Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 10.6s

Alternatives: 8

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. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-neg-in N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 67.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + \frac{1}{\frac{1}{x}}\right) - z\\ t_1 := \left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ y (/ 1.0 (/ 1.0 x))) z))
        (t_1 (- (+ y (- x (* (log y) (+ y 0.5)))) z)))
   (if (<= t_1 -1e+30) t_0 (if (<= t_1 500.0) (* (log y) -0.5) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + (1.0 / (1.0 / x))) - z;
	double t_1 = (y + (x - (log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_1 <= -1e+30) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = log(y) * -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + (1.0d0 / (1.0d0 / x))) - z
    t_1 = (y + (x - (log(y) * (y + 0.5d0)))) - z
    if (t_1 <= (-1d+30)) then
        tmp = t_0
    else if (t_1 <= 500.0d0) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + (1.0 / (1.0 / x))) - z;
	double t_1 = (y + (x - (Math.log(y) * (y + 0.5)))) - z;
	double tmp;
	if (t_1 <= -1e+30) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + (1.0 / (1.0 / x))) - z
	t_1 = (y + (x - (math.log(y) * (y + 0.5)))) - z
	tmp = 0
	if t_1 <= -1e+30:
		tmp = t_0
	elif t_1 <= 500.0:
		tmp = math.log(y) * -0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + Float64(1.0 / Float64(1.0 / x))) - z)
	t_1 = Float64(Float64(y + Float64(x - Float64(log(y) * Float64(y + 0.5)))) - z)
	tmp = 0.0
	if (t_1 <= -1e+30)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + (1.0 / (1.0 / x))) - z;
	t_1 = (y + (x - (log(y) * (y + 0.5)))) - z;
	tmp = 0.0;
	if (t_1 <= -1e+30)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = log(y) * -0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y + N[(x - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+30], t$95$0, If[LessEqual[t$95$1, 500.0], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]

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) < -1e30 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 68.2

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

   7. Applied rewrites 68.2%

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

   if -1e30 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 97.3

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

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{y + \mathsf{fma}\left(\log y, -0.5 - y, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 88.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 86.3%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq -1 \cdot 10^{+30}:\\ \;\;\;\;\left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \mathbf{elif}\;\left(y + \left(x - \log y \cdot \left(y + 0.5\right)\right)\right) - z \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \mathbf{if}\;z \leq -9 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ y (/ 1.0 (/ 1.0 x))) z)))
   (if (<= z -9e+48) t_0 (if (<= z 9.8e+20) (fma (log y) -0.5 x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + (1.0 / (1.0 / x))) - z;
	double tmp;
	if (z <= -9e+48) {
		tmp = t_0;
	} else if (z <= 9.8e+20) {
		tmp = fma(log(y), -0.5, x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + Float64(1.0 / Float64(1.0 / x))) - z)
	tmp = 0.0
	if (z <= -9e+48)
		tmp = t_0;
	elseif (z <= 9.8e+20)
		tmp = fma(log(y), -0.5, x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -9e+48], t$95$0, If[LessEqual[z, 9.8e+20], N[(N[Log[y], $MachinePrecision] * -0.5 + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999991e48 or 9.8e20 < 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. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 81.4

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

   7. Applied rewrites 81.4%

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

   if -8.99999999999999991e48 < z < 9.8e20

   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 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 63.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+48}:\\ \;\;\;\;\left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \mathbf{elif}\;z \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.15e+81) (- (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 <= 1.15e+81) {
		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 <= 1.15e+81)
		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, 1.15e+81], 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 < 1.1499999999999999e81

   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. lower-fma.f64 N/A

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

      7. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if 1.1499999999999999e81 < 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 z around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 86.2

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

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{y + \mathsf{fma}\left(\log y, -0.5 - y, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 86.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 86.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.55e+152) (- (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 <= 1.55e+152) {
		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 <= 1.55e+152)
		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, 1.55e+152], 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 < 1.55e152

   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. lower-fma.f64 N/A

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

      7. lower-log.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if 1.55e152 < y

   1. Initial program 99.4%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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. lower-fma.f64 N/A

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

      12. lower-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. lower-neg.f64 86.6

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

   5. Applied rewrites 86.6%

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

Alternative 6: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+152}:\\ \;\;\;\;\left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+152) (- (+ y (/ 1.0 (/ 1.0 x))) z) (fma (log y) (- y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+152) {
		tmp = (y + (1.0 / (1.0 / x))) - z;
	} else {
		tmp = fma(log(y), -y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+152)
		tmp = Float64(Float64(y + Float64(1.0 / Float64(1.0 / x))) - z);
	else
		tmp = fma(log(y), Float64(-y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+152], N[(N[(y + N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000007e152

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 72.2

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

   7. Applied rewrites 72.2%

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

   if 1.35000000000000007e152 < y

   1. Initial program 99.4%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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. lower-fma.f64 N/A

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

      12. lower-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. lower-neg.f64 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+152}:\\ \;\;\;\;\left(y + \frac{1}{\frac{1}{x}}\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(y + \frac{1}{\frac{1}{x}}\right) - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (+ y (/ 1.0 (/ 1.0 x))) z))

C

double code(double x, double y, double z) {
	return (y + (1.0 / (1.0 / 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 = (y + (1.0d0 / (1.0d0 / x))) - z
end function

Java

public static double code(double x, double y, double z) {
	return (y + (1.0 / (1.0 / x))) - z;
}

Python

def code(x, y, z):
	return (y + (1.0 / (1.0 / x))) - z

Julia

function code(x, y, z)
	return Float64(Float64(y + Float64(1.0 / Float64(1.0 / x))) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y + (1.0 / (1.0 / x))) - z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y + N[(1.0 / N[(1.0 / x), $MachinePrecision]), $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. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 59.1

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

7. Applied rewrites 59.1%

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

8. Final simplification 59.1%

   \[\leadsto \left(y + \frac{1}{\frac{1}{x}}\right) - z \]
9. Add Preprocessing

Alternative 8: 30.2% accurate, 39.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. lower-neg.f64 32.0

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

5. Applied rewrites 32.0%

   \[\leadsto \color{blue}{-z} \]
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 2024221 
(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))