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

Percentage Accurate: 77.4% → 99.5%

Time: 6.9s

Alternatives: 10

Speedup: 0.6×

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: 77.4% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-1 \cdot x\right) + \log \left(\frac{-1}{y}\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-x, \log x, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (+ (log (* -1.0 x)) (log (/ -1.0 y)))) z)
   (- (fma (log y) x (fma (- x) (log x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log((-1.0 * x)) + log((-1.0 / y)))) - z;
	} else {
		tmp = -fma(log(y), x, fma(-x, log(x), z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(-1.0 * x)) + log(Float64(-1.0 / y)))) - z);
	else
		tmp = Float64(-fma(log(y), x, fma(Float64(-x), log(x), z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-310], N[(N[(x * N[(N[Log[N[(-1.0 * x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] * x + N[((-x) * N[Log[x], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 77.4%

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

   2. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 49.9

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

   4. Applied rewrites 49.9%

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

   if -4.999999999999985e-310 < y

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

   7. Applied rewrites 49.6%

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

Alternative 2: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+149}:\\ \;\;\;\;\log \left(-x\right) \cdot x - \log \left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(-x, \log x, z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+149)
   (- (* (log (- x)) x) (* (log (- y)) x))
   (if (<= x -4.5e-82)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-309) (- z) (- (fma (log y) x (fma (- x) (log x) z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+149) {
		tmp = (log(-x) * x) - (log(-y) * x);
	} else if (x <= -4.5e-82) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = -fma(log(y), x, fma(-x, log(x), z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e+149)
		tmp = Float64(Float64(log(Float64(-x)) * x) - Float64(log(Float64(-y)) * x));
	elseif (x <= -4.5e-82)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(-fma(log(y), x, fma(Float64(-x), log(x), z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+149], N[(N[(N[Log[(-x)], $MachinePrecision] * x), $MachinePrecision] - N[(N[Log[(-y)], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-82], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), (-N[(N[Log[y], $MachinePrecision] * x + N[((-x) * N[Log[x], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.50000000000000002e149

   1. Initial program 77.4%

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

   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 39.6

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

   4. Applied rewrites 39.6%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. distribute-lft-in N/A

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

      16. mul0-lft N/A

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

      17. div0 N/A

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

      18. div0 N/A

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

      19. div0 N/A

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

      20. div0 N/A

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

      21. mul0-rgt N/A

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

      22. mul0-lft N/A

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

      23. div0 N/A

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

      24. div0 N/A

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

      25. mul0-lft N/A

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

      26. distribute-rgt-in N/A

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lift-log.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. mul-1-neg N/A

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

      20. associate-*l* N/A

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

      21. lift-*.f64 N/A

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

      22. sum-log N/A

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

      23. lift-log.f64 N/A

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

      24. lift-log.f64 N/A

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

   8. Applied rewrites 25.5%

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

   if -1.50000000000000002e149 < x < -4.4999999999999998e-82

   1. Initial program 77.4%

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

   if -4.4999999999999998e-82 < x < -4.9999999999999995e-309

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

   7. Applied rewrites 49.6%

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

Alternative 3: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+149}:\\ \;\;\;\;\log \left(-x\right) \cdot x - \log \left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log x \cdot x - \mathsf{fma}\left(x, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+149)
   (- (* (log (- x)) x) (* (log (- y)) x))
   (if (<= x -4.5e-82)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-309) (- z) (- (* (log x) x) (fma x (log y) z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+149) {
		tmp = (log(-x) * x) - (log(-y) * x);
	} else if (x <= -4.5e-82) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (log(x) * x) - fma(x, log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e+149)
		tmp = Float64(Float64(log(Float64(-x)) * x) - Float64(log(Float64(-y)) * x));
	elseif (x <= -4.5e-82)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-309)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(log(x) * x) - fma(x, log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+149], N[(N[(N[Log[(-x)], $MachinePrecision] * x), $MachinePrecision] - N[(N[Log[(-y)], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-82], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-z), N[(N[(N[Log[x], $MachinePrecision] * x), $MachinePrecision] - N[(x * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.50000000000000002e149

   1. Initial program 77.4%

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

   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 39.6

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

   4. Applied rewrites 39.6%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. distribute-lft-in N/A

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

      16. mul0-lft N/A

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

      17. div0 N/A

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

      18. div0 N/A

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

      19. div0 N/A

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

      20. div0 N/A

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

      21. mul0-rgt N/A

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

      22. mul0-lft N/A

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

      23. div0 N/A

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

      24. div0 N/A

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

      25. mul0-lft N/A

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

      26. distribute-rgt-in N/A

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lift-log.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. mul-1-neg N/A

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

      20. associate-*l* N/A

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

      21. lift-*.f64 N/A

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

      22. sum-log N/A

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

      23. lift-log.f64 N/A

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

      24. lift-log.f64 N/A

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

   8. Applied rewrites 25.5%

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

   if -1.50000000000000002e149 < x < -4.4999999999999998e-82

   1. Initial program 77.4%

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

   if -4.4999999999999998e-82 < x < -4.9999999999999995e-309

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.3

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

   5. Applied rewrites 77.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. +-commutative N/A

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

      9. add-flip N/A

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

   7. Applied rewrites 49.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. lift-neg.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. associate-+l+ N/A

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

   9. Applied rewrites 49.5%

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

Alternative 4: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+149}:\\ \;\;\;\;\log \left(-x\right) \cdot x - \log \left(-y\right) \cdot x\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+149)
   (- (* (log (- x)) x) (* (log (- y)) x))
   (if (<= x -4.5e-82)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+149) {
		tmp = (log(-x) * x) - (log(-y) * x);
	} else if (x <= -4.5e-82) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-309) {
		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 <= (-1.5d+149)) then
        tmp = (log(-x) * x) - (log(-y) * x)
    else if (x <= (-4.5d-82)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-309)) 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 <= -1.5e+149) {
		tmp = (Math.log(-x) * x) - (Math.log(-y) * x);
	} else if (x <= -4.5e-82) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e+149:
		tmp = (math.log(-x) * x) - (math.log(-y) * x)
	elif x <= -4.5e-82:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-309:
		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 <= -1.5e+149)
		tmp = Float64(Float64(log(Float64(-x)) * x) - Float64(log(Float64(-y)) * x));
	elseif (x <= -4.5e-82)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-309)
		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 <= -1.5e+149)
		tmp = (log(-x) * x) - (log(-y) * x);
	elseif (x <= -4.5e-82)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+149], N[(N[(N[Log[(-x)], $MachinePrecision] * x), $MachinePrecision] - N[(N[Log[(-y)], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-82], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-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 < -1.50000000000000002e149

   1. Initial program 77.4%

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

   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 39.6

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

   4. Applied rewrites 39.6%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. *-rgt-identity N/A

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

      12. metadata-eval N/A

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. distribute-lft-in N/A

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

      16. mul0-lft N/A

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

      17. div0 N/A

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

      18. div0 N/A

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

      19. div0 N/A

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

      20. div0 N/A

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

      21. mul0-rgt N/A

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

      22. mul0-lft N/A

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

      23. div0 N/A

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

      24. div0 N/A

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

      25. mul0-lft N/A

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

      26. distribute-rgt-in N/A

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

   6. Applied rewrites 32.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. associate-*l* N/A

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

      10. lft-mult-inverse N/A

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

      11. *-rgt-identity N/A

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

      12. lift-log.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. metadata-eval N/A

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

      16. distribute-frac-neg N/A

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

      17. lift-/.f64 N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. mul-1-neg N/A

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

      20. associate-*l* N/A

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

      21. lift-*.f64 N/A

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

      22. sum-log N/A

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

      23. lift-log.f64 N/A

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

      24. lift-log.f64 N/A

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

   8. Applied rewrites 25.5%

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

   if -1.50000000000000002e149 < x < -4.4999999999999998e-82

   1. Initial program 77.4%

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

   if -4.4999999999999998e-82 < x < -4.9999999999999995e-309

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. sum-log N/A

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

      5. lift-log.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. sub-flip N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 49.6

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

   8. Applied rewrites 49.6%

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

Alternative 5: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+149}:\\ \;\;\;\;-x \cdot \left(\log \left(-y\right) - \log \left(-x\right)\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e+149)
   (- (* x (- (log (- y)) (log (- x)))))
   (if (<= x -4.5e-82)
     (- (* x (log (/ x y))) z)
     (if (<= x -5e-309) (- z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e+149) {
		tmp = -(x * (log(-y) - log(-x)));
	} else if (x <= -4.5e-82) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-309) {
		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 <= (-1.5d+149)) then
        tmp = -(x * (log(-y) - log(-x)))
    else if (x <= (-4.5d-82)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-309)) 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 <= -1.5e+149) {
		tmp = -(x * (Math.log(-y) - Math.log(-x)));
	} else if (x <= -4.5e-82) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-309) {
		tmp = -z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e+149:
		tmp = -(x * (math.log(-y) - math.log(-x)))
	elif x <= -4.5e-82:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-309:
		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 <= -1.5e+149)
		tmp = Float64(-Float64(x * Float64(log(Float64(-y)) - log(Float64(-x)))));
	elseif (x <= -4.5e-82)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-309)
		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 <= -1.5e+149)
		tmp = -(x * (log(-y) - log(-x)));
	elseif (x <= -4.5e-82)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-309)
		tmp = -z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e+149], (-N[(x * N[(N[Log[(-y)], $MachinePrecision] - N[Log[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, -4.5e-82], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-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 < -1.50000000000000002e149

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   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 40.1

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

   6. Applied rewrites 40.1%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. diff-log N/A

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

      6. remove-double-neg N/A

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

      7. log-rec N/A

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

      8. metadata-eval N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift-log.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. frac-2neg N/A

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

      20. metadata-eval N/A

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

      21. log-rec N/A

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

      22. remove-double-neg N/A

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

      23. lower-log.f64 N/A

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

      24. lower-neg.f64 25.6

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

      25. lift-*.f64 N/A

          \[\leadsto -x \cdot \left(\log \left(-y\right) - \log \left(-1 \cdot x\right)\right) \]

      26. mul-1-neg N/A

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

      27. lift-neg.f64 25.6

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

   8. Applied rewrites 25.6%

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

   if -1.50000000000000002e149 < x < -4.4999999999999998e-82

   1. Initial program 77.4%

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

   if -4.4999999999999998e-82 < x < -4.9999999999999995e-309

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. sum-log N/A

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

      5. lift-log.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. sub-flip N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 49.6

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

   8. Applied rewrites 49.6%

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

Alternative 6: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-309}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e-82)
   (- (* x (log (/ x y))) z)
   (if (<= x -5e-309) (- z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e-82) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -5e-309) {
		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-82)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-5d-309)) 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-82) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -5e-309) {
		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-82:
		tmp = (x * math.log((x / y))) - z
	elif x <= -5e-309:
		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-82)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -5e-309)
		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-82)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -5e-309)
		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-82], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -5e-309], (-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 < -4.4999999999999998e-82

   1. Initial program 77.4%

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

   if -4.4999999999999998e-82 < x < -4.9999999999999995e-309

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -4.9999999999999995e-309 < x

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. sum-log N/A

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

      5. lift-log.f64 N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. sub-flip N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-log.f64 49.6

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

   8. Applied rewrites 49.6%

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

Alternative 7: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+290}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 1e+290) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+290) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+290) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+290:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+290)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+290)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+290], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000006e290

   1. Initial program 77.4%

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

   if 1.00000000000000006e290 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 77.4%

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

   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 39.6

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

   4. Applied rewrites 39.6%

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{x}{y}\right)} \]
   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. diff-log N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. sub-flip-reverse N/A

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

      7. mul-1-neg N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower--.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity 25.2

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

   6. Applied rewrites 25.2%

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

Alternative 8: 86.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 10^{+294}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 1e+294) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 1e+294) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 1e+294) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 1e+294:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 1e+294)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 1e+294)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 1e+294], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1.00000000000000007e294 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.00000000000000007e294

   1. Initial program 77.4%

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

Alternative 9: 66.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-87}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 6.3 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -7.6e-87) (- z) (if (<= z 6.3e+35) (* x (log (/ x y))) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -7.6e-87) {
		tmp = -z;
	} else if (z <= 6.3e+35) {
		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 <= (-7.6d-87)) then
        tmp = -z
    else if (z <= 6.3d+35) 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 <= -7.6e-87) {
		tmp = -z;
	} else if (z <= 6.3e+35) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -7.6e-87:
		tmp = -z
	elif z <= 6.3e+35:
		tmp = x * math.log((x / y))
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -7.6e-87)
		tmp = Float64(-z);
	elseif (z <= 6.3e+35)
		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 <= -7.6e-87)
		tmp = -z;
	elseif (z <= 6.3e+35)
		tmp = x * log((x / y));
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -7.6e-87], (-z), If[LessEqual[z, 6.3e+35], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6e-87 or 6.29999999999999969e35 < z

   1. Initial program 77.4%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 77.3

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

   3. Applied rewrites 77.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 50.1%

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

   if -7.6e-87 < z < 6.29999999999999969e35

   1. Initial program 77.4%

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

   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 39.6

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

   4. Applied rewrites 39.6%

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

Alternative 10: 50.1% accurate, 8.0× 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 77.4%

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]
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 77.3

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

3. Applied rewrites 77.3%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 50.1%

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

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64
  (- (* x (log (/ x y))) z))