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

Percentage Accurate: 77.0% → 99.5%

Time: 12.3s

Alternatives: 8

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: 77.0% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(0 - x\right) - \log \left(0 - y\right), 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-310)
   (fma x (- (log (- 0.0 x)) (log (- 0.0 y))) (- 0.0 z))
   (- (* x (- (log x) (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2e-310) {
		tmp = fma(x, (log((0.0 - x)) - log((0.0 - y))), (0.0 - z));
	} else {
		tmp = (x * (log(x) - log(y))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2e-310)
		tmp = fma(x, Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y))), 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[y, -2e-310], N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $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 < -1.999999999999994e-310

   1. Initial program 74.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-in N/A

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

      3. distribute-neg-in N/A

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

      4. *-commutative N/A

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

      5. unsub-neg N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. cancel-sign-sub-inv N/A

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

      10. *-rgt-identity N/A

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(-\log \left(0 - y\right)\right) + \log \left(0 - x\right), 0 - z\right)} \]

   if -1.999999999999994e-310 < y

   1. Initial program 82.4%

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

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

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

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

      4. log-lowering-log.f64 99.2

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(0 - x\right) - \log \left(0 - y\right), 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= t_0 (- INFINITY))
     (- 0.0 z)
     (if (<= t_0 5e+307) (- 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 <= 5e+307) {
		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 <= 5e+307) {
		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 <= 5e+307:
		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 <= 5e+307)
		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 <= 5e+307)
		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, 5e+307], 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 5e307 < (*.f64 x (log.f64 (/.f64 x y)))

   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-sub0 N/A

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

      3. --lowering--.f64 48.4

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

   5. Simplified 48.4%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 48.4

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

   7. Applied egg-rr 48.4%

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

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

   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. Final simplification 87.9%

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

Alternative 3: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(0 - x\right) - \log \left(0 - y\right), 0\right)\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-221}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.2e+146)
   (fma x (- (log (- 0.0 x)) (log (- 0.0 y))) 0.0)
   (if (<= x -2.15e-221)
     (- (* x (log (/ x y))) z)
     (if (<= x -4e-308) (- 0.0 z) (- (* x (- (log x) (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.2e+146) {
		tmp = fma(x, (log((0.0 - x)) - log((0.0 - y))), 0.0);
	} else if (x <= -2.15e-221) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -4e-308) {
		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 <= -1.2e+146)
		tmp = fma(x, Float64(log(Float64(0.0 - x)) - log(Float64(0.0 - y))), 0.0);
	elseif (x <= -2.15e-221)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -4e-308)
		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, -1.2e+146], N[(x * N[(N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision] - N[Log[N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision], If[LessEqual[x, -2.15e-221], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-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 4 regimes

2. if x < -1.2000000000000001e146

   1. Initial program 56.2%

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

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x + \log \left(\frac{1}{y}\right), 0\right)} \]

      9. log-rec N/A

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

      10. unsub-neg N/A

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

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

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

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

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

      13. log-lowering-log.f64 0.0

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

   5. Simplified 0.0%

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

      1. diff-log N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log \left(\frac{x}{y}\right)}, 0\right) \]

      2. frac-2neg N/A

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

      3. log-div N/A

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

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

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

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

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

      6. neg-sub0 N/A

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

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

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

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

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

      9. neg-sub0 N/A

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

      10. --lowering--.f64 79.7

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

   7. Applied egg-rr 79.7%

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

   if -1.2000000000000001e146 < x < -2.1499999999999999e-221

   1. Initial program 92.9%

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

   if -2.1499999999999999e-221 < x < -4.00000000000000013e-308

   1. Initial program 43.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 100.0

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

   5. Simplified 100.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -4.00000000000000013e-308 < x

   1. Initial program 82.4%

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

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

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

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

      4. log-lowering-log.f64 99.2

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(x, \log \left(0 - x\right) - \log \left(0 - y\right), 0\right)\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-221}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-221}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.15e-221)
   (- (* x (log (/ x y))) z)
   (if (<= x -4e-308) (- 0.0 z) (- (* x (- (log x) (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.15e-221) {
		tmp = (x * log((x / y))) - z;
	} else if (x <= -4e-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 <= (-2.15d-221)) then
        tmp = (x * log((x / y))) - z
    else if (x <= (-4d-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 <= -2.15e-221) {
		tmp = (x * Math.log((x / y))) - z;
	} else if (x <= -4e-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 <= -2.15e-221:
		tmp = (x * math.log((x / y))) - z
	elif x <= -4e-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 <= -2.15e-221)
		tmp = Float64(Float64(x * log(Float64(x / y))) - z);
	elseif (x <= -4e-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 <= -2.15e-221)
		tmp = (x * log((x / y))) - z;
	elseif (x <= -4e-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, -2.15e-221], N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, -4e-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 < -2.1499999999999999e-221

   1. Initial program 78.9%

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

   if -2.1499999999999999e-221 < x < -4.00000000000000013e-308

   1. Initial program 43.3%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 100.0

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

   5. Simplified 100.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if -4.00000000000000013e-308 < x

   1. Initial program 82.4%

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

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

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

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

      4. log-lowering-log.f64 99.2

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-221}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-308}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(0 - x\right) - \log \left(0 - y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2e-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 <= -2e-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 <= (-2d-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 <= -2e-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 <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 74.3%

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

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

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

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

      5. neg-sub0 N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

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

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

      10. neg-sub0 N/A

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

      11. metadata-eval N/A

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

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

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

      13. metadata-eval 99.3

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

   4. Applied egg-rr 99.3%

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

   if -1.999999999999994e-310 < y

   1. Initial program 82.4%

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

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

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

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

      4. log-lowering-log.f64 99.2

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

   4. Applied egg-rr 99.2%

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

Alternative 6: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (log (/ x y)))))
   (if (<= x -3.8e+56) t_0 (if (<= x 7.2e-87) (- 0.0 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * log((x / y));
	double tmp;
	if (x <= -3.8e+56) {
		tmp = t_0;
	} else if (x <= 7.2e-87) {
		tmp = 0.0 - 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 <= (-3.8d+56)) then
        tmp = t_0
    else if (x <= 7.2d-87) then
        tmp = 0.0d0 - 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 <= -3.8e+56) {
		tmp = t_0;
	} else if (x <= 7.2e-87) {
		tmp = 0.0 - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * math.log((x / y))
	tmp = 0
	if x <= -3.8e+56:
		tmp = t_0
	elif x <= 7.2e-87:
		tmp = 0.0 - 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 <= -3.8e+56)
		tmp = t_0;
	elseif (x <= 7.2e-87)
		tmp = Float64(0.0 - 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 <= -3.8e+56)
		tmp = t_0;
	elseif (x <= 7.2e-87)
		tmp = 0.0 - 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, -3.8e+56], t$95$0, If[LessEqual[x, 7.2e-87], N[(0.0 - z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999996e56 or 7.19999999999999986e-87 < x

   1. Initial program 77.8%

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

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

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log x + \log \left(\frac{1}{y}\right), 0\right)} \]

      9. log-rec N/A

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

      10. unsub-neg N/A

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

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

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

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

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

      13. log-lowering-log.f64 42.9

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

   5. Simplified 42.9%

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

      1. +-rgt-identity N/A

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

      2. diff-log N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 60.1

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

   7. Applied egg-rr 60.1%

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

   if -3.79999999999999996e56 < x < 7.19999999999999986e-87

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 82.8

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

   5. Simplified 82.8%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 82.8

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

   7. Applied egg-rr 82.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+56}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.3% accurate, 30.0× 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 78.3%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 48.6

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

5. Simplified 48.6%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 48.6

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

7. Applied egg-rr 48.6%

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

8. Final simplification 48.6%

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

Alternative 8: 2.3% 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 78.3%

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

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 48.6

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

5. Simplified 48.6%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 48.6

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

7. Applied egg-rr 48.6%

   \[\leadsto \color{blue}{-z} \]
8. 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. +-rgt-identity N/A

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

   5. +-lft-identity N/A

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

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

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

   7. +-lft-identity N/A

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

   8. +-commutative N/A

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

   9. sub0-neg N/A

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

   10. +-rgt-identity N/A

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

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

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

   12. neg-sub0 N/A

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

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

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(z, 0 - z, 0\right)}}{z} \]

   14. --lowering--.f64 26.7

       \[\leadsto \frac{\mathsf{fma}\left(z, \color{blue}{0 - z}, 0\right)}{z} \]

9. Applied egg-rr 26.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(z, 0 - z, 0\right)}{z}} \]

11. Applied egg-rr 2.2%

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

Developer Target 1: 88.2% 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 2024195 
(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))