Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 9.7s

Alternatives: 17

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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. associate--r+ N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 74.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{\frac{1}{x}} + y\right) - z\\ t_1 := \left(x - \left(0.5 + y\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+203}:\\ \;\;\;\;y - \log y \cdot y\\ \mathbf{elif}\;t\_1 \leq -1.5 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (/ 1.0 (/ 1.0 x)) y) z))
        (t_1 (+ (- x (* (+ 0.5 y) (log y))) y)))
   (if (<= t_1 -5e+203)
     (- y (* (log y) y))
     (if (<= t_1 -1.5e+49)
       t_0
       (if (<= t_1 -5e+32)
         (* (- 1.0 (log y)) y)
         (if (<= t_1 500.0) (- (* -0.5 (log y)) z) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 / (1.0 / x)) + y) - z;
	double t_1 = (x - ((0.5 + y) * log(y))) + y;
	double tmp;
	if (t_1 <= -5e+203) {
		tmp = y - (log(y) * y);
	} else if (t_1 <= -1.5e+49) {
		tmp = t_0;
	} else if (t_1 <= -5e+32) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_1 <= 500.0) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((1.0d0 / (1.0d0 / x)) + y) - z
    t_1 = (x - ((0.5d0 + y) * log(y))) + y
    if (t_1 <= (-5d+203)) then
        tmp = y - (log(y) * y)
    else if (t_1 <= (-1.5d+49)) then
        tmp = t_0
    else if (t_1 <= (-5d+32)) then
        tmp = (1.0d0 - log(y)) * y
    else if (t_1 <= 500.0d0) then
        tmp = ((-0.5d0) * log(y)) - z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 / (1.0 / x)) + y) - z;
	double t_1 = (x - ((0.5 + y) * Math.log(y))) + y;
	double tmp;
	if (t_1 <= -5e+203) {
		tmp = y - (Math.log(y) * y);
	} else if (t_1 <= -1.5e+49) {
		tmp = t_0;
	} else if (t_1 <= -5e+32) {
		tmp = (1.0 - Math.log(y)) * y;
	} else if (t_1 <= 500.0) {
		tmp = (-0.5 * Math.log(y)) - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 / (1.0 / x)) + y) - z
	t_1 = (x - ((0.5 + y) * math.log(y))) + y
	tmp = 0
	if t_1 <= -5e+203:
		tmp = y - (math.log(y) * y)
	elif t_1 <= -1.5e+49:
		tmp = t_0
	elif t_1 <= -5e+32:
		tmp = (1.0 - math.log(y)) * y
	elif t_1 <= 500.0:
		tmp = (-0.5 * math.log(y)) - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 / x)) + y) - z)
	t_1 = Float64(Float64(x - Float64(Float64(0.5 + y) * log(y))) + y)
	tmp = 0.0
	if (t_1 <= -5e+203)
		tmp = Float64(y - Float64(log(y) * y));
	elseif (t_1 <= -1.5e+49)
		tmp = t_0;
	elseif (t_1 <= -5e+32)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_1 <= 500.0)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 / (1.0 / x)) + y) - z;
	t_1 = (x - ((0.5 + y) * log(y))) + y;
	tmp = 0.0;
	if (t_1 <= -5e+203)
		tmp = y - (log(y) * y);
	elseif (t_1 <= -1.5e+49)
		tmp = t_0;
	elseif (t_1 <= -5e+32)
		tmp = (1.0 - log(y)) * y;
	elseif (t_1 <= 500.0)
		tmp = (-0.5 * log(y)) - z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+203], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -1.5e+49], t$95$0, If[LessEqual[t$95$1, -5e+32], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, 500.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4.99999999999999994e203

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 65.2

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

   7. Applied rewrites 65.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 55.1%

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

   if -4.99999999999999994e203 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.5000000000000001e49 or 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. 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 99.8

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

      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. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   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 82.2

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

   7. Applied rewrites 82.2%

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

   if -1.5000000000000001e49 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4.9999999999999997e32

   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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -4.9999999999999997e32 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. 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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 94.9%

       \[\leadsto -0.5 \cdot \color{blue}{\log y} - z \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \left(0.5 + y\right) \cdot \log y\right) + y \leq -5 \cdot 10^{+203}:\\ \;\;\;\;y - \log y \cdot y\\ \mathbf{elif}\;\left(x - \left(0.5 + y\right) \cdot \log y\right) + y \leq -1.5 \cdot 10^{+49}:\\ \;\;\;\;\left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \mathbf{elif}\;\left(x - \left(0.5 + y\right) \cdot \log y\right) + y \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \left(0.5 + y\right) \cdot \log y\right) + y \leq 500:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 15.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 15.2)
   (- (fma -0.5 (log y) x) z)
   (if (<= y 1.4e+76)
     (fma (- -0.5 y) (log y) (+ x y))
     (- y (fma (+ 0.5 y) (log y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 15.2) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (y <= 1.4e+76) {
		tmp = fma((-0.5 - y), log(y), (x + y));
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 15.2)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (y <= 1.4e+76)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 15.2], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.4e+76], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 15.199999999999999

   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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 15.199999999999999 < y < 1.3999999999999999e76

   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 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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if 1.3999999999999999e76 < y

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 15.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e+26)
   (- (fma -0.5 (log y) x) z)
   (if (<= y 1.4e+76)
     (fma (- y) (log y) (+ x y))
     (- y (fma (+ 0.5 y) (log y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+26) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (y <= 1.4e+76) {
		tmp = fma(-y, log(y), (x + y));
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e+26)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (y <= 1.4e+76)
		tmp = fma(Float64(-y), log(y), Float64(x + y));
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e+26], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.4e+76], N[((-y) * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.40000000000000005e26

   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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2.40000000000000005e26 < y < 1.3999999999999999e76

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 55.3

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

   7. Applied rewrites 55.3%

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

   8. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 91.1

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 91.1%

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

   if 1.3999999999999999e76 < y

   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 x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e+26)
   (- (fma -0.5 (log y) x) z)
   (if (<= y 1.4e+76)
     (fma (- y) (log y) (+ x y))
     (- (* (- 1.0 (log y)) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+26) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (y <= 1.4e+76) {
		tmp = fma(-y, log(y), (x + y));
	} else {
		tmp = ((1.0 - log(y)) * y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e+26)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (y <= 1.4e+76)
		tmp = fma(Float64(-y), log(y), Float64(x + y));
	else
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e+26], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.4e+76], N[((-y) * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.40000000000000005e26

   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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2.40000000000000005e26 < y < 1.3999999999999999e76

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 55.3

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

   7. Applied rewrites 55.3%

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

   8. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 91.1

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

   10. Applied rewrites 91.1%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 91.1%

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

   if 1.3999999999999999e76 < y

   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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \mathbf{if}\;x \leq -550000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-68}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (/ 1.0 (/ 1.0 x)) y) z)))
   (if (<= x -550000000.0)
     t_0
     (if (<= x 1.2e-68) (fma (- -0.5 y) (log y) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 / (1.0 / x)) + y) - z;
	double tmp;
	if (x <= -550000000.0) {
		tmp = t_0;
	} else if (x <= 1.2e-68) {
		tmp = fma((-0.5 - y), log(y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 / x)) + y) - z)
	tmp = 0.0
	if (x <= -550000000.0)
		tmp = t_0;
	elseif (x <= 1.2e-68)
		tmp = fma(Float64(-0.5 - y), log(y), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -550000000.0], t$95$0, If[LessEqual[x, 1.2e-68], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5e8 or 1.19999999999999996e-68 < x

   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. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   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 80.6

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

   7. Applied rewrites 80.6%

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

   if -5.5e8 < x < 1.19999999999999996e-68

   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 68.6%

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

Alternative 7: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, z, -z\right)\\ \mathbf{if}\;z \leq -545000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 205:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) z (- z))))
   (if (<= z -545000000000.0) t_0 (if (<= z 205.0) (fma -0.5 (log y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), z, -z);
	double tmp;
	if (z <= -545000000000.0) {
		tmp = t_0;
	} else if (z <= 205.0) {
		tmp = fma(-0.5, log(y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), z, Float64(-z))
	tmp = 0.0
	if (z <= -545000000000.0)
		tmp = t_0;
	elseif (z <= 205.0)
		tmp = fma(-0.5, log(y), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * z + (-z)), $MachinePrecision]}, If[LessEqual[z, -545000000000.0], t$95$0, If[LessEqual[z, 205.0], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.45e11 or 205 < 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. 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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. distribute-rgt-in N/A

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

      6. div-sub N/A

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

      7. distribute-lft-in N/A

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, z, -z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 75.2%

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

   if -5.45e11 < z < 205

   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 69.7

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

   7. Applied rewrites 69.7%

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

   8. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   10. Applied rewrites 99.1%

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

   11. Taylor expanded in y around 0

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

   13. Applied rewrites 69.1%

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

Alternative 8: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.4e+26) (- (fma -0.5 (log y) x) z) (fma (- y) (log y) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.4e+26) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(-y, log(y), (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.4e+26)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(Float64(-y), log(y), Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.4e+26], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[((-y) * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.40000000000000005e26

   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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if 2.40000000000000005e26 < y

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

   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. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 40.6

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

   7. Applied rewrites 40.6%

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

   8. Taylor expanded in z around 0

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

      1. distribute-rgt-in N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-in N/A

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

      10. metadata-eval N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 81.2

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

   10. Applied rewrites 81.2%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 81.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $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. Final simplification 99.8%

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

Alternative 10: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+205}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+205) (- (fma -0.5 (log y) x) z) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+205) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+205)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+205], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999997e205

   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. distribute-lft-neg-in N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 4.3999999999999997e205 < 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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 93.4

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

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 82.4%

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

Alternative 11: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+205) (- (+ (/ 1.0 (/ 1.0 x)) y) z) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+205) {
		tmp = ((1.0 / (1.0 / x)) + y) - z;
	} else {
		tmp = y - (log(y) * 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 <= 4.4d+205) then
        tmp = ((1.0d0 / (1.0d0 / x)) + y) - z
    else
        tmp = y - (log(y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+205) {
		tmp = ((1.0 / (1.0 / x)) + y) - z;
	} else {
		tmp = y - (Math.log(y) * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e+205:
		tmp = ((1.0 / (1.0 / x)) + y) - z
	else:
		tmp = y - (math.log(y) * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+205)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 / x)) + y) - z);
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e+205)
		tmp = ((1.0 / (1.0 / x)) + y) - z;
	else
		tmp = y - (log(y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+205], N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999997e205

   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. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   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 69.9

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

   7. Applied rewrites 69.9%

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

   if 4.3999999999999997e205 < 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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. distribute-rgt-in N/A

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

      2. lower--.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-log.f64 93.4

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

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 82.4%

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

Alternative 12: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+205}:\\ \;\;\;\;\left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+205) (- (+ (/ 1.0 (/ 1.0 x)) y) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+205) {
		tmp = ((1.0 / (1.0 / x)) + y) - z;
	} else {
		tmp = (1.0 - log(y)) * 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 <= 4.4d+205) then
        tmp = ((1.0d0 / (1.0d0 / x)) + y) - z
    else
        tmp = (1.0d0 - log(y)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.4e+205) {
		tmp = ((1.0 / (1.0 / x)) + y) - z;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.4e+205:
		tmp = ((1.0 / (1.0 / x)) + y) - z
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.4e+205)
		tmp = Float64(Float64(Float64(1.0 / Float64(1.0 / x)) + y) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.4e+205)
		tmp = ((1.0 / (1.0 / x)) + y) - z;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.4e+205], N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3999999999999997e205

   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. distribute-lft-neg-in N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   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 69.9

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

   7. Applied rewrites 69.9%

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

   if 4.3999999999999997e205 < 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. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 82.3

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

   5. Applied rewrites 82.3%

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

Alternative 13: 56.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{x}{z}, z, -z\right)\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{-188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ x z) z (- z))))
   (if (<= z -2.9e-188) t_0 (if (<= z 3.9e-7) (/ 1.0 (/ 1.0 x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((x / z), z, -z);
	double tmp;
	if (z <= -2.9e-188) {
		tmp = t_0;
	} else if (z <= 3.9e-7) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(x / z), z, Float64(-z))
	tmp = 0.0
	if (z <= -2.9e-188)
		tmp = t_0;
	elseif (z <= 3.9e-7)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x / z), $MachinePrecision] * z + (-z)), $MachinePrecision]}, If[LessEqual[z, -2.9e-188], t$95$0, If[LessEqual[z, 3.9e-7], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.9000000000000001e-188 or 3.90000000000000025e-7 < 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. 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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. associate--r+ N/A

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

      10. lower--.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. distribute-rgt-in N/A

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

      6. div-sub N/A

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

      7. distribute-lft-in N/A

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

   7. Applied rewrites 95.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{x}{z}, z, -z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 67.5%

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

   if -2.9000000000000001e-188 < z < 3.90000000000000025e-7

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.5

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 43.0

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 43.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((1.0d0 / (1.0d0 / x)) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $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. distribute-lft-neg-in N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

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 58.7

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

7. Applied rewrites 58.7%

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

Alternative 15: 48.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{z}, z, -z\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] * z + (-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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-lft-in N/A

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

   9. associate--r+ N/A

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

   10. lower--.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in z around -inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. div-sub N/A

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

   5. distribute-rgt-in N/A

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

   6. div-sub N/A

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

   7. distribute-lft-in N/A

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

7. Applied rewrites 84.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\frac{x}{z}, z, -z\right) \]
9. Step-by-step derivation

10. Applied rewrites 52.2%

    \[\leadsto \mathsf{fma}\left(\frac{x}{z}, z, -z\right) \]
11. Add Preprocessing

Alternative 16: 29.9% 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 29.4

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

5. Applied rewrites 29.4%

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

Alternative 17: 2.3% accurate, 118.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 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 29.4

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

5. Applied rewrites 29.4%

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

7. Applied rewrites 11.0%

   \[\leadsto \frac{0 - z \cdot z}{\color{blue}{0 + z}} \]
8. Step-by-step derivation

9. Applied rewrites 2.4%

   \[\leadsto z \]
10. 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 2024277 
(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))