Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.6s

Alternatives: 11

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

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

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

Wolfram

code[x_, y_, z_] := N[(N[(x + N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] - 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. Step-by-step derivation

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. unsub-neg N/A

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

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

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

   12. metadata-eval 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;t\_0 \leq -400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 325:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+106)
     (fma (log y) (- y) y)
     (if (<= t_0 -400000.0)
       (- x z)
       (if (<= t_0 325.0) (- (* (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 <= -2e+106) {
		tmp = fma(log(y), -y, y);
	} else if (t_0 <= -400000.0) {
		tmp = x - z;
	} else if (t_0 <= 325.0) {
		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 <= -2e+106)
		tmp = fma(log(y), Float64(-y), y);
	elseif (t_0 <= -400000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 325.0)
		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, -2e+106], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t$95$0, -400000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 325.0], 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) < -2.00000000000000018e106

   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 63.6

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

   5. Simplified 63.6%

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

   if -2.00000000000000018e106 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4e5 or 325 < (+.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 85.8%

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

   if -4e5 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 325

   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 97.5

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

   8. Simplified 97.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 325:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+106}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;t\_0 \leq -400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 325:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))))
   (if (<= t_0 -2e+106)
     (- y (* y (log y)))
     (if (<= t_0 -400000.0)
       (- x z)
       (if (<= t_0 325.0) (- (* (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 <= -2e+106) {
		tmp = y - (y * log(y));
	} else if (t_0 <= -400000.0) {
		tmp = x - z;
	} else if (t_0 <= 325.0) {
		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 <= (-2d+106)) then
        tmp = y - (y * log(y))
    else if (t_0 <= (-400000.0d0)) then
        tmp = x - z
    else if (t_0 <= 325.0d0) 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 <= -2e+106) {
		tmp = y - (y * Math.log(y));
	} else if (t_0 <= -400000.0) {
		tmp = x - z;
	} else if (t_0 <= 325.0) {
		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 <= -2e+106:
		tmp = y - (y * math.log(y))
	elif t_0 <= -400000.0:
		tmp = x - z
	elif t_0 <= 325.0:
		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 <= -2e+106)
		tmp = Float64(y - Float64(y * log(y)));
	elseif (t_0 <= -400000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 325.0)
		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 <= -2e+106)
		tmp = y - (y * log(y));
	elseif (t_0 <= -400000.0)
		tmp = x - z;
	elseif (t_0 <= 325.0)
		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, -2e+106], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -400000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 325.0], 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) < -2.00000000000000018e106

   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 63.6

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

   5. Simplified 63.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -y, y\right)} \]
   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 63.5

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

   7. Applied egg-rr 63.5%

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

   if -2.00000000000000018e106 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4e5 or 325 < (+.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 85.8%

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

   if -4e5 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 325

   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 97.5

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

   8. Simplified 97.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -2 \cdot 10^{+106}:\\ \;\;\;\;y - y \cdot \log y\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq 325:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+218}:\\ \;\;\;\;x + t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+50}:\\ \;\;\;\;t\_1 - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ y (- x (* (log y) (+ y 0.5))))) (t_1 (fma (log y) (- y) y)))
   (if (<= t_0 -5e+218)
     (+ x t_1)
     (if (<= t_0 -1e+50) (- t_1 z) (- (fma (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)));
	double t_1 = fma(log(y), -y, y);
	double tmp;
	if (t_0 <= -5e+218) {
		tmp = x + t_1;
	} else if (t_0 <= -1e+50) {
		tmp = t_1 - z;
	} else {
		tmp = fma(log(y), -0.5, x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5))))
	t_1 = fma(log(y), Float64(-y), y)
	tmp = 0.0
	if (t_0 <= -5e+218)
		tmp = Float64(x + t_1);
	elseif (t_0 <= -1e+50)
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(fma(log(y), -0.5, 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]}, Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+218], N[(x + t$95$1), $MachinePrecision], If[LessEqual[t$95$0, -1e+50], N[(t$95$1 - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - 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) < -4.99999999999999983e218

   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. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

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

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

      12. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 96.4

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

   7. Simplified 96.4%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 96.4

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

   10. Simplified 96.4%

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

   if -4.99999999999999983e218 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.0000000000000001e50

   1. Initial program 99.7%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 88.8

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

   5. Simplified 88.8%

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

   if -1.0000000000000001e50 < (+.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 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 96.8

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

   5. Simplified 96.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} - z \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -5 \cdot 10^{+218}:\\ \;\;\;\;x + \mathsf{fma}\left(\log y, -y, y\right)\\ \mathbf{elif}\;y + \left(x - \log y \cdot \left(y + 0.5\right)\right) \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -16600000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 210:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -16600000000000.0)
   (- x z)
   (if (<= z 210.0) (fma (log y) -0.5 x) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -16600000000000.0) {
		tmp = x - z;
	} else if (z <= 210.0) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -16600000000000.0)
		tmp = Float64(x - z);
	elseif (z <= 210.0)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -16600000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 210.0], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.66e13 or 210 < z

   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 82.7%

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

   if -1.66e13 < z < 210

   1. Initial program 99.7%

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

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

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

      12. metadata-eval 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. --lowering--.f64 56.7

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

   7. Simplified 56.7%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

      4. log-lowering-log.f64 56.7

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

   10. Simplified 56.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -0.5, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.7e+56) (- (fma (log y) -0.5 x) z) (+ x (fma (log y) (- y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.7e+56) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = x + fma(log(y), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.7e+56)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = Float64(x + fma(log(y), Float64(-y), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.7e+56], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7e56

   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 96.8

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

   5. Simplified 96.8%

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

   if 1.7e56 < 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. sub-neg N/A

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

      2. associate-+l+ N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

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

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

      12. metadata-eval 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

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

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

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

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

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

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. --lowering--.f64 84.8

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

   7. Simplified 84.8%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-lft-identity N/A

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

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

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

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

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 84.8

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

   10. Simplified 84.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e+91) (- (fma (log y) -0.5 x) z) (fma (log y) (- y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e+91) {
		tmp = fma(log(y), -0.5, x) - z;
	} else {
		tmp = fma(log(y), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e+91)
		tmp = Float64(fma(log(y), -0.5, x) - z);
	else
		tmp = fma(log(y), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e+91], N[(N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e91

   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 93.0

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

   5. Simplified 93.0%

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

   if 1.1e91 < 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

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 67.1

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

   5. Simplified 67.1%

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

Alternative 8: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{+69}:\\ \;\;\;\;\left(x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.8e+69) (- (+ x y) z) (- y (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.8e+69) {
		tmp = (x + y) - 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 <= 4.8d+69) then
        tmp = (x + y) - 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 <= 4.8e+69) {
		tmp = (x + y) - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.8e+69:
		tmp = (x + y) - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.8e+69)
		tmp = Float64(Float64(x + y) - 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 <= 4.8e+69)
		tmp = (x + y) - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.8e+69], N[(N[(x + y), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.8000000000000003e69

   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

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

   5. Simplified 80.7%

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

   if 4.8000000000000003e69 < 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

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 65.6

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

   5. Simplified 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, -y, y\right)} \]
   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 65.5

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

   7. Applied egg-rr 65.5%

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

Alternative 9: 48.1% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+80}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+80}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -8.2e+80) (- z) (if (<= z 3.7e+80) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -8.2e+80) {
		tmp = -z;
	} else if (z <= 3.7e+80) {
		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 <= (-8.2d+80)) then
        tmp = -z
    else if (z <= 3.7d+80) 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 <= -8.2e+80) {
		tmp = -z;
	} else if (z <= 3.7e+80) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -8.2e+80:
		tmp = -z
	elif z <= 3.7e+80:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -8.2e+80)
		tmp = Float64(-z);
	elseif (z <= 3.7e+80)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -8.2e+80)
		tmp = -z;
	elseif (z <= 3.7e+80)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -8.2e+80], (-z), If[LessEqual[z, 3.7e+80], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -8.20000000000000003e80 or 3.69999999999999996e80 < z

   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 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 70.1

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

   5. Simplified 70.1%

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

   if -8.20000000000000003e80 < z < 3.69999999999999996e80

   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 39.1%

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

Alternative 10: 57.9% 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 61.3%

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

Alternative 11: 30.0% 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 29.7%

   \[\leadsto \color{blue}{x} \]
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 2024199 
(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))