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

Percentage Accurate: 77.2% → 99.3%

Time: 11.3s

Alternatives: 5

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4e-310)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (- (+ (* x (log x)) (* x (log (/ 1.0 y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	} else {
		tmp = ((x * log(x)) + (x * log((1.0 / y)))) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-4d-310)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - y)))) - z
    else
        tmp = ((x * log(x)) + (x * log((1.0d0 / y)))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -4e-310) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else {
		tmp = ((x * Math.log(x)) + (x * Math.log((1.0 / y)))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4e-310:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - y)))) - z
	else:
		tmp = ((x * math.log(x)) + (x * math.log((1.0 / y)))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y)))) - z);
	else
		tmp = Float64(Float64(Float64(x * log(x)) + Float64(x * log(Float64(1.0 / y)))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -4e-310)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	else
		tmp = ((x * log(x)) + (x * log((1.0 / y)))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4e-310], N[(N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x * N[Log[x], $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.999999999999988e-310

   1. Initial program 75.5%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log \left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(y\right)}\right)\right), z\right) \]

      2. log-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\log \left(\mathsf{neg}\left(x\right)\right) - \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log \left(\mathsf{neg}\left(x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\log 1 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), z\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(0 - y\right)\right)\right)\right), z\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\log 1 - y\right)\right)\right)\right), z\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, y\right)\right)\right)\right), z\right) \]

      13. metadata-eval 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

   4. Applied egg-rr 99.6%

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

   if -3.999999999999988e-310 < y

   1. Initial program 78.4%

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

      1. log-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\log x - \log y\right)\right), z\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\log x + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right), z\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\log x \cdot x + \left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right), z\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(\log x \cdot x\right), \left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)\right), z\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\log x, x\right), \left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)\right), z\right) \]

      6. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \left(\left(\mathsf{neg}\left(\log y\right)\right) \cdot x\right)\right), z\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\left(\mathsf{neg}\left(\log y\right)\right), x\right)\right), z\right) \]

      8. neg-log N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\log \left(\frac{1}{y}\right), x\right)\right), z\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{y}\right)\right), x\right)\right), z\right) \]

      10. /-lowering-/.f64 98.2%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(x\right), x\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(1, y\right)\right), x\right)\right), z\right) \]

   4. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\left(\log x \cdot x + \log \left(\frac{1}{y}\right) \cdot x\right)} - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log x + x \cdot \log \left(\frac{1}{y}\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- 0.0 z)
     (if (<= t_0 INFINITY) (- t_0 z) (- 0.0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0 - z;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 - z;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - z;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 - z;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.0 - z
	elif t_0 <= math.inf:
		tmp = t_0 - z
	else:
		tmp = 0.0 - z
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(t$95$0 - z), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 46.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 46.5%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 46.5%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 46.5%

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

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

   1. Initial program 85.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq \infty:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-305}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-183}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+172}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1e-305)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (if (<= x 7.6e-183)
     (- 0.0 z)
     (if (<= x 4.1e+172)
       (- (* (- 0.0 x) (log (/ y x))) z)
       (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-305) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	} else if (x <= 7.6e-183) {
		tmp = 0.0 - z;
	} else if (x <= 4.1e+172) {
		tmp = ((0.0 - x) * log((y / x))) - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1d-305)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - y)))) - z
    else if (x <= 7.6d-183) then
        tmp = 0.0d0 - z
    else if (x <= 4.1d+172) then
        tmp = ((0.0d0 - x) * log((y / x))) - z
    else
        tmp = x * (log(x) - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1e-305) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else if (x <= 7.6e-183) {
		tmp = 0.0 - z;
	} else if (x <= 4.1e+172) {
		tmp = ((0.0 - x) * Math.log((y / x))) - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1e-305:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - y)))) - z
	elif x <= 7.6e-183:
		tmp = 0.0 - z
	elif x <= 4.1e+172:
		tmp = ((0.0 - x) * math.log((y / x))) - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1e-305)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y)))) - z);
	elseif (x <= 7.6e-183)
		tmp = Float64(0.0 - z);
	elseif (x <= 4.1e+172)
		tmp = Float64(Float64(Float64(0.0 - x) * log(Float64(y / x))) - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1e-305)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	elseif (x <= 7.6e-183)
		tmp = 0.0 - z;
	elseif (x <= 4.1e+172)
		tmp = ((0.0 - x) * log((y / x))) - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1e-305], N[(N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 7.6e-183], N[(0.0 - z), $MachinePrecision], If[LessEqual[x, 4.1e+172], N[(N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.99999999999999996e-306

   1. Initial program 76.1%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log \left(\frac{\mathsf{neg}\left(x\right)}{\mathsf{neg}\left(y\right)}\right)\right), z\right) \]

      2. log-div N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\log \left(\mathsf{neg}\left(x\right)\right) - \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log \left(\mathsf{neg}\left(x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\left(\log 1 - x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \log \left(\mathsf{neg}\left(y\right)\right)\right)\right), z\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), z\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(0 - y\right)\right)\right)\right), z\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\left(\log 1 - y\right)\right)\right)\right), z\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\log 1, y\right)\right)\right)\right), z\right) \]

      13. metadata-eval 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), z\right) \]

   4. Applied egg-rr 99.6%

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

   if -9.99999999999999996e-306 < x < 7.5999999999999993e-183

   1. Initial program 55.3%

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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 100.0%

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

   if 7.5999999999999993e-183 < x < 4.1e172

   1. Initial program 89.2%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log \left(\frac{1}{\frac{y}{x}}\right)\right), z\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\log \left(\frac{y}{x}\right)\right)\right)\right), z\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\log \left(\frac{y}{x}\right)\right)\right), z\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{y}{x}\right)\right)\right)\right), z\right) \]

      5. /-lowering-/.f64 91.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right), z\right) \]

   4. Applied egg-rr 91.3%

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

   if 4.1e172 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)}\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\log x + \left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\log x - \color{blue}{\log y}\right)\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right) - \log \color{blue}{y}\right)\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right) - \log y\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(-1 \cdot \log \left(\frac{1}{x}\right) - \log \color{blue}{y}\right)\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(-1 \cdot \log \left(\frac{1}{x}\right)\right), \color{blue}{\log y}\right)\right) \]

      14. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), \log \color{blue}{y}\right)\right) \]

      15. log-rec N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), \log y\right)\right) \]

      16. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\log x, \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \log \color{blue}{y}\right)\right) \]

      18. log-lowering-log.f64 92.2%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{log.f64}\left(x\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   5. Simplified 92.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-305}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-183}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+172}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.1e-52)
   (- 0.0 z)
   (if (<= z 5.4e-18) (* x (log (/ x y))) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1e-52) {
		tmp = 0.0 - z;
	} else if (z <= 5.4e-18) {
		tmp = x * log((x / y));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.1d-52)) then
        tmp = 0.0d0 - z
    else if (z <= 5.4d-18) then
        tmp = x * log((x / y))
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.1e-52) {
		tmp = 0.0 - z;
	} else if (z <= 5.4e-18) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.1e-52:
		tmp = 0.0 - z
	elif z <= 5.4e-18:
		tmp = x * math.log((x / y))
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.1e-52)
		tmp = Float64(0.0 - z);
	elseif (z <= 5.4e-18)
		tmp = Float64(x * log(Float64(x / y)));
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.1e-52)
		tmp = 0.0 - z;
	elseif (z <= 5.4e-18)
		tmp = x * log((x / y));
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.1e-52], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 5.4e-18], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000005e-52 or 5.39999999999999977e-18 < z

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{z} \]

      3. --lowering--.f64 75.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   5. Simplified 75.2%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 75.2%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   7. Applied egg-rr 75.2%

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

   if -1.10000000000000005e-52 < z < 5.39999999999999977e-18

   1. Initial program 77.5%

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log \left(\frac{x}{y}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(\left(\frac{x}{y}\right)\right)\right) \]

      3. /-lowering-/.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(\mathsf{/.f64}\left(x, y\right)\right)\right) \]

   5. Simplified 69.2%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-52}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.3% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 0 - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- 0.0 z))

C

double code(double x, double y, double z) {
	return 0.0 - z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0d0 - z
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 - z;
}

Python

def code(x, y, z):
	return 0.0 - z

Julia

function code(x, y, z)
	return Float64(0.0 - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 - z;
end

Wolfram

code[x_, y_, z_] := N[(0.0 - z), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{z} \]

   3. --lowering--.f64 48.3%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

5. Simplified 48.3%

   \[\leadsto \color{blue}{0 - z} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 48.3%

      \[\leadsto \mathsf{neg.f64}\left(z\right) \]

7. Applied egg-rr 48.3%

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

8. Final simplification 48.3%

   \[\leadsto 0 - z \]
9. Add Preprocessing

Developer Target 1: 88.5% accurate, 0.5× 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

real(8) function code(x, y, z)
    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 2024138 
(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))