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

Percentage Accurate: 77.1% → 95.7%

Time: 9.5s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-y\right)\\ t_1 := \log \left(-x\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{t\_1}^{3} - {t\_0}^{3}}{{t\_1}^{2} + \left({t\_0}^{2} + t\_1 \cdot t\_0\right)} - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\frac{\log x}{z} - \frac{\log y}{z}\right), z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- y))) (t_1 (log (- x))))
   (if (<= x -5e-310)
     (-
      (*
       x
       (/
        (- (pow t_1 3.0) (pow t_0 3.0))
        (+ (pow t_1 2.0) (+ (pow t_0 2.0) (* t_1 t_0)))))
      z)
     (if (<= x 1.1e+145)
       (fma (* x (- (/ (log x) z) (/ (log y) z))) z (- z))
       (* (- (log x) (log y)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-y);
	double t_1 = log(-x);
	double tmp;
	if (x <= -5e-310) {
		tmp = (x * ((pow(t_1, 3.0) - pow(t_0, 3.0)) / (pow(t_1, 2.0) + (pow(t_0, 2.0) + (t_1 * t_0))))) - z;
	} else if (x <= 1.1e+145) {
		tmp = fma((x * ((log(x) / z) - (log(y) / z))), z, -z);
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-y))
	t_1 = log(Float64(-x))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((t_1 ^ 3.0) - (t_0 ^ 3.0)) / Float64((t_1 ^ 2.0) + Float64((t_0 ^ 2.0) + Float64(t_1 * t_0))))) - z);
	elseif (x <= 1.1e+145)
		tmp = fma(Float64(x * Float64(Float64(log(x) / z) - Float64(log(y) / z))), z, Float64(-z));
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-y)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x)], $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[(x * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + N[(N[Power[t$95$0, 2.0], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.1e+145], N[(N[(x * N[(N[(N[Log[x], $MachinePrecision] / z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 80.0%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. flip3-- N/A

         \[\leadsto x \cdot \color{blue}{\frac{{\log \left(\mathsf{neg}\left(x\right)\right)}^{3} - {\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{{\log \left(\mathsf{neg}\left(x\right)\right)}^{3} - {\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

      7. lower--.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{3} - {\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      8. lower-pow.f64 N/A

         \[\leadsto x \cdot \frac{\color{blue}{{\log \left(\mathsf{neg}\left(x\right)\right)}^{3}} - {\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      9. lower-log.f64 N/A

         \[\leadsto x \cdot \frac{{\color{blue}{\log \left(\mathsf{neg}\left(x\right)\right)}}^{3} - {\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      10. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \color{blue}{\left(-x\right)}}^{3} - {\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      11. lower-pow.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{3} - \color{blue}{{\log \left(\mathsf{neg}\left(y\right)\right)}^{3}}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      12. lower-log.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{3} - {\color{blue}{\log \left(\mathsf{neg}\left(y\right)\right)}}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      13. lower-neg.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{3} - {\log \color{blue}{\left(-y\right)}}^{3}}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)} - z \]

      14. lower-+.f64 N/A

          \[\leadsto x \cdot \frac{{\log \left(-x\right)}^{3} - {\log \left(-y\right)}^{3}}{\color{blue}{\log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(x\right)\right) + \left(\log \left(\mathsf{neg}\left(y\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right) + \log \left(\mathsf{neg}\left(x\right)\right) \cdot \log \left(\mathsf{neg}\left(y\right)\right)\right)}} - z \]

   4. Applied rewrites 99.6%

      \[\leadsto x \cdot \color{blue}{\frac{{\log \left(-x\right)}^{3} - {\log \left(-y\right)}^{3}}{{\log \left(-x\right)}^{2} + \left({\log \left(-y\right)}^{2} + \log \left(-x\right) \cdot \log \left(-y\right)\right)}} - z \]

   if -4.999999999999985e-310 < x < 1.10000000000000004e145

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.3%

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

   12. Applied rewrites 96.5%

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

   if 1.10000000000000004e145 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

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

Alternative 2: 89.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right)\\ t_1 := \log \left(-y\right)\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{t\_0}^{3} - {t\_1}^{3}}{\mathsf{fma}\left(t\_1, \log \left(y \cdot x\right), {t\_0}^{2}\right)}, x, -z\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\frac{\log x}{z} - \frac{\log y}{z}\right), z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- x))) (t_1 (log (- y))))
   (if (<= x -5e-310)
     (fma
      (/ (- (pow t_0 3.0) (pow t_1 3.0)) (fma t_1 (log (* y x)) (pow t_0 2.0)))
      x
      (- z))
     (if (<= x 1.1e+145)
       (fma (* x (- (/ (log x) z) (/ (log y) z))) z (- z))
       (* (- (log x) (log y)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x);
	double t_1 = log(-y);
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(((pow(t_0, 3.0) - pow(t_1, 3.0)) / fma(t_1, log((y * x)), pow(t_0, 2.0))), x, -z);
	} else if (x <= 1.1e+145) {
		tmp = fma((x * ((log(x) / z) - (log(y) / z))), z, -z);
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-x))
	t_1 = log(Float64(-y))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = fma(Float64(Float64((t_0 ^ 3.0) - (t_1 ^ 3.0)) / fma(t_1, log(Float64(y * x)), (t_0 ^ 2.0))), x, Float64(-z));
	elseif (x <= 1.1e+145)
		tmp = fma(Float64(x * Float64(Float64(log(x) / z) - Float64(log(y) / z))), z, Float64(-z));
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[Log[(-y)], $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, 1.1e+145], N[(N[(x * N[(N[(N[Log[x], $MachinePrecision] / z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 80.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 91.9%

      \[\leadsto \mathsf{fma}\left(\frac{{\log \left(-x\right)}^{3} - {\log \left(-y\right)}^{3}}{\mathsf{fma}\left(\log \left(-y\right), \log \left(\left(-y\right) \cdot \left(-x\right)\right), {\log \left(-x\right)}^{2}\right)}, x, -z\right) \]

   if -4.999999999999985e-310 < x < 1.10000000000000004e145

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.3%

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

   12. Applied rewrites 96.5%

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

   if 1.10000000000000004e145 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\log \left(-x\right)}^{3} - {\log \left(-y\right)}^{3}}{\mathsf{fma}\left(\log \left(-y\right), \log \left(y \cdot x\right), {\log \left(-x\right)}^{2}\right)}, x, -z\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\frac{\log x}{z} - \frac{\log y}{z}\right), z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(x \cdot y\right)} - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\frac{\log x}{z} - \frac{\log y}{z}\right), z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e-310)
   (-
    (* x (/ (- (pow (log (- x)) 2.0) (pow (log (- y)) 2.0)) (log (* x y))))
    z)
   (if (<= x 1.1e+145)
     (fma (* x (- (/ (log x) z) (/ (log y) z))) z (- z))
     (* (- (log x) (log y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (x * ((pow(log(-x), 2.0) - pow(log(-y), 2.0)) / log((x * y)))) - z;
	} else if (x <= 1.1e+145) {
		tmp = fma((x * ((log(x) / z) - (log(y) / z))), z, -z);
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(Float64(x * Float64(Float64((log(Float64(-x)) ^ 2.0) - (log(Float64(-y)) ^ 2.0)) / log(Float64(x * y)))) - z);
	elseif (x <= 1.1e+145)
		tmp = fma(Float64(x * Float64(Float64(log(x) / z) - Float64(log(y) / z))), z, Float64(-z));
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e-310], N[(N[(x * N[(N[(N[Power[N[Log[(-x)], $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[Log[(-y)], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.1e+145], N[(N[(x * N[(N[(N[Log[x], $MachinePrecision] / z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 80.0%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. log-div N/A

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

      5. flip-- N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-neg.f64 91.8

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

   4. Applied rewrites 91.8%

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

   if -4.999999999999985e-310 < x < 1.10000000000000004e145

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.3%

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

   12. Applied rewrites 96.5%

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

   if 1.10000000000000004e145 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \frac{{\log \left(-x\right)}^{2} - {\log \left(-y\right)}^{2}}{\log \left(x \cdot y\right)} - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\frac{\log x}{z} - \frac{\log y}{z}\right), z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (- (* x t_0) z)))
   (if (<= t_1 (- INFINITY))
     (- z)
     (if (<= t_1 5e+307) (fma t_0 x (- z)) (* (- (log x) (log y)) x)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = (x * t_0) - z;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_1 <= 5e+307) {
		tmp = fma(t_0, x, -z);
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(Float64(x * t_0) - z)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_1 <= 5e+307)
		tmp = fma(t_0, x, Float64(-z));
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], (-z), If[LessEqual[t$95$1, 5e+307], N[(t$95$0 * x + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.4%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.0

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

   5. Applied rewrites 63.0%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 5.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 56.4

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

   5. Applied rewrites 56.4%

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

Alternative 5: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0 - z\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 10^{+299}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (- (* x t_0) z)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+299)))
     (- z)
     (fma t_0 x (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = (x * t_0) - z;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+299)) {
		tmp = -z;
	} else {
		tmp = fma(t_0, x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(Float64(x * t_0) - z)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+299))
		tmp = Float64(-z);
	else
		tmp = fma(t_0, x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * t$95$0), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+299]], $MachinePrecision]], (-z), N[(t$95$0 * x + (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < -inf.0 or 1.0000000000000001e299 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z)

   1. Initial program 7.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   if -inf.0 < (-.f64 (*.f64 x (log.f64 (/.f64 x y))) z) < 1.0000000000000001e299

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 99.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) - z \leq -\infty \lor \neg \left(x \cdot \log \left(\frac{x}{y}\right) - z \leq 10^{+299}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x - \log y\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+227}:\\ \;\;\;\;\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot x\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-145}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{t\_0}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log x) (log y))))
   (if (<= x -6.2e+227)
     (* (- (log (- x)) (log (- y))) x)
     (if (<= x -7e-145)
       (fma (log (/ x y)) x (- z))
       (if (<= x -1e-308)
         (- z)
         (if (<= x 1.1e+145) (fma (* x (/ t_0 z)) z (- z)) (* t_0 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) - log(y);
	double tmp;
	if (x <= -6.2e+227) {
		tmp = (log(-x) - log(-y)) * x;
	} else if (x <= -7e-145) {
		tmp = fma(log((x / y)), x, -z);
	} else if (x <= -1e-308) {
		tmp = -z;
	} else if (x <= 1.1e+145) {
		tmp = fma((x * (t_0 / z)), z, -z);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) - log(y))
	tmp = 0.0
	if (x <= -6.2e+227)
		tmp = Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * x);
	elseif (x <= -7e-145)
		tmp = fma(log(Float64(x / y)), x, Float64(-z));
	elseif (x <= -1e-308)
		tmp = Float64(-z);
	elseif (x <= 1.1e+145)
		tmp = fma(Float64(x * Float64(t_0 / z)), z, Float64(-z));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e+227], N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, -7e-145], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + (-z)), $MachinePrecision], If[LessEqual[x, -1e-308], (-z), If[LessEqual[x, 1.1e+145], N[(N[(x * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.1999999999999997e227

   1. Initial program 45.1%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.7%

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

   if -6.1999999999999997e227 < x < -6.99999999999999994e-145

   1. Initial program 89.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 89.0%

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

   if -6.99999999999999994e-145 < x < -9.9999999999999991e-309

   1. Initial program 74.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.7

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

   5. Applied rewrites 96.7%

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

   if -9.9999999999999991e-309 < x < 1.10000000000000004e145

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.5%

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

   if 1.10000000000000004e145 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

      \[\leadsto \color{blue}{\left(\log x - \log y\right) \cdot x} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot \frac{x}{z}, z, -z\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(\frac{\log x}{z} - \frac{\log y}{z}\right), z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log x - \log y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5e-310)
   (fma (* (- (log (- x)) (log (- y))) (/ x z)) z (- z))
   (if (<= x 1.1e+145)
     (fma (* x (- (/ (log x) z) (/ (log y) z))) z (- z))
     (* (- (log x) (log y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(((log(-x) - log(-y)) * (x / z)), z, -z);
	} else if (x <= 1.1e+145) {
		tmp = fma((x * ((log(x) / z) - (log(y) / z))), z, -z);
	} else {
		tmp = (log(x) - log(y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = fma(Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * Float64(x / z)), z, Float64(-z));
	elseif (x <= 1.1e+145)
		tmp = fma(Float64(x * Float64(Float64(log(x) / z) - Float64(log(y) / z))), z, Float64(-z));
	else
		tmp = Float64(Float64(log(x) - log(y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5e-310], N[(N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], If[LessEqual[x, 1.1e+145], N[(N[(x * N[(N[(N[Log[x], $MachinePrecision] / z), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 80.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 71.7%

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

   10. Applied rewrites 88.0%

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

   if -4.999999999999985e-310 < x < 1.10000000000000004e145

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.3%

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

   12. Applied rewrites 96.5%

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

   if 1.10000000000000004e145 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

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

Alternative 8: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log x - \log y\\ \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(\left(\log \left(-x\right) - \log \left(-y\right)\right) \cdot \frac{x}{z}, z, -z\right)\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{t\_0}{z}, z, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log x) (log y))))
   (if (<= x -5e-310)
     (fma (* (- (log (- x)) (log (- y))) (/ x z)) z (- z))
     (if (<= x 1.1e+145) (fma (* x (/ t_0 z)) z (- z)) (* t_0 x)))))

C

double code(double x, double y, double z) {
	double t_0 = log(x) - log(y);
	double tmp;
	if (x <= -5e-310) {
		tmp = fma(((log(-x) - log(-y)) * (x / z)), z, -z);
	} else if (x <= 1.1e+145) {
		tmp = fma((x * (t_0 / z)), z, -z);
	} else {
		tmp = t_0 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(log(x) - log(y))
	tmp = 0.0
	if (x <= -5e-310)
		tmp = fma(Float64(Float64(log(Float64(-x)) - log(Float64(-y))) * Float64(x / z)), z, Float64(-z));
	elseif (x <= 1.1e+145)
		tmp = fma(Float64(x * Float64(t_0 / z)), z, Float64(-z));
	else
		tmp = Float64(t_0 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-310], N[(N[(N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], If[LessEqual[x, 1.1e+145], N[(N[(x * N[(t$95$0 / z), $MachinePrecision]), $MachinePrecision] * z + (-z)), $MachinePrecision], N[(t$95$0 * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 80.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 71.7%

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

   10. Applied rewrites 88.0%

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

   if -4.999999999999985e-310 < x < 1.10000000000000004e145

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-/l* N/A

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

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

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

      10. distribute-rgt-neg-in N/A

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

      11. distribute-lft-neg-in N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. mul-1-neg N/A

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

      15. distribute-rgt-neg-in N/A

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

      16. distribute-lft-neg-in N/A

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

      17. metadata-eval N/A

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

   8. Applied rewrites 74.0%

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

   10. Applied rewrites 74.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.5%

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

   if 1.10000000000000004e145 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

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

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. lower--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. lower-log.f64 N/A

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

      14. lower-log.f64 91.7

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

   5. Applied rewrites 91.7%

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

Alternative 9: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-114} \lor \neg \left(z \leq 9.2 \cdot 10^{-191}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05e-114) (not (<= z 9.2e-191))) (- z) (* (log (/ x y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e-114) || !(z <= 9.2e-191)) {
		tmp = -z;
	} else {
		tmp = log((x / y)) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.05d-114)) .or. (.not. (z <= 9.2d-191))) then
        tmp = -z
    else
        tmp = log((x / y)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e-114) || !(z <= 9.2e-191)) {
		tmp = -z;
	} else {
		tmp = Math.log((x / y)) * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.05e-114) or not (z <= 9.2e-191):
		tmp = -z
	else:
		tmp = math.log((x / y)) * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05e-114) || !(z <= 9.2e-191))
		tmp = Float64(-z);
	else
		tmp = Float64(log(Float64(x / y)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.05e-114) || ~((z <= 9.2e-191)))
		tmp = -z;
	else
		tmp = log((x / y)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05e-114], N[Not[LessEqual[z, 9.2e-191]], $MachinePrecision]], (-z), N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.04999999999999996e-114 or 9.20000000000000042e-191 < z

   1. Initial program 73.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -1.04999999999999996e-114 < z < 9.20000000000000042e-191

   1. Initial program 85.7%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-114} \lor \neg \left(z \leq 9.2 \cdot 10^{-191}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{x}{y}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.6% accurate, 40.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 56.2

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

5. Applied rewrites 56.2%

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

Developer Target 1: 88.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024363 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

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

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