Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 10.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   10. associate-+l- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

   17. log-lowering-log.f64 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+163}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 20000000:\\ \;\;\;\;\left(x + y\right) - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.18e+163)
   (- x z)
   (if (<= z 20000000.0)
     (- (+ x y) (* (+ y 0.5) (log y)))
     (+ x (- (* (log y) -0.5) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.18e+163) {
		tmp = x - z;
	} else if (z <= 20000000.0) {
		tmp = (x + y) - ((y + 0.5) * log(y));
	} else {
		tmp = x + ((log(y) * -0.5) - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.18d+163)) then
        tmp = x - z
    else if (z <= 20000000.0d0) then
        tmp = (x + y) - ((y + 0.5d0) * log(y))
    else
        tmp = x + ((log(y) * (-0.5d0)) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.18e+163) {
		tmp = x - z;
	} else if (z <= 20000000.0) {
		tmp = (x + y) - ((y + 0.5) * Math.log(y));
	} else {
		tmp = x + ((Math.log(y) * -0.5) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.18e+163:
		tmp = x - z
	elif z <= 20000000.0:
		tmp = (x + y) - ((y + 0.5) * math.log(y))
	else:
		tmp = x + ((math.log(y) * -0.5) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.18e+163)
		tmp = Float64(x - z);
	elseif (z <= 20000000.0)
		tmp = Float64(Float64(x + y) - Float64(Float64(y + 0.5) * log(y)));
	else
		tmp = Float64(x + Float64(Float64(log(y) * -0.5) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.18e+163)
		tmp = x - z;
	elseif (z <= 20000000.0)
		tmp = (x + y) - ((y + 0.5) * log(y));
	else
		tmp = x + ((log(y) * -0.5) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.18e+163], N[(x - z), $MachinePrecision], If[LessEqual[z, 20000000.0], N[(N[(x + y), $MachinePrecision] - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.18000000000000005e163

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 90.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 90.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{z}\right) \]
   9. Step-by-step derivation

   10. Simplified 90.4%

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

   if -1.18000000000000005e163 < z < 2e7

   1. Initial program 99.8%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x + y\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(y, \frac{1}{2}\right)}, \mathsf{log.f64}\left(y\right)\right)\right) \]

      2. +-lowering-+.f64 93.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(y, \frac{1}{2}\right)}, \mathsf{log.f64}\left(y\right)\right)\right) \]

   7. Simplified 93.8%

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

   if 2e7 < z

   1. Initial program 99.9%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 84.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 84.0%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+163}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 20000000:\\ \;\;\;\;\left(x + y\right) - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13500000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 90000000:\\ \;\;\;\;x - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -13500000000.0)
   (- x z)
   (if (<= z 90000000.0) (- x (* (+ y 0.5) (log y))) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -13500000000.0) {
		tmp = x - z;
	} else if (z <= 90000000.0) {
		tmp = x - ((y + 0.5) * log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-13500000000.0d0)) then
        tmp = x - z
    else if (z <= 90000000.0d0) then
        tmp = x - ((y + 0.5d0) * log(y))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -13500000000.0) {
		tmp = x - z;
	} else if (z <= 90000000.0) {
		tmp = x - ((y + 0.5) * Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -13500000000.0:
		tmp = x - z
	elif z <= 90000000.0:
		tmp = x - ((y + 0.5) * math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -13500000000.0)
		tmp = Float64(x - z);
	elseif (z <= 90000000.0)
		tmp = Float64(x - Float64(Float64(y + 0.5) * log(y)));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -13500000000.0)
		tmp = x - z;
	elseif (z <= 90000000.0)
		tmp = x - ((y + 0.5) * log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -13500000000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 90000000.0], N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e10 or 9e7 < z

   1. Initial program 99.9%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 79.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 79.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{z}\right) \]
   9. Step-by-step derivation

   10. Simplified 78.3%

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

   if -1.35e10 < z < 9e7

   1. Initial program 99.8%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.2%

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

Alternative 4: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2400000.0)
   (- x z)
   (if (<= z 175.0) (- x (* 0.5 (log y))) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2400000.0) {
		tmp = x - z;
	} else if (z <= 175.0) {
		tmp = x - (0.5 * log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2400000.0d0)) then
        tmp = x - z
    else if (z <= 175.0d0) then
        tmp = x - (0.5d0 * log(y))
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2400000.0) {
		tmp = x - z;
	} else if (z <= 175.0) {
		tmp = x - (0.5 * Math.log(y));
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2400000.0:
		tmp = x - z
	elif z <= 175.0:
		tmp = x - (0.5 * math.log(y))
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2400000.0)
		tmp = Float64(x - z);
	elseif (z <= 175.0)
		tmp = Float64(x - Float64(0.5 * log(y)));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2400000.0)
		tmp = x - z;
	elseif (z <= 175.0)
		tmp = x - (0.5 * log(y));
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2400000.0], N[(x - z), $MachinePrecision], If[LessEqual[z, 175.0], N[(x - N[(0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4e6 or 175 < z

   1. Initial program 99.9%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 78.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 78.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{z}\right) \]
   9. Step-by-step derivation

   10. Simplified 77.6%

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

   if -2.4e6 < z < 175

   1. Initial program 99.8%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{x}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 75.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \log y\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\log y \cdot \color{blue}{\frac{1}{2}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{1}{2}}\right)\right) \]

      3. log-lowering-log.f64 65.7%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{1}{2}\right)\right) \]

   10. Simplified 65.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2400000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x - 0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.045:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(y - z\right)\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.045)
   (+ x (- (* (log y) -0.5) z))
   (- (+ x (- y z)) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.045) {
		tmp = x + ((log(y) * -0.5) - z);
	} else {
		tmp = (x + (y - z)) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 0.045d0) then
        tmp = x + ((log(y) * (-0.5d0)) - z)
    else
        tmp = (x + (y - z)) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.045) {
		tmp = x + ((Math.log(y) * -0.5) - z);
	} else {
		tmp = (x + (y - z)) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 0.045:
		tmp = x + ((math.log(y) * -0.5) - z)
	else:
		tmp = (x + (y - z)) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.045)
		tmp = Float64(x + Float64(Float64(log(y) * -0.5) - z));
	else
		tmp = Float64(Float64(x + Float64(y - z)) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 0.045)
		tmp = x + ((log(y) * -0.5) - z);
	else
		tmp = (x + (y - z)) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.045], N[(x + N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(y - z), $MachinePrecision]), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.044999999999999998

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 98.6%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 98.6%

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

   if 0.044999999999999998 < y

   1. Initial program 99.7%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \color{blue}{\left(-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(y \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(y \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(y, \color{blue}{\log y}\right)\right) \]

      6. log-lowering-log.f64 99.1%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(y, \mathsf{log.f64}\left(y\right)\right)\right) \]

   7. Simplified 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.045:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(y - z\right)\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1300000000:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1300000000.0)
   (+ x (- (* (log y) -0.5) z))
   (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1300000000.0) {
		tmp = x + ((log(y) * -0.5) - z);
	} else {
		tmp = (x + y) - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1300000000.0d0) then
        tmp = x + ((log(y) * (-0.5d0)) - z)
    else
        tmp = (x + y) - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1300000000.0) {
		tmp = x + ((Math.log(y) * -0.5) - z);
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1300000000.0:
		tmp = x + ((math.log(y) * -0.5) - z)
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1300000000.0)
		tmp = Float64(x + Float64(Float64(log(y) * -0.5) - z));
	else
		tmp = Float64(Float64(x + y) - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1300000000.0)
		tmp = x + ((log(y) * -0.5) - z);
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1300000000.0], N[(x + N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e9

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 98.0%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 98.0%

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

   if 1.3e9 < y

   1. Initial program 99.8%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x + y\right)}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(y + x\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(y, \frac{1}{2}\right)}, \mathsf{log.f64}\left(y\right)\right)\right) \]

      2. +-lowering-+.f64 79.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(y, \frac{1}{2}\right)}, \mathsf{log.f64}\left(y\right)\right)\right) \]

   7. Simplified 79.0%

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

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \color{blue}{\left(-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(\mathsf{neg}\left(y \cdot \log \left(\frac{1}{y}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right)\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \left(y \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(y, \color{blue}{\log y}\right)\right) \]

      6. log-lowering-log.f64 78.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, x\right), \mathsf{*.f64}\left(y, \mathsf{log.f64}\left(y\right)\right)\right) \]

   10. Simplified 78.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1300000000:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+176}:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+176) (+ x (- (* (log y) -0.5) z)) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+176) {
		tmp = x + ((log(y) * -0.5) - z);
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.8d+176) then
        tmp = x + ((log(y) * (-0.5d0)) - z)
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+176) {
		tmp = x + ((Math.log(y) * -0.5) - z);
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.8e+176:
		tmp = x + ((math.log(y) * -0.5) - z)
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e+176)
		tmp = Float64(x + Float64(Float64(log(y) * -0.5) - z));
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.8e+176)
		tmp = x + ((log(y) * -0.5) - z);
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e+176], N[(x + N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8000000000000002e176

   1. Initial program 99.9%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 83.9%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 83.9%

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

   if 2.8000000000000002e176 < y

   1. Initial program 99.6%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

         \[\leadsto y \cdot \left(1 - \log y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right) \]

      6. log-lowering-log.f64 78.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right) \]

   7. Simplified 78.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+176}:\\ \;\;\;\;x + \left(\log y \cdot -0.5 - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 9: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{+176}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.25e+176) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e+176) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.25d+176) then
        tmp = x - z
    else
        tmp = y * (1.0d0 - log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.25e+176) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.25e+176:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.25e+176)
		tmp = Float64(x - z);
	else
		tmp = Float64(y * Float64(1.0 - log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.25e+176)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.25e+176], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.25000000000000002e176

   1. Initial program 99.9%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

      9. log-lowering-log.f64 83.9%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

   7. Simplified 83.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{z}\right) \]
   9. Step-by-step derivation

   10. Simplified 65.1%

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

   if 2.25000000000000002e176 < y

   1. Initial program 99.6%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.6%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

         \[\leadsto y \cdot \left(1 - \log y\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(1 - \log y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \color{blue}{\log y}\right)\right) \]

      6. log-lowering-log.f64 78.0%

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{log.f64}\left(y\right)\right)\right) \]

   7. Simplified 78.0%

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

Alternative 10: 48.2% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -72000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+129}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -72000000.0) x (if (<= x 5.5e+129) (- 0.0 z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -72000000.0) {
		tmp = x;
	} else if (x <= 5.5e+129) {
		tmp = 0.0 - z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-72000000.0d0)) then
        tmp = x
    else if (x <= 5.5d+129) then
        tmp = 0.0d0 - z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -72000000.0) {
		tmp = x;
	} else if (x <= 5.5e+129) {
		tmp = 0.0 - z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -72000000.0:
		tmp = x
	elif x <= 5.5e+129:
		tmp = 0.0 - z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -72000000.0)
		tmp = x;
	elseif (x <= 5.5e+129)
		tmp = Float64(0.0 - z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -72000000.0)
		tmp = x;
	elseif (x <= 5.5e+129)
		tmp = 0.0 - z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -72000000.0], x, If[LessEqual[x, 5.5e+129], N[(0.0 - z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e7 or 5.49999999999999984e129 < x

   1. Initial program 100.0%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 100.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 70.8%

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

   if -7.2e7 < x < 5.49999999999999984e129

   1. Initial program 99.8%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r- N/A

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

      4. +-commutative N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      10. associate-+l- N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

      17. log-lowering-log.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 34.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{z}\right) \]

   7. Simplified 34.5%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 34.5%

         \[\leadsto \mathsf{neg.f64}\left(z\right) \]

   9. Applied egg-rr 34.5%

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

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -72000000:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+129}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.6% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   10. associate-+l- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

   17. log-lowering-log.f64 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in y around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(z + \frac{1}{2} \cdot \log y\right)}\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)\right)\right)\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(z - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \log y\right)\right)}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\mathsf{neg}\left(\log y \cdot \frac{1}{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \left(\log y \cdot \frac{-1}{2}\right)\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\log y, \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   9. log-lowering-log.f64 71.7%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, \mathsf{*.f64}\left(\mathsf{log.f64}\left(y\right), \frac{-1}{2}\right)\right)\right) \]

7. Simplified 71.7%

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

8. Taylor expanded in z around inf

   \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{z}\right) \]
9. Step-by-step derivation

10. Simplified 56.5%

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

Alternative 12: 30.7% accurate, 111.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.9%

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+r- N/A

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

   4. +-commutative N/A

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

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

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(y - z\right)\right), \color{blue}{\left(\left(y + \frac{1}{2}\right) \cdot \log y\right)}\right) \]

   6. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x - \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{neg}\left(\left(y - z\right)\right)\right)\right), \left(\color{blue}{\left(y + \frac{1}{2}\right)} \cdot \log y\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(0 - \left(y - z\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   10. associate-+l- N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(0 - y\right) + z\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   11. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(\left(\mathsf{neg}\left(y\right)\right) + z\right)\right), \left(\left(y + \frac{1}{2}\right) \cdot \log y\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \left(z - y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \left(\left(y + \color{blue}{\frac{1}{2}}\right) \cdot \log y\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\left(y + \frac{1}{2}\right), \color{blue}{\log y}\right)\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \log \color{blue}{y}\right)\right) \]

   17. log-lowering-log.f64 99.9%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{\_.f64}\left(z, y\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, \frac{1}{2}\right), \mathsf{log.f64}\left(y\right)\right)\right) \]

3. Simplified 99.9%

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

5. Taylor expanded in x around inf

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

7. Simplified 31.6%

   \[\leadsto \color{blue}{x} \]
8. 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 2024138 
(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))