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

Percentage Accurate: 77.1% → 99.4%

Time: 12.6s

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

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: 77.1% 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

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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-y\right)\\ t_1 := \log \left(-x\right)\\ t_2 := {t\_1}^{2} + \left({t\_0}^{2} + t\_1 \cdot t\_0\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\frac{{t\_1}^{3}}{t\_2} - \frac{{t\_0}^{3}}{t\_2}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- y)))
        (t_1 (log (- x)))
        (t_2 (+ (pow t_1 2.0) (+ (pow t_0 2.0) (* t_1 t_0)))))
   (if (<= y -4e-311)
     (- (* x (- (/ (pow t_1 3.0) t_2) (/ (pow t_0 3.0) t_2))) z)
     (- (fma (log x) x (* (- x) (log y))) z))))

C

double code(double x, double y, double z) {
	double t_0 = log(-y);
	double t_1 = log(-x);
	double t_2 = pow(t_1, 2.0) + (pow(t_0, 2.0) + (t_1 * t_0));
	double tmp;
	if (y <= -4e-311) {
		tmp = (x * ((pow(t_1, 3.0) / t_2) - (pow(t_0, 3.0) / t_2))) - z;
	} else {
		tmp = fma(log(x), x, (-x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-y))
	t_1 = log(Float64(-x))
	t_2 = Float64((t_1 ^ 2.0) + Float64((t_0 ^ 2.0) + Float64(t_1 * t_0)))
	tmp = 0.0
	if (y <= -4e-311)
		tmp = Float64(Float64(x * Float64(Float64((t_1 ^ 3.0) / t_2) - Float64((t_0 ^ 3.0) / t_2))) - z);
	else
		tmp = Float64(fma(log(x), x, Float64(Float64(-x) * log(y))) - z);
	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]}, Block[{t$95$2 = 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]}, If[LessEqual[y, -4e-311], N[(N[(x * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[Power[t$95$0, 3.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * x + N[((-x) * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.99999999999979e-311

   1. Initial program 79.6%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

      3. flip3-- N/A

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

      4. div-sub N/A

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

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

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

   4. Applied egg-rr 99.2%

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

   if -3.99999999999979e-311 < y

   1. Initial program 76.2%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

      8. log-lowering-log.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-311}:\\ \;\;\;\;x \cdot \left(\frac{{\log \left(-x\right)}^{3}}{{\log \left(-x\right)}^{2} + \left({\log \left(-y\right)}^{2} + \log \left(-x\right) \cdot \log \left(-y\right)\right)} - \frac{{\log \left(-y\right)}^{3}}{{\log \left(-x\right)}^{2} + \left({\log \left(-y\right)}^{2} + \log \left(-x\right) \cdot \log \left(-y\right)\right)}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- z)
     (if (<= t_0 2e+304) (- t_0 z) (* x (- (log x) (log y)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+304) {
		tmp = t_0 - z;
	} else {
		tmp = x * (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+304) {
		tmp = t_0 - z;
	} else {
		tmp = x * (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+304:
		tmp = t_0 - z
	else:
		tmp = x * (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+304)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(x * Float64(log(x) - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+304)
		tmp = t_0 - z;
	else
		tmp = x * (log(x) - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+304], N[(t$95$0 - z), $MachinePrecision], N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 6.6%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. neg-lowering-neg.f64 56.4

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

   5. Simplified 56.4%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.9999999999999999e304

   1. Initial program 99.7%

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

   if 1.9999999999999999e304 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 8.3%

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

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

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

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

      8. log-rec N/A

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

      9. unsub-neg N/A

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

      10. --lowering--.f64 N/A

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

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

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

      12. log-lowering-log.f64 59.3

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

   5. Simplified 59.3%

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

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY)) (- z) (if (<= t_0 2e+307) (- t_0 z) (- z)))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = -z;
	} else if (t_0 <= 2e+307) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = -z;
	} else if (t_0 <= 2e+307) {
		tmp = t_0 - z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = -z
	elif t_0 <= 2e+307:
		tmp = t_0 - z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(-z);
	elseif (t_0 <= 2e+307)
		tmp = Float64(t_0 - z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = -z;
	elseif (t_0 <= 2e+307)
		tmp = t_0 - z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], (-z), If[LessEqual[t$95$0, 2e+307], N[(t$95$0 - z), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (log.f64 (/.f64 x y))) < -inf.0 or 1.99999999999999997e307 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 6.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 49.3

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

   5. Simplified 49.3%

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

   if -inf.0 < (*.f64 x (log.f64 (/.f64 x y))) < 1.99999999999999997e307

   1. Initial program 99.7%

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

Alternative 4: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right)\\ t_1 := \log \left(-y\right)\\ \mathbf{if}\;x \leq -6 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, \left(-x\right) \cdot t\_1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-z\right) \cdot \mathsf{fma}\left(\frac{t\_0 - t\_1}{z}, -x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- x))) (t_1 (log (- y))))
   (if (<= x -6e+185)
     (fma t_0 x (* (- x) t_1))
     (if (<= x -5e-310)
       (* (- z) (fma (/ (- t_0 t_1) z) (- x) 1.0))
       (- (fma (log x) x (* (- x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x);
	double t_1 = log(-y);
	double tmp;
	if (x <= -6e+185) {
		tmp = fma(t_0, x, (-x * t_1));
	} else if (x <= -5e-310) {
		tmp = -z * fma(((t_0 - t_1) / z), -x, 1.0);
	} else {
		tmp = fma(log(x), x, (-x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-x))
	t_1 = log(Float64(-y))
	tmp = 0.0
	if (x <= -6e+185)
		tmp = fma(t_0, x, Float64(Float64(-x) * t_1));
	elseif (x <= -5e-310)
		tmp = Float64(Float64(-z) * fma(Float64(Float64(t_0 - t_1) / z), Float64(-x), 1.0));
	else
		tmp = Float64(fma(log(x), x, Float64(Float64(-x) * log(y))) - z);
	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, -6e+185], N[(t$95$0 * x + N[((-x) * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-310], N[((-z) * N[(N[(N[(t$95$0 - t$95$1), $MachinePrecision] / z), $MachinePrecision] * (-x) + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * x + N[((-x) * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.99999999999999988e185

   1. Initial program 78.2%

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

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

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

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

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

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 91.1

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

   5. Simplified 91.1%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 91.2

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

   7. Applied egg-rr 91.2%

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

   if -5.99999999999999988e185 < 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. clear-num N/A

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

      2. log-rec N/A

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

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

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

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

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

      5. /-lowering-/.f64 78.1

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

   4. Applied egg-rr 78.1%

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

   5. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. accelerator-lowering-fma.f64 N/A

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

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

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

      16. /-lowering-/.f64 N/A

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

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

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

      18. neg-lowering-neg.f64 74.7

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

   7. Simplified 74.7%

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

   8. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. neg-mul-1 N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lft-mult-inverse N/A

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

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

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

   10. Simplified 95.3%

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

   if -4.999999999999985e-310 < x

   1. Initial program 76.2%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

      8. log-lowering-log.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+185}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \left(-x\right) \cdot \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(-z\right) \cdot \mathsf{fma}\left(\frac{\log \left(-x\right) - \log \left(-y\right)}{z}, -x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(-x\right)\\ t_1 := \log \left(-y\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, \left(-x\right) \cdot t\_1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(t\_0 - t\_1, \frac{x}{z}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (- x))) (t_1 (log (- y))))
   (if (<= x -6.5e+189)
     (fma t_0 x (* (- x) t_1))
     (if (<= x -5e-310)
       (* z (fma (- t_0 t_1) (/ x z) -1.0))
       (- (fma (log x) x (* (- x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double t_0 = log(-x);
	double t_1 = log(-y);
	double tmp;
	if (x <= -6.5e+189) {
		tmp = fma(t_0, x, (-x * t_1));
	} else if (x <= -5e-310) {
		tmp = z * fma((t_0 - t_1), (x / z), -1.0);
	} else {
		tmp = fma(log(x), x, (-x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(-x))
	t_1 = log(Float64(-y))
	tmp = 0.0
	if (x <= -6.5e+189)
		tmp = fma(t_0, x, Float64(Float64(-x) * t_1));
	elseif (x <= -5e-310)
		tmp = Float64(z * fma(Float64(t_0 - t_1), Float64(x / z), -1.0));
	else
		tmp = Float64(fma(log(x), x, Float64(Float64(-x) * log(y))) - z);
	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, -6.5e+189], N[(t$95$0 * x + N[((-x) * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-310], N[(z * N[(N[(t$95$0 - t$95$1), $MachinePrecision] * N[(x / z), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[x], $MachinePrecision] * x + N[((-x) * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.50000000000000027e189

   1. Initial program 80.4%

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

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

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

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

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

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 90.5

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

   5. Simplified 90.5%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. neg-mul-1 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. neg-mul-1 N/A

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

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

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

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

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

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

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

      14. neg-lowering-neg.f64 90.6

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

   7. Applied egg-rr 90.6%

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

   if -6.50000000000000027e189 < x < -4.999999999999985e-310

   1. Initial program 79.4%

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

      1. *-commutative N/A

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

      2. log-div N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

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

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

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

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

      7. difference-of-squares N/A

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

      8. log-div N/A

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

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

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

      10. sum-log N/A

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

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

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

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

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

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

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

      14. /-lowering-/.f64 N/A

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

      15. sum-log N/A

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

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

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

      17. *-lowering-*.f64 58.7

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

   4. Applied egg-rr 58.7%

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

   5. Taylor expanded in z around inf

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

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

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

      2. sub-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. /-lowering-/.f64 76.1

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

   7. Simplified 76.1%

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

      1. frac-2neg N/A

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

      2. log-div N/A

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

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

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

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

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

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

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

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

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

      7. neg-lowering-neg.f64 95.4

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

   9. Applied egg-rr 95.4%

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

   if -4.999999999999985e-310 < x

   1. Initial program 76.2%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

      8. log-lowering-log.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(-x\right), x, \left(-x\right) \cdot \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(\log \left(-x\right) - \log \left(-y\right), \frac{x}{z}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-208}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7.6e+76)
   (* x (- (log (- x)) (log (- y))))
   (if (<= x -7.5e-208)
     (- (* x (log (/ x y))) z)
     (if (<= x -2e-308) (- z) (- (fma (log x) x (* (- x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7.6e+76) {
		tmp = x * (log(-x) - log(-y));
	} else if (x <= -7.5e-208) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -2e-308) {
		tmp = -z;
	} else {
		tmp = fma(log(x), x, (-x * log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7.6e+76)
		tmp = Float64(x * Float64(log(Float64(-x)) - log(Float64(-y))));
	elseif (x <= -7.5e-208)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -2e-308)
		tmp = Float64(-z);
	else
		tmp = Float64(fma(log(x), x, Float64(Float64(-x) * log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7.6e+76], N[(x * N[(N[Log[(-x)], $MachinePrecision] - N[Log[(-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-208], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], (-z), N[(N[(N[Log[x], $MachinePrecision] * x + N[((-x) * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.60000000000000049e76

   1. Initial program 75.4%

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

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

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

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

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

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 86.4

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

   5. Simplified 86.4%

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

      1. +-commutative N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

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

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

      7. neg-lowering-neg.f64 86.4

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

   7. Applied egg-rr 86.4%

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

   if -7.60000000000000049e76 < x < -7.4999999999999999e-208

   1. Initial program 90.4%

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

   if -7.4999999999999999e-208 < x < -1.9999999999999998e-308

   1. Initial program 53.8%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. neg-lowering-neg.f64 96.2

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

   5. Simplified 96.2%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 76.2%

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

      1. log-div N/A

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

      2. sub-neg N/A

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

      3. distribute-rgt-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

      8. log-lowering-log.f64 99.3

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right)\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-208}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x, \left(-x\right) \cdot \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{+74}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+108}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.75e+74)
   (* x (log (/ x y)))
   (if (<= x 3.1e+108) (- z) (* (- x) (log (/ y x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.75e+74) {
		tmp = x * log((x / y));
	} else if (x <= 3.1e+108) {
		tmp = -z;
	} else {
		tmp = -x * log((y / x));
	}
	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 <= (-1.75d+74)) then
        tmp = x * log((x / y))
    else if (x <= 3.1d+108) then
        tmp = -z
    else
        tmp = -x * log((y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.75e+74) {
		tmp = x * Math.log((x / y));
	} else if (x <= 3.1e+108) {
		tmp = -z;
	} else {
		tmp = -x * Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.75e+74:
		tmp = x * math.log((x / y))
	elif x <= 3.1e+108:
		tmp = -z
	else:
		tmp = -x * math.log((y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.75e+74)
		tmp = Float64(x * log(Float64(x / y)));
	elseif (x <= 3.1e+108)
		tmp = Float64(-z);
	else
		tmp = Float64(Float64(-x) * log(Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.75e+74)
		tmp = x * log((x / y));
	elseif (x <= 3.1e+108)
		tmp = -z;
	else
		tmp = -x * log((y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.75e+74], N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+108], (-z), N[((-x) * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.75000000000000007e74

   1. Initial program 76.4%

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

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

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

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

      3. /-lowering-/.f64 70.5

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

   5. Simplified 70.5%

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

   if -1.75000000000000007e74 < x < 3.1000000000000001e108

   1. Initial program 84.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. neg-lowering-neg.f64 73.7

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

   5. Simplified 73.7%

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

   if 3.1000000000000001e108 < x

   1. Initial program 61.7%

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

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

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

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

      3. remove-double-neg N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

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

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

      13. metadata-eval N/A

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

      14. associate-/r* N/A

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

      15. remove-double-div N/A

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

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

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

      17. mul-1-neg N/A

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

      18. neg-lowering-neg.f64 0.0

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

   5. Simplified 0.0%

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

      1. neg-sub0 N/A

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

      2. associate-+l- N/A

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

      3. log-div N/A

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

      4. frac-2neg N/A

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

      5. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

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

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

      11. /-lowering-/.f64 59.2

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

   7. Applied egg-rr 59.2%

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

Alternative 8: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+108}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= x -4.2e+72) t_0 (if (<= x 8e+108) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (x <= -4.2e+72) {
		tmp = t_0;
	} else if (x <= 8e+108) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = x * log((x / y))
    if (x <= (-4.2d+72)) then
        tmp = t_0
    else if (x <= 8d+108) then
        tmp = -z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * Math.log((x / y));
	double tmp;
	if (x <= -4.2e+72) {
		tmp = t_0;
	} else if (x <= 8e+108) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if x <= -4.2e+72:
		tmp = t_0
	elif x <= 8e+108:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * log(Float64(x / y)))
	tmp = 0.0
	if (x <= -4.2e+72)
		tmp = t_0;
	elseif (x <= 8e+108)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * log((x / y));
	tmp = 0.0;
	if (x <= -4.2e+72)
		tmp = t_0;
	elseif (x <= 8e+108)
		tmp = -z;
	else
		tmp = t_0;
	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[x, -4.2e+72], t$95$0, If[LessEqual[x, 8e+108], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000003e72 or 8.0000000000000003e108 < x

   1. Initial program 68.6%

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

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

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

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

      3. /-lowering-/.f64 64.0

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

   5. Simplified 64.0%

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

   if -4.2000000000000003e72 < x < 8.0000000000000003e108

   1. Initial program 84.1%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. neg-lowering-neg.f64 73.7

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

   5. Simplified 73.7%

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

Alternative 9: 50.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 49.1

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

5. Simplified 49.1%

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

Alternative 10: 2.2% accurate, 120.0× 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

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 49.1

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

5. Simplified 49.1%

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

   1. neg-sub0 N/A

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

   2. flip-- N/A

      \[\leadsto \color{blue}{\frac{0 \cdot 0 - z \cdot z}{0 + z}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0} - z \cdot z}{0 + z} \]

   4. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(z \cdot z\right)}}{0 + z} \]

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

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(z \cdot z\right)}{0 + z}} \]

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

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(z \cdot z\right)}}{0 + z} \]

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

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{z \cdot z}\right)}{0 + z} \]

   8. +-lowering-+.f64 27.1

      \[\leadsto \frac{-z \cdot z}{\color{blue}{0 + z}} \]

7. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{\frac{-z \cdot z}{0 + z}} \]
8. Step-by-step derivation

   1. neg-sub0 N/A

      \[\leadsto \frac{\color{blue}{0 - z \cdot z}}{0 + z} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0 \cdot 0} - z \cdot z}{0 + z} \]

   3. flip-- N/A

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

   4. neg-sub0 N/A

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

   5. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(0 + z\right)}\right) \]

   6. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{{0}^{3} + {z}^{3}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)}}\right) \]

   7. distribute-neg-frac N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left({0}^{3} + {z}^{3}\right)\right)}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{0} + {z}^{3}\right)\right)}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   9. +-lft-identity N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{{z}^{3}}\right)}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   10. cube-neg N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{3}}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   11. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(z\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   12. pow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(\mathsf{neg}\left(z\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   13. sqr-neg N/A

       \[\leadsto \frac{{\color{blue}{\left(z \cdot z\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   14. pow-prod-down N/A

       \[\leadsto \frac{\color{blue}{{z}^{\left(\frac{3}{2}\right)} \cdot {z}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   15. sqr-pow N/A

       \[\leadsto \frac{\color{blue}{{z}^{3}}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   16. +-lft-identity N/A

       \[\leadsto \frac{\color{blue}{0 + {z}^{3}}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{{0}^{3}} + {z}^{3}}{0 \cdot 0 + \left(z \cdot z - 0 \cdot z\right)} \]

   18. flip3-+ N/A

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

   19. +-lft-identity 2.2

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

9. Applied egg-rr 2.2%

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

Developer Target 1: 87.8% 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

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 2024204 
(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))