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

Percentage Accurate: 78.0% → 99.5%

Time: 10.1s

Alternatives: 10

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 78.0% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-310)
   (- (* x (- (log (- 0.0 x)) (log (- 0.0 y)))) z)
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-310) {
		tmp = (x * (log((0.0 - x)) - log((0.0 - 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 <= (-5d-310)) then
        tmp = (x * (log((0.0d0 - x)) - log((0.0d0 - 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 <= -5e-310) {
		tmp = (x * (Math.log((0.0 - x)) - Math.log((0.0 - y)))) - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-310:
		tmp = (x * (math.log((0.0 - x)) - math.log((0.0 - 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 <= -5e-310)
		tmp = Float64(Float64(x * Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - 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 <= -5e-310)
		tmp = (x * (log((0.0 - x)) - log((0.0 - y)))) - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -4.999999999999985e-310

   1. Initial program 76.9%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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 99.5%

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

   4. Applied egg-rr 99.5%

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

   if -4.999999999999985e-310 < y

   1. Initial program 80.7%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

Alternative 2: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- 0.0 z)
     (if (<= t_1 2e+304) (fma t_0 x (- 0.0 z)) (- 0.0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - z;
	} else if (t_1 <= 2e+304) {
		tmp = fma(t_0, x, (0.0 - z));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - z);
	elseif (t_1 <= 2e+304)
		tmp = fma(t_0, x, Float64(0.0 - z));
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(t$95$0 * x + N[(0.0 - z), $MachinePrecision]), $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 1.9999999999999999e304 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 46.2%

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

   5. Simplified 46.2%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 46.2%

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

   7. Applied egg-rr 46.2%

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

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

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. fmm-def N/A

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval 99.5%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.5%

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

   6. Applied egg-rr 99.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{x}{\frac{1}{t\_0}} - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (/ x y))) (t_1 (* x t_0)))
   (if (<= t_1 (- INFINITY))
     (- 0.0 z)
     (if (<= t_1 2e+304) (- (/ x (/ 1.0 t_0)) z) (- 0.0 z)))))

C

double code(double x, double y, double z) {
	double t_0 = log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 0.0 - z;
	} else if (t_1 <= 2e+304) {
		tmp = (x / (1.0 / t_0)) - z;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log((x / y));
	double t_1 = x * t_0;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0 - z;
	} else if (t_1 <= 2e+304) {
		tmp = (x / (1.0 / t_0)) - z;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log((x / y))
	t_1 = x * t_0
	tmp = 0
	if t_1 <= -math.inf:
		tmp = 0.0 - z
	elif t_1 <= 2e+304:
		tmp = (x / (1.0 / t_0)) - z
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	t_0 = log(Float64(x / y))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(0.0 - z);
	elseif (t_1 <= 2e+304)
		tmp = Float64(Float64(x / Float64(1.0 / t_0)) - z);
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log((x / y));
	t_1 = x * t_0;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = 0.0 - z;
	elseif (t_1 <= 2e+304)
		tmp = (x / (1.0 / t_0)) - z;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(0.0 - z), $MachinePrecision], If[LessEqual[t$95$1, 2e+304], N[(N[(x / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 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 1.9999999999999999e304 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 46.2%

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

   5. Simplified 46.2%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 46.2%

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

   7. Applied egg-rr 46.2%

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

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

   1. Initial program 99.5%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 53.9%

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

   4. Applied egg-rr 53.9%

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

      1. diff-log N/A

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

      2. *-commutative N/A

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

      3. remove-double-div N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. remove-double-div N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 99.5%

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

   8. Applied egg-rr 99.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;\frac{x}{\frac{1}{\log \left(\frac{x}{y}\right)}} - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- 0.0 z)
     (if (<= t_0 2e+304) (- 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 <= 2e+304) {
		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 <= 2e+304) {
		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 <= 2e+304:
		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 <= 2e+304)
		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 <= 2e+304)
		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, 2e+304], 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 1.9999999999999999e304 < (*.f64 x (log.f64 (/.f64 x y)))

   1. Initial program 4.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 46.2%

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

   5. Simplified 46.2%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 46.2%

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

   7. Applied egg-rr 46.2%

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

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

   1. Initial program 99.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log \left(\frac{x}{y}\right) \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;x \cdot \log \left(\frac{x}{y}\right) \leq 2 \cdot 10^{+304}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-307}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.7e+128)
   (* x (+ (log (- 0.0 x)) (log (/ -1.0 y))))
   (if (<= x -1.25e-104)
     (fma (log (/ x y)) x (- 0.0 z))
     (if (<= x -1e-307) (- 0.0 z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.7e+128) {
		tmp = x * (log((0.0 - x)) + log((-1.0 / y)));
	} else if (x <= -1.25e-104) {
		tmp = fma(log((x / y)), x, (0.0 - z));
	} else if (x <= -1e-307) {
		tmp = 0.0 - z;
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.7e+128)
		tmp = Float64(x * Float64(log(Float64(0.0 - x)) + log(Float64(-1.0 / y))));
	elseif (x <= -1.25e-104)
		tmp = fma(log(Float64(x / y)), x, Float64(0.0 - z));
	elseif (x <= -1e-307)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.7e+128], N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.25e-104], N[(N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision] * x + N[(0.0 - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-307], N[(0.0 - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.6999999999999999e128

   1. Initial program 62.9%

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. metadata-eval N/A

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

      8. associate-/r* N/A

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

      9. remove-double-div N/A

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

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

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

      11. mul-1-neg N/A

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

      12. neg-sub0 N/A

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

      13. --lowering--.f64 87.8%

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

   5. Simplified 87.8%

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

   if -1.6999999999999999e128 < x < -1.24999999999999995e-104

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. fmm-def N/A

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

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

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

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

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

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

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

      6. neg-sub0 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)} \]
   5. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 99.7%

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

   6. Applied egg-rr 99.7%

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

   if -1.24999999999999995e-104 < x < -9.99999999999999909e-308

   1. Initial program 66.8%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 83.8%

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

   5. Simplified 83.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.8%

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

   7. Applied egg-rr 83.8%

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

   if -9.99999999999999909e-308 < x

   1. Initial program 80.7%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;x \leq -1.25 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, 0 - z\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-307}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{\frac{1}{\log \left(\frac{x}{y}\right)}} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.8e-105)
   (- (/ x (/ 1.0 (log (/ x y)))) z)
   (if (<= x -2e-308) (- 0.0 z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e-105) {
		tmp = (x / (1.0 / log((x / y)))) - z;
	} else if (x <= -2e-308) {
		tmp = 0.0 - 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 (x <= (-6.8d-105)) then
        tmp = (x / (1.0d0 / log((x / y)))) - z
    else if (x <= (-2d-308)) then
        tmp = 0.0d0 - z
    else
        tmp = (x * (log(x) - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e-105) {
		tmp = (x / (1.0 / Math.log((x / y)))) - z;
	} else if (x <= -2e-308) {
		tmp = 0.0 - z;
	} else {
		tmp = (x * (Math.log(x) - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.8e-105:
		tmp = (x / (1.0 / math.log((x / y)))) - z
	elif x <= -2e-308:
		tmp = 0.0 - z
	else:
		tmp = (x * (math.log(x) - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.8e-105)
		tmp = Float64(Float64(x / Float64(1.0 / log(Float64(x / y)))) - z);
	elseif (x <= -2e-308)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(Float64(x * Float64(log(x) - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.8e-105)
		tmp = (x / (1.0 / log((x / y)))) - z;
	elseif (x <= -2e-308)
		tmp = 0.0 - z;
	else
		tmp = (x * (log(x) - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.8e-105], N[(N[(x / N[(1.0 / N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -2e-308], N[(0.0 - z), $MachinePrecision], N[(N[(x * N[(N[Log[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.79999999999999984e-105

   1. Initial program 83.3%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 0.0%

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

   4. Applied egg-rr 0.0%

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

      1. diff-log N/A

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

      2. *-commutative N/A

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

      3. remove-double-div N/A

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

      4. un-div-inv N/A

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

      5. clear-num N/A

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 83.2%

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

   6. Applied egg-rr 83.2%

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

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. remove-double-div N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 83.3%

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

   8. Applied egg-rr 83.3%

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

   if -6.79999999999999984e-105 < x < -1.9999999999999998e-308

   1. Initial program 66.8%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 83.8%

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

   5. Simplified 83.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.8%

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

   7. Applied egg-rr 83.8%

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

   if -1.9999999999999998e-308 < x

   1. Initial program 80.7%

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

      1. log-div N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{\frac{1}{\log \left(\frac{x}{y}\right)}} - z\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -4.8e+26)
   (- 0.0 z)
   (if (<= z 1.35e-108) (* (- 0.0 x) (log (/ y x))) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -4.8e+26) {
		tmp = 0.0 - z;
	} else if (z <= 1.35e-108) {
		tmp = (0.0 - x) * log((y / x));
	} 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 <= (-4.8d+26)) then
        tmp = 0.0d0 - z
    else if (z <= 1.35d-108) then
        tmp = (0.0d0 - x) * log((y / x))
    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 <= -4.8e+26) {
		tmp = 0.0 - z;
	} else if (z <= 1.35e-108) {
		tmp = (0.0 - x) * Math.log((y / x));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -4.8e+26:
		tmp = 0.0 - z
	elif z <= 1.35e-108:
		tmp = (0.0 - x) * math.log((y / x))
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -4.8e+26)
		tmp = Float64(0.0 - z);
	elseif (z <= 1.35e-108)
		tmp = Float64(Float64(0.0 - x) * log(Float64(y / x)));
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -4.8e+26)
		tmp = 0.0 - z;
	elseif (z <= 1.35e-108)
		tmp = (0.0 - x) * log((y / x));
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -4.8e+26], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 1.35e-108], N[(N[(0.0 - x), $MachinePrecision] * N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.80000000000000009e26 or 1.35000000000000002e-108 < z

   1. Initial program 83.6%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 74.4%

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

   5. Simplified 74.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.4%

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

   7. Applied egg-rr 74.4%

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

   if -4.80000000000000009e26 < z < 1.35000000000000002e-108

   1. Initial program 73.7%

      \[x \cdot \log \left(\frac{x}{y}\right) - z \]
   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. 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. remove-double-neg N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. unsub-neg N/A

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

      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 44.2%

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

   5. Simplified 44.2%

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

      1. diff-log N/A

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

      2. clear-num N/A

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

      3. log-rec N/A

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

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

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

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

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

      6. /-lowering-/.f64 63.3%

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

   7. Applied egg-rr 63.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+26}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;\left(0 - x\right) \cdot \log \left(\frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+28}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.2e+28)
   (- 0.0 z)
   (if (<= z 1.35e-108) (* x (log (/ x y))) (- 0.0 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.2e+28) {
		tmp = 0.0 - z;
	} else if (z <= 1.35e-108) {
		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 <= (-6.2d+28)) then
        tmp = 0.0d0 - z
    else if (z <= 1.35d-108) 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 <= -6.2e+28) {
		tmp = 0.0 - z;
	} else if (z <= 1.35e-108) {
		tmp = x * Math.log((x / y));
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.2e+28:
		tmp = 0.0 - z
	elif z <= 1.35e-108:
		tmp = x * math.log((x / y))
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.2e+28)
		tmp = Float64(0.0 - z);
	elseif (z <= 1.35e-108)
		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 <= -6.2e+28)
		tmp = 0.0 - z;
	elseif (z <= 1.35e-108)
		tmp = x * log((x / y));
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.2e+28], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 1.35e-108], 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 < -6.2000000000000001e28 or 1.35000000000000002e-108 < z

   1. Initial program 83.6%

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

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 74.4%

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

   5. Simplified 74.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.4%

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

   7. Applied egg-rr 74.4%

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

   if -6.2000000000000001e28 < z < 1.35000000000000002e-108

   1. Initial program 73.7%

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

   3. Taylor expanded in z around 0

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

      1. *-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 63.1%

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

   5. Simplified 63.1%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+28}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.8% 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 79.0%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 49.0%

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

5. Simplified 49.0%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 49.0%

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

7. Applied egg-rr 49.0%

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

8. Final simplification 49.0%

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

Alternative 10: 2.3% accurate, 107.0× 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 79.0%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 49.0%

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

5. Simplified 49.0%

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

   1. flip3-- N/A

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

   2. metadata-eval N/A

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

   3. sub0-neg N/A

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

   4. cube-neg N/A

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

   5. sub0-neg N/A

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

   6. sqr-pow N/A

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

   7. unpow-prod-down N/A

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

   8. sub0-neg N/A

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

   9. sub0-neg N/A

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

   10. sqr-neg N/A

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

   11. unpow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. metadata-eval N/A

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

   14. +-lft-identity N/A

       \[\leadsto \frac{{z}^{3}}{z \cdot z + \color{blue}{0 \cdot z}} \]

   15. distribute-rgt-in N/A

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

   16. +-rgt-identity N/A

       \[\leadsto \frac{{z}^{3}}{z \cdot z} \]

   17. pow2 N/A

       \[\leadsto \frac{{z}^{3}}{{z}^{\color{blue}{2}}} \]

   18. pow-div N/A

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

   19. metadata-eval N/A

       \[\leadsto {z}^{1} \]

   20. unpow1 2.5%

       \[\leadsto z \]

7. Applied egg-rr 2.5%

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

Developer Target 1: 88.9% 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 2024161 
(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))