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

Percentage Accurate: 77.2% → 99.5%

Time: 7.1s

Alternatives: 10

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: 77.2% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (fma (- (log (- x)) (log (- y))) x (- z))
   (fma (- (log x) (log y)) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = fma((log(-x) - log(-y)), x, -z);
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-309)
		tmp = fma(Float64(log(Float64(-x)) - log(Float64(-y))), x, Float64(-z));
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-309], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 75.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 75.9

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

   4. Applied rewrites 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. frac-2neg N/A

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

      4. diff-log N/A

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

      5. lift-log.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. lift--.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if -1.000000000000002e-309 < y

   1. Initial program 79.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 2: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0 - z\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \end{array} \end{array} \]

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 5e+307)))
     (- 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 <= 5e+307)) {
		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 <= 5e+307))
		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, 5e+307]], $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 5e307 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 6.0%

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

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

      2. lower-neg.f64 48.9

         \[\leadsto -z \]

   5. Applied rewrites 48.9%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 5e307

   1. Initial program 99.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) - z \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) - z \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* x (log (/ x y))) z)))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 5e+307))) (- z) t_0)))

C

double code(double x, double y, double z) {
	double t_0 = (x * log((x / y))) - z;
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 5e+307)) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) || !(t_0 <= 5e+307)) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * math.log((x / y))) - z
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 5e+307):
		tmp = -z
	else:
		tmp = t_0
	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)) || !(t_0 <= 5e+307))
		tmp = Float64(-z);
	else
		tmp = t_0;
	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) || ~((t_0 <= 5e+307)))
		tmp = -z;
	else
		tmp = t_0;
	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[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 5e+307]], $MachinePrecision]], (-z), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -inf.0 or 5e307 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 6.0%

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

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

      2. lower-neg.f64 48.9

         \[\leadsto -z \]

   5. Applied rewrites 48.9%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 5e307

   1. Initial program 99.7%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) - z \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) - z \leq 5 \cdot 10^{+307}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+197}:\\ \;\;\;\;\left(-x\right) \cdot \left(\log \left(-y\right) - \log \left(-x\right)\right)\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.28e+197)
   (* (- x) (- (log (- y)) (log (- x))))
   (if (<= x -2.85e-117)
     (fma (log (/ x y)) x (- z))
     (if (<= x -5e-307) (- z) (fma (- (log x) (log y)) x (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.28e+197) {
		tmp = -x * (log(-y) - log(-x));
	} else if (x <= -2.85e-117) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -5e-307) {
		tmp = -z;
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.28e+197)
		tmp = Float64(Float64(-x) * Float64(log(Float64(-y)) - log(Float64(-x))));
	elseif (x <= -2.85e-117)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -5e-307)
		tmp = Float64(-z);
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.28e+197], N[((-x) * N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.85e-117], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -5e-307], (-z), N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.27999999999999997e197

   1. Initial program 66.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   4. Applied rewrites 66.5%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 62.2

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

   7. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift-neg.f64 91.3

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

   9. Applied rewrites 91.3%

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

   if -1.27999999999999997e197 < x < -2.85e-117

   1. Initial program 91.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if -2.85e-117 < x < -5.00000000000000014e-307

   1. Initial program 59.5%

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

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

      2. lower-neg.f64 90.8

         \[\leadsto -z \]

   5. Applied rewrites 90.8%

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

   if -5.00000000000000014e-307 < x

   1. Initial program 79.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 5: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-175}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\log y - \log x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (log (/ x y)) x (- z))))
   (if (<= x -2.85e-117)
     t_0
     (if (<= x 1.6e-175)
       (- z)
       (if (<= x 4.5e+141) t_0 (* (- x) (- (log y) (log x))))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(log((x / y)), x, -z);
	double tmp;
	if (x <= -2.85e-117) {
		tmp = t_0;
	} else if (x <= 1.6e-175) {
		tmp = -z;
	} else if (x <= 4.5e+141) {
		tmp = t_0;
	} else {
		tmp = -x * (log(y) - log(x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(log(Float64(x / y)), x, Float64(-z))
	tmp = 0.0
	if (x <= -2.85e-117)
		tmp = t_0;
	elseif (x <= 1.6e-175)
		tmp = Float64(-z);
	elseif (x <= 4.5e+141)
		tmp = t_0;
	else
		tmp = Float64(Float64(-x) * Float64(log(y) - log(x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision]}, If[LessEqual[x, -2.85e-117], t$95$0, If[LessEqual[x, 1.6e-175], (-z), If[LessEqual[x, 4.5e+141], t$95$0, N[((-x) * N[(N[Log[y], $MachinePrecision] - N[Log[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.85e-117 or 1.6e-175 < x < 4.5000000000000002e141

   1. Initial program 89.2%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 89.2

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

   4. Applied rewrites 89.2%

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

   if -2.85e-117 < x < 1.6e-175

   1. Initial program 61.7%

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

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

      2. lower-neg.f64 92.4

         \[\leadsto -z \]

   5. Applied rewrites 92.4%

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

   if 4.5000000000000002e141 < x

   1. Initial program 66.2%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   4. Applied rewrites 69.7%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 64.9

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

   7. Applied rewrites 64.9%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 93.9

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

   9. Applied rewrites 93.9%

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

Alternative 6: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.85e-117)
   (fma (log (/ x y)) x (- z))
   (if (<= x -5e-307) (- z) (fma (- (log x) (log y)) x (- z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.85e-117) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -5e-307) {
		tmp = -z;
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.85e-117)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -5e-307)
		tmp = Float64(-z);
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.85e-117], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -5e-307], (-z), N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.85e-117

   1. Initial program 83.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -2.85e-117 < x < -5.00000000000000014e-307

   1. Initial program 59.5%

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

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

      2. lower-neg.f64 90.8

         \[\leadsto -z \]

   5. Applied rewrites 90.8%

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

   if -5.00000000000000014e-307 < x

   1. Initial program 79.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 7: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-117}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-307}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.85e-117)
   (fma (log (/ x y)) x (- z))
   (if (<= x -5e-307) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.85e-117) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -5e-307) {
		tmp = -z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.85e-117)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -5e-307)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.85e-117], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -5e-307], (-z), N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.85e-117

   1. Initial program 83.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 83.9

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

   4. Applied rewrites 83.9%

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

   if -2.85e-117 < x < -5.00000000000000014e-307

   1. Initial program 59.5%

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

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

      2. lower-neg.f64 90.8

         \[\leadsto -z \]

   5. Applied rewrites 90.8%

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

   if -5.00000000000000014e-307 < x

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. sum-log N/A

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

      11. +-commutative N/A

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

      12. flip-+ N/A

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

      13. log-rec N/A

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

      14. mul-1-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. log-rec N/A

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

      18. mul-1-neg N/A

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

      19. flip-+ N/A

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

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

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

   4. Applied rewrites 99.4%

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

Alternative 8: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x - \log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-309)
   (- (* x (- (log (- x)) (log (- y)))) z)
   (fma (- (log x) (log y)) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-309) {
		tmp = (x * (log(-x) - log(-y))) - z;
	} else {
		tmp = fma((log(x) - log(y)), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-309)
		tmp = Float64(Float64(x * Float64(log(Float64(-x)) - log(Float64(-y)))) - z);
	else
		tmp = fma(Float64(log(x) - log(y)), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-309], N[(N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.000000000000002e-309

   1. Initial program 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      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. mul-1-neg N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if -1.000000000000002e-309 < y

   1. Initial program 79.5%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. *-lft-identity N/A

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

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

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 79.5

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

   4. Applied rewrites 79.5%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. log-div N/A

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

      4. lower--.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 9: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+45} \lor \neg \left(x \leq 1.5 \cdot 10^{-82}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e+45) (not (<= x 1.5e-82))) (* (- x) (log (/ y x))) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e+45) || !(x <= 1.5e-82)) {
		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 ((x <= (-5d+45)) .or. (.not. (x <= 1.5d-82))) 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 ((x <= -5e+45) || !(x <= 1.5e-82)) {
		tmp = -x * Math.log((y / x));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5e+45) or not (x <= 1.5e-82):
		tmp = -x * math.log((y / x))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e+45) || !(x <= 1.5e-82))
		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 ((x <= -5e+45) || ~((x <= 1.5e-82)))
		tmp = -x * log((y / x));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e+45], N[Not[LessEqual[x, 1.5e-82]], $MachinePrecision]], N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -5e45 or 1.4999999999999999e-82 < x

   1. Initial program 80.3%

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

      1. lift-/.f64 N/A

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

      2. lift-log.f64 N/A

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

      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. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. frac-2neg N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. log-div N/A

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

      13. sum-log N/A

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

      14. remove-double-neg N/A

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

      15. log-rec N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{0}\right) - \log y\right) - z \]

      18. associate-+r- N/A

          \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \left(0 - \log y\right)\right)} - z \]

      19. metadata-eval N/A

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

      20. log-div N/A

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

      21. +-commutative N/A

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

      22. log-div N/A

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

      23. metadata-eval N/A

          \[\leadsto x \cdot \left(\left(\color{blue}{0} - \log y\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) - z \]

      24. associate-+l- N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

      25. lower--.f64 N/A

          \[\leadsto x \cdot \color{blue}{\left(0 - \left(\log y - -1 \cdot \log \left(\frac{1}{x}\right)\right)\right)} - z \]

   4. Applied rewrites 81.3%

      \[\leadsto x \cdot \color{blue}{\left(0 - \log \left(\frac{y}{x}\right)\right)} - z \]

   5. Taylor expanded in z around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-/.f64 62.0

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

   7. Applied rewrites 62.0%

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

   if -5e45 < x < 1.4999999999999999e-82

   1. Initial program 74.7%

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

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

      2. lower-neg.f64 86.9

         \[\leadsto -z \]

   5. Applied rewrites 86.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+45} \lor \neg \left(x \leq 1.5 \cdot 10^{-82}\right):\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.9% 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 77.7%

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

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

   2. lower-neg.f64 50.9

      \[\leadsto -z \]

5. Applied rewrites 50.9%

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

Developer Target 1: 88.5% 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 2025051 
(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))