Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + 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 - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))

C

double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * log(y))) + 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 - ((y + 0.5d0) * log(y))) + y) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((x - ((y + 0.5) * Math.log(y))) + y) - z;
}

Python

def code(x, y, z):
	return ((x - ((y + 0.5) * math.log(y))) + y) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - ((y + 0.5) * log(y))) + y) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (fma (log y) (- -0.5 y) (- y z))))

C

double code(double x, double y, double z) {
	return x + fma(log(y), (-0.5 - y), (y - z));
}

Julia

function code(x, y, z)
	return Float64(x + fma(log(y), Float64(-0.5 - y), Float64(y - z)))
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto \color{blue}{\left(x - \left(y + \frac{1}{2}\right) \cdot \log y\right) + \left(y - z\right)} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right)\right)} + \left(y - z\right) \]

   3. associate-+l+ N/A

      \[\leadsto \color{blue}{x + \left(\left(\mathsf{neg}\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) + \left(y - z\right)\right)} \]

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

      \[\leadsto \color{blue}{x + \left(\left(\mathsf{neg}\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) + \left(y - z\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto x + \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \left(y + \frac{1}{2}\right)}\right)\right) + \left(y - z\right)\right) \]

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

      \[\leadsto x + \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(y + \frac{1}{2}\right)\right)\right)} + \left(y - z\right)\right) \]

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

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(y + \frac{1}{2}\right)\right), y - z\right)} \]

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

      \[\leadsto x + \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(y + \frac{1}{2}\right)\right), y - z\right) \]

   9. +-commutative N/A

      \[\leadsto x + \mathsf{fma}\left(\log y, \mathsf{neg}\left(\color{blue}{\left(\frac{1}{2} + y\right)}\right), y - z\right) \]

   10. distribute-neg-in N/A

       \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y - z\right) \]

   11. unsub-neg N/A

       \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) - y}, y - z\right) \]

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

       \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) - y}, y - z\right) \]

   13. metadata-eval N/A

       \[\leadsto x + \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} - y, y - z\right) \]

   14. --lowering--.f64 99.9

       \[\leadsto x + \mathsf{fma}\left(\log y, -0.5 - y, \color{blue}{y - z}\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{x + \mathsf{fma}\left(\log y, -0.5 - y, y - z\right)} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -1e+75)
     (fma (log y) (- y) y)
     (if (<= t_0 -1e+14)
       (- x z)
       (if (<= t_0 500.0) (- (* (log y) -0.5) z) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -1e+75) {
		tmp = fma(log(y), -y, y);
	} else if (t_0 <= -1e+14) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -1e+75)
		tmp = fma(log(y), Float64(-y), y);
	elseif (t_0 <= -1e+14)
		tmp = Float64(x - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+75], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t$95$0, -1e+14], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999927e74

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 63.5

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

   5. Simplified 63.5%

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

   if -9.99999999999999927e74 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1e14 or 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 89.0%

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

   if -1e14 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(y + \left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right) + y\right)} - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)\right)} + y\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. --lowering--.f64 98.2

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{-0.5 - y}, y\right) - z \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)} - z \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \log y - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)} - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) - z} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \log y} - z \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \log y - z \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

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

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

      8. log-lowering-log.f64 94.5

         \[\leadsto \color{blue}{\log y} \cdot -0.5 - z \]

   8. Simplified 94.5%

      \[\leadsto \color{blue}{\log y \cdot -0.5 - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+75}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -1e+75)
     (- y (* y (log y)))
     (if (<= t_0 -1e+14)
       (- x z)
       (if (<= t_0 500.0) (- (* (log y) -0.5) z) (- x z))))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -1e+75) {
		tmp = y - (y * log(y));
	} else if (t_0 <= -1e+14) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - 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) :: t_0
    real(8) :: tmp
    t_0 = y + (x - (log(y) * (y + 0.5d0)))
    if (t_0 <= (-1d+75)) then
        tmp = y - (y * log(y))
    else if (t_0 <= (-1d+14)) then
        tmp = x - z
    else if (t_0 <= 500.0d0) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y + (x - (Math.log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -1e+75) {
		tmp = y - (y * Math.log(y));
	} else if (t_0 <= -1e+14) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y + (x - (math.log(y) * (y + 0.5)))
	tmp = 0
	if t_0 <= -1e+75:
		tmp = y - (y * math.log(y))
	elif t_0 <= -1e+14:
		tmp = x - z
	elif t_0 <= 500.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -1e+75)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= -1e+14)
		tmp = Float64(x - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y + (x - (log(y) * (y + 0.5)));
	tmp = 0.0;
	if (t_0 <= -1e+75)
		tmp = y - (y * log(y));
	elseif (t_0 <= -1e+14)
		tmp = x - z;
	elseif (t_0 <= 500.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+75], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -1e+14], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999927e74

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 63.5

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

   5. Simplified 63.5%

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

      1. +-commutative N/A

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

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

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

      3. unsub-neg N/A

         \[\leadsto \color{blue}{y - \log y \cdot y} \]

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

         \[\leadsto \color{blue}{y - \log y \cdot y} \]

      5. *-commutative N/A

         \[\leadsto y - \color{blue}{y \cdot \log y} \]

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

         \[\leadsto y - \color{blue}{y \cdot \log y} \]

      7. log-lowering-log.f64 63.4

         \[\leadsto y - y \cdot \color{blue}{\log y} \]

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{y - y \cdot \log y} \]

   if -9.99999999999999927e74 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1e14 or 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 89.0%

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

   if -1e14 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(y + \left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right) + y\right)} - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)\right)} + y\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. --lowering--.f64 98.2

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{-0.5 - y}, y\right) - z \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)} - z \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \log y - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)} - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) - z} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \log y} - z \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \log y - z \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

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

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

      8. log-lowering-log.f64 94.5

         \[\leadsto \color{blue}{\log y} \cdot -0.5 - z \]

   8. Simplified 94.5%

      \[\leadsto \color{blue}{\log y \cdot -0.5 - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1 \cdot 10^{+75}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ y (- x (* (log y) (+ y 0.5)))) z)))
   (if (<= t_0 -1e+14) (- x z) (if (<= t_0 500.0) (* (log y) -0.5) (- x z)))))

C

double code(double x, double y, double z) {
	double t_0 = (y + (x - (log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_0 <= -1e+14) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = log(y) * -0.5;
	} else {
		tmp = x - 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) :: t_0
    real(8) :: tmp
    t_0 = (y + (x - (log(y) * (y + 0.5d0)))) - z
    if (t_0 <= (-1d+14)) then
        tmp = x - z
    else if (t_0 <= 500.0d0) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + (x - (Math.log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_0 <= -1e+14) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + (x - (math.log(y) * (y + 0.5)))) - z
	tmp = 0
	if t_0 <= -1e+14:
		tmp = x - z
	elif t_0 <= 500.0:
		tmp = math.log(y) * -0.5
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5)))) - z)
	tmp = 0.0
	if (t_0 <= -1e+14)
		tmp = Float64(x - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + (x - (log(y) * (y + 0.5)))) - z;
	tmp = 0.0;
	if (t_0 <= -1e+14)
		tmp = x - z;
	elseif (t_0 <= 500.0)
		tmp = log(y) * -0.5;
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+14], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -1e14 or 500 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 61.4%

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

   if -1e14 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(y + \left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right) + y\right)} - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)\right)} + y\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. --lowering--.f64 96.5

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{-0.5 - y}, y\right) - z \]

   5. Simplified 96.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)} - z \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \log y - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)} - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) - z} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \log y} - z \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \log y - z \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

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

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

      8. log-lowering-log.f64 89.2

         \[\leadsto \color{blue}{\log y} \cdot -0.5 - z \]

   8. Simplified 89.2%

      \[\leadsto \color{blue}{\log y \cdot -0.5 - z} \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} \]

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

          \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} \]

       3. log-lowering-log.f64 86.6

          \[\leadsto \color{blue}{\log y} \cdot -0.5 \]

   11. Simplified 86.6%

       \[\leadsto \color{blue}{\log y \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2000000000:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2000000000.0)
     (- y (fma y (log y) z))
     (if (<= t_0 500.0) (- (* (log y) -0.5) z) (- x z)))))

C

double code(double x, double y, double z) {
	double t_0 = y + (x - (log(y) * (y + 0.5)));
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = y - fma(y, log(y), z);
	} else if (t_0 <= 500.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	tmp = 0.0
	if (t_0 <= -2000000000.0)
		tmp = Float64(y - fma(y, log(y), z));
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000.0], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e9

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. log-rec N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-rgt-identity N/A

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

      7. associate--l- N/A

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. log-lowering-log.f64 74.9

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

   8. Simplified 74.9%

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

   if -2e9 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(y + \left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\log y \cdot \left(\frac{1}{2} + y\right)\right)\right) + y\right)} - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right)\right)} + y\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right)} - z \]

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

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, \mathsf{neg}\left(\left(\frac{1}{2} + y\right)\right), y\right) - z \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, y\right) - z \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2}} + \left(\mathsf{neg}\left(y\right)\right), y\right) - z \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{\frac{-1}{2} - y}, y\right) - z \]

      9. --lowering--.f64 98.2

         \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{-0.5 - y}, y\right) - z \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5 - y, y\right)} - z \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log y - z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \log y - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)} - z \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) - z} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \log y} - z \]

      5. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot \log y - z \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

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

         \[\leadsto \color{blue}{\log y \cdot \frac{-1}{2}} - z \]

      8. log-lowering-log.f64 96.1

         \[\leadsto \color{blue}{\log y} \cdot -0.5 - z \]

   8. Simplified 96.1%

      \[\leadsto \color{blue}{\log y \cdot -0.5 - z} \]

   if 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 98.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2000000000:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28) (- (fma (log y) -0.5 x) z) (- (+ y (- x (* y (log y)))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = (y + (x - (y * log(y)))) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(Float64(y + Float64(x - Float64(y * log(y)))) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(y + N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   1. Initial program 100.0%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 98.9

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 0.28000000000000003 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

      6. log-lowering-log.f64 99.4

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

   5. Simplified 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(x - y \cdot \log y\right)\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \log y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.6e+66) (- (fma (log y) -0.5 x) z) (fma y (- 1.0 (log y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.6e+66) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(y, (1.0 - log(y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.6e+66)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = fma(y, Float64(1.0 - log(y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.6e+66], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000012e66

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 94.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 94.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 2.60000000000000012e66 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - \color{blue}{\log y \cdot \left(y + \frac{1}{2}\right)}\right) + y\right) - z \]

      2. flip-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{y \cdot y - \frac{1}{2} \cdot \frac{1}{2}}{y - \frac{1}{2}}}\right) + y\right) - z \]

      3. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y - \frac{1}{2}}{y \cdot y - \frac{1}{2} \cdot \frac{1}{2}}}}\right) + y\right) - z \]

      4. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y - \frac{1}{2}}{y \cdot y - \frac{1}{2} \cdot \frac{1}{2}}}}\right) + y\right) - z \]

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

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y - \frac{1}{2}}{y \cdot y - \frac{1}{2} \cdot \frac{1}{2}}}}\right) + y\right) - z \]

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

         \[\leadsto \left(\left(x - \frac{\color{blue}{\log y}}{\frac{y - \frac{1}{2}}{y \cdot y - \frac{1}{2} \cdot \frac{1}{2}}}\right) + y\right) - z \]

      7. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{y \cdot y - \frac{1}{2} \cdot \frac{1}{2}}{y - \frac{1}{2}}}}}\right) + y\right) - z \]

      8. flip-+ N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{\color{blue}{y + \frac{1}{2}}}}\right) + y\right) - z \]

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

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{y + \frac{1}{2}}}}\right) + y\right) - z \]

      10. +-lowering-+.f64 99.5

          \[\leadsto \left(\left(x - \frac{\log y}{\frac{1}{\color{blue}{y + 0.5}}}\right) + y\right) - z \]

   4. Applied egg-rr 99.5%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{1}{y + 0.5}}}\right) + y\right) - z \]

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 99.5

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

   7. Simplified 99.5%

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

   8. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-out-- N/A

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

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

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

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

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

      11. log-lowering-log.f64 91.6

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

   10. Simplified 91.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, 1 - \log y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+104) (- (fma (log y) -0.5 x) z) (- y (fma y (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+104) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = y - fma(y, log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.5e+104)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(y - fma(y, log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.5e+104], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5e104

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x - \frac{1}{2} \cdot \log y\right)} - z \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(x + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)} - z \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right) + x\right)} - z \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\log y \cdot \frac{1}{2}}\right)\right) + x\right) - z \]

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

         \[\leadsto \left(\color{blue}{\log y \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} + x\right) - z \]

      5. metadata-eval N/A

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{-1}{2}} + x\right) - z \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, \frac{-1}{2}, x\right)} - z \]

      7. log-lowering-log.f64 89.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, -0.5, x\right) - z \]

   5. Simplified 89.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]

   if 9.5e104 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 83.2

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

   5. Simplified 83.2%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. log-rec N/A

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

      4. sub-neg N/A

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

      5. distribute-lft-out-- N/A

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

      6. *-rgt-identity N/A

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

      7. associate--l- N/A

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. log-lowering-log.f64 83.1

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

   8. Simplified 83.1%

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

Alternative 9: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+132}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+132) (- x z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+132) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.15d+132) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.15e+132) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.15e+132:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.15e+132)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.15e+132)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.15e+132], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1500000000000001e132

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Simplified 71.7%

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

   if 1.1500000000000001e132 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. log-rec N/A

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

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

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

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

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. mul-1-neg N/A

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

      14. neg-lowering-neg.f64 77.6

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

   5. Simplified 77.6%

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

      1. +-commutative N/A

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

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

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

      3. unsub-neg N/A

         \[\leadsto \color{blue}{y - \log y \cdot y} \]

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

         \[\leadsto \color{blue}{y - \log y \cdot y} \]

      5. *-commutative N/A

         \[\leadsto y - \color{blue}{y \cdot \log y} \]

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

         \[\leadsto y - \color{blue}{y \cdot \log y} \]

      7. log-lowering-log.f64 77.5

         \[\leadsto y - y \cdot \color{blue}{\log y} \]

   7. Applied egg-rr 77.5%

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

Alternative 10: 47.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+121}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 10^{+144}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.05e+121) (- z) (if (<= z 1e+144) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+121) {
		tmp = -z;
	} else if (z <= 1e+144) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.05d+121)) then
        tmp = -z
    else if (z <= 1d+144) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.05e+121) {
		tmp = -z;
	} else if (z <= 1e+144) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.05e+121:
		tmp = -z
	elif z <= 1e+144:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.05e+121)
		tmp = Float64(-z);
	elseif (z <= 1e+144)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.05e+121)
		tmp = -z;
	elseif (z <= 1e+144)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.05e+121], (-z), If[LessEqual[z, 1e+144], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0500000000000001e121 or 1.00000000000000002e144 < z

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 67.2

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

   5. Simplified 67.2%

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

   if -1.0500000000000001e121 < z < 1.00000000000000002e144

   1. Initial program 99.8%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 40.2%

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

Alternative 11: 58.8% accurate, 29.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x - 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 - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

5. Simplified 55.2%

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

Alternative 12: 30.7% accurate, 118.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 32.7%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (- (+ y x) z) (* (+ y 0.5) (log y))))

C

double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * log(y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y + x) - z) - ((y + 0.5d0) * log(y))
end function

Java

public static double code(double x, double y, double z) {
	return ((y + x) - z) - ((y + 0.5) * Math.log(y));
}

Python

def code(x, y, z):
	return ((y + x) - z) - ((y + 0.5) * math.log(y))

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y + x) - z) - Float64(Float64(y + 0.5) * log(y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y + x) - z) - ((y + 0.5) * log(y));
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y + x), $MachinePrecision] - z), $MachinePrecision] - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024204 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (- (+ y x) z) (* (+ y 1/2) (log y))))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))