-step derivation

   6. Applied rewrites 49.3%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 2e297

   1. Initial program 77.0%

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

Alternative 5: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+18}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;-x \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.6e+18) (- z) (if (<= z 1.8e-36) (- (* x (log (/ y x)))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.6e+18) {
		tmp = -z;
	} else if (z <= 1.8e-36) {
		tmp = -(x * log((y / x)));
	} else {
		tmp = -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 (z <= (-5.6d+18)) then
        tmp = -z
    else if (z <= 1.8d-36) then
        tmp = -(x * log((y / x)))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.6e+18) {
		tmp = -z;
	} else if (z <= 1.8e-36) {
		tmp = -(x * Math.log((y / x)));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.6e+18:
		tmp = -z
	elif z <= 1.8e-36:
		tmp = -(x * math.log((y / x)))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.6e+18)
		tmp = Float64(-z);
	elseif (z <= 1.8e-36)
		tmp = Float64(-Float64(x * log(Float64(y / x))));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.6e+18)
		tmp = -z;
	elseif (z <= 1.8e-36)
		tmp = -(x * log((y / x)));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.6e+18], (-z), If[LessEqual[z, 1.8e-36], (-N[(x * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -5.6e18 or 1.80000000000000016e-36 < z

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. neg-log N/A

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

      14. lower-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lower-/.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto -\color{blue}{z} \]
   5. Step-by-step derivation

   6. Applied rewrites 49.3%

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

   if -5.6e18 < z < 1.80000000000000016e-36

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. neg-log N/A

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

      14. lower-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lower-/.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 40.5%

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

   6. Applied rewrites 40.5%

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

Alternative 6: 66.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+18}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.6e+18) (- z) (if (<= z 1.8e-36) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.6e+18) {
		tmp = -z;
	} else if (z <= 1.8e-36) {
		tmp = x * log((x / y));
	} else {
		tmp = -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 (z <= (-5.6d+18)) then
        tmp = -z
    else if (z <= 1.8d-36) then
        tmp = x * log((x / y))
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.6e+18) {
		tmp = -z;
	} else if (z <= 1.8e-36) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -5.6e+18:
		tmp = -z
	elif z <= 1.8e-36:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.6e+18)
		tmp = Float64(-z);
	elseif (z <= 1.8e-36)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -5.6e+18)
		tmp = -z;
	elseif (z <= 1.8e-36)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.6e+18], (-z), If[LessEqual[z, 1.8e-36], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -5.6e18 or 1.80000000000000016e-36 < z

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lower-neg.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-neg-out N/A

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-log.f64 N/A

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

      13. neg-log N/A

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

      14. lower-log.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. div-flip-rev N/A

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

      17. lower-/.f64 76.9%

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in x around 0

      \[\leadsto -\color{blue}{z} \]
   5. Step-by-step derivation

   6. Applied rewrites 49.3%

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

   if -5.6e18 < z < 1.80000000000000016e-36

   1. Initial program 77.0%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 40.0%

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

   4. Applied rewrites 40.0%

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

Alternative 7: 49.3% accurate, 8.0× speedup

Math

\[-z \]

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 77.0%

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

   1. lift--.f64 N/A

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

   2. sub-negate-rev N/A

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

   3. lower-neg.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. sub-flip N/A

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

   6. distribute-neg-in N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. remove-double-neg N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. neg-log N/A

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

   14. lower-log.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. div-flip-rev N/A

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

   17. lower-/.f64 76.9%

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

3. Applied rewrites 76.9%

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

4. Taylor expanded in x around 0

   \[\leadsto -\color{blue}{z} \]
5. Step-by-step derivation

6. Applied rewrites 49.3%

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

Reproduce

herbie shell --seed 2025183 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64
  (- (* x (log (/ x y))) z))