Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 13.1s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-279}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-206}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+177}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e-279)
   (- x z)
   (if (<= y 4.4e-206)
     (- x (* (log y) 0.5))
     (if (<= y 6.5e+177) (- x z) (- (* y (- 1.0 (log y))) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e-279) {
		tmp = x - z;
	} else if (y <= 4.4e-206) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 6.5e+177) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - log(y))) - 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 (y <= 3.8d-279) then
        tmp = x - z
    else if (y <= 4.4d-206) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 6.5d+177) then
        tmp = x - z
    else
        tmp = (y * (1.0d0 - log(y))) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e-279) {
		tmp = x - z;
	} else if (y <= 4.4e-206) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 6.5e+177) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.8e-279:
		tmp = x - z
	elif y <= 4.4e-206:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 6.5e+177:
		tmp = x - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e-279)
		tmp = Float64(x - z);
	elseif (y <= 4.4e-206)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 6.5e+177)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(y * Float64(1.0 - log(y))) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.8e-279)
		tmp = x - z;
	elseif (y <= 4.4e-206)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 6.5e+177)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e-279], N[(x - z), $MachinePrecision], If[LessEqual[y, 4.4e-206], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+177], N[(x - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.80000000000000033e-279 or 4.3999999999999997e-206 < y < 6.5000000000000002e177

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 96.5%

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

   6. Taylor expanded in y around inf 84.7%

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

      1. log-rec 84.7%

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

      2. distribute-frac-neg 84.7%

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

      3. unsub-neg 84.7%

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

   8. Simplified 84.7%

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

   9. Taylor expanded in y around inf 84.7%

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

       1. cancel-sign-sub-inv 84.7%

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

       2. metadata-eval 84.7%

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

       3. log-rec 84.7%

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

       4. distribute-neg-frac 84.7%

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

       5. *-lft-identity 84.7%

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

       6. sub-neg 84.7%

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

       7. div-sub 84.7%

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

   11. Simplified 84.7%

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

   12. Taylor expanded in y around 0 76.8%

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

       1. neg-mul-1 76.8%

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

       2. sub-neg 76.8%

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

   14. Simplified 76.8%

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

   if 3.80000000000000033e-279 < y < 4.3999999999999997e-206

   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+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.3%

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

   6. Taylor expanded in y around 0 72.3%

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

   if 6.5000000000000002e177 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 93.2%

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

      1. *-commutative 93.2%

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

      2. log-rec 93.2%

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

      3. distribute-lft-neg-in 93.2%

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

      4. distribute-rgt-neg-in 93.2%

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

   7. Simplified 93.2%

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

   8. Taylor expanded in y around 0 93.4%

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

      1. neg-mul-1 93.4%

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

      2. sub-neg 93.4%

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

   10. Simplified 93.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-279}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-206}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+177}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-280}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-205}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+92}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e-280)
   (- x z)
   (if (<= y 3.4e-205)
     (- x (* (log y) 0.5))
     (if (<= y 6e+92) (- x z) (- (+ x y) (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e-280) {
		tmp = x - z;
	} else if (y <= 3.4e-205) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 6e+92) {
		tmp = x - 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 <= 2.1d-280) then
        tmp = x - z
    else if (y <= 3.4d-205) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 6d+92) then
        tmp = x - 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 <= 2.1e-280) {
		tmp = x - z;
	} else if (y <= 3.4e-205) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 6e+92) {
		tmp = x - z;
	} else {
		tmp = (x + y) - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.1e-280:
		tmp = x - z
	elif y <= 3.4e-205:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 6e+92:
		tmp = x - z
	else:
		tmp = (x + y) - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e-280)
		tmp = Float64(x - z);
	elseif (y <= 3.4e-205)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 6e+92)
		tmp = Float64(x - 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 <= 2.1e-280)
		tmp = x - z;
	elseif (y <= 3.4e-205)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 6e+92)
		tmp = x - z;
	else
		tmp = (x + y) - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e-280], N[(x - z), $MachinePrecision], If[LessEqual[y, 3.4e-205], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+92], N[(x - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.10000000000000001e-280 or 3.4000000000000002e-205 < y < 6.00000000000000026e92

   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+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 98.5%

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

   6. Taylor expanded in y around inf 83.7%

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

      1. log-rec 83.7%

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

      2. distribute-frac-neg 83.7%

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

      3. unsub-neg 83.7%

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

   8. Simplified 83.7%

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

   9. Taylor expanded in y around inf 83.7%

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

       1. cancel-sign-sub-inv 83.7%

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

       2. metadata-eval 83.7%

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

       3. log-rec 83.7%

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

       4. distribute-neg-frac 83.7%

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

       5. *-lft-identity 83.7%

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

       6. sub-neg 83.7%

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

       7. div-sub 83.7%

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

   11. Simplified 83.7%

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

   12. Taylor expanded in y around 0 79.4%

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

       1. neg-mul-1 79.4%

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

       2. sub-neg 79.4%

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

   14. Simplified 79.4%

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

   if 2.10000000000000001e-280 < y < 3.4000000000000002e-205

   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+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.3%

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

   6. Taylor expanded in y around 0 72.3%

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

   if 6.00000000000000026e92 < 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+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around 0 86.4%

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

   6. Taylor expanded in y around inf 86.4%

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

      1. mul-1-neg 86.4%

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

      2. distribute-rgt-neg-in 86.4%

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

      3. log-rec 86.4%

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

      4. remove-double-neg 86.4%

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

   8. Simplified 86.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-280}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-205}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+92}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-279}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-206}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+179}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e-279)
   (- x z)
   (if (<= y 3.1e-206)
     (- x (* (log y) 0.5))
     (if (<= y 7.4e+179) (- x z) (- y (* y (log y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e-279) {
		tmp = x - z;
	} else if (y <= 3.1e-206) {
		tmp = x - (log(y) * 0.5);
	} else if (y <= 7.4e+179) {
		tmp = x - z;
	} else {
		tmp = y - (y * log(y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.8d-279) then
        tmp = x - z
    else if (y <= 3.1d-206) then
        tmp = x - (log(y) * 0.5d0)
    else if (y <= 7.4d+179) then
        tmp = x - z
    else
        tmp = y - (y * log(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e-279) {
		tmp = x - z;
	} else if (y <= 3.1e-206) {
		tmp = x - (Math.log(y) * 0.5);
	} else if (y <= 7.4e+179) {
		tmp = x - z;
	} else {
		tmp = y - (y * Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.8e-279:
		tmp = x - z
	elif y <= 3.1e-206:
		tmp = x - (math.log(y) * 0.5)
	elif y <= 7.4e+179:
		tmp = x - z
	else:
		tmp = y - (y * math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e-279)
		tmp = Float64(x - z);
	elseif (y <= 3.1e-206)
		tmp = Float64(x - Float64(log(y) * 0.5));
	elseif (y <= 7.4e+179)
		tmp = Float64(x - z);
	else
		tmp = Float64(y - Float64(y * log(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.8e-279)
		tmp = x - z;
	elseif (y <= 3.1e-206)
		tmp = x - (log(y) * 0.5);
	elseif (y <= 7.4e+179)
		tmp = x - z;
	else
		tmp = y - (y * log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e-279], N[(x - z), $MachinePrecision], If[LessEqual[y, 3.1e-206], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.4e+179], N[(x - z), $MachinePrecision], N[(y - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.80000000000000033e-279 or 3.1000000000000003e-206 < y < 7.3999999999999999e179

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 96.5%

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

   6. Taylor expanded in y around inf 84.7%

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

      1. log-rec 84.7%

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

      2. distribute-frac-neg 84.7%

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

      3. unsub-neg 84.7%

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

   8. Simplified 84.7%

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

   9. Taylor expanded in y around inf 84.7%

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

       1. cancel-sign-sub-inv 84.7%

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

       2. metadata-eval 84.7%

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

       3. log-rec 84.7%

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

       4. distribute-neg-frac 84.7%

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

       5. *-lft-identity 84.7%

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

       6. sub-neg 84.7%

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

       7. div-sub 84.7%

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

   11. Simplified 84.7%

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

   12. Taylor expanded in y around 0 76.8%

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

       1. neg-mul-1 76.8%

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

       2. sub-neg 76.8%

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

   14. Simplified 76.8%

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

   if 3.80000000000000033e-279 < y < 3.1000000000000003e-206

   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+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 72.3%

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

   6. Taylor expanded in y around 0 72.3%

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

   if 7.3999999999999999e179 < y

   1. Initial program 99.5%

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

      1. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around 0 91.8%

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

   6. Taylor expanded in y around inf 91.8%

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

      1. mul-1-neg 91.8%

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

      2. distribute-rgt-neg-in 91.8%

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

      3. log-rec 91.8%

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

      4. remove-double-neg 91.8%

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

   8. Simplified 91.8%

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

   9. Taylor expanded in x around 0 85.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-279}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-206}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \mathbf{elif}\;y \leq 7.4 \cdot 10^{+179}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -600 \lor \neg \left(z \leq 255\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -600.0) (not (<= z 255.0))) (- x z) (- x (* (log y) 0.5))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -600.0) || !(z <= 255.0)) {
		tmp = x - z;
	} else {
		tmp = x - (log(y) * 0.5);
	}
	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 <= (-600.0d0)) .or. (.not. (z <= 255.0d0))) then
        tmp = x - z
    else
        tmp = x - (log(y) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -600.0) || !(z <= 255.0)) {
		tmp = x - z;
	} else {
		tmp = x - (Math.log(y) * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -600.0) or not (z <= 255.0):
		tmp = x - z
	else:
		tmp = x - (math.log(y) * 0.5)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -600.0) || !(z <= 255.0))
		tmp = Float64(x - z);
	else
		tmp = Float64(x - Float64(log(y) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -600.0) || ~((z <= 255.0)))
		tmp = x - z;
	else
		tmp = x - (log(y) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -600.0], N[Not[LessEqual[z, 255.0]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -600 or 255 < 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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 99.8%

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

   6. Taylor expanded in y around inf 99.1%

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

      1. log-rec 99.1%

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

      2. distribute-frac-neg 99.1%

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

      3. unsub-neg 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in y around inf 99.1%

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

       1. cancel-sign-sub-inv 99.1%

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

       2. metadata-eval 99.1%

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

       3. log-rec 99.1%

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

       4. distribute-neg-frac 99.1%

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

       5. *-lft-identity 99.1%

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

       6. sub-neg 99.1%

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

       7. div-sub 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in y around 0 74.0%

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

       1. neg-mul-1 74.0%

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

       2. sub-neg 74.0%

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

   14. Simplified 74.0%

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

   if -600 < z < 255

   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+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 99.2%

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

   6. Taylor expanded in y around 0 70.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -600 \lor \neg \left(z \leq 255\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;x - \log y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.8e-8)
   (- (+ x (* (log y) -0.5)) z)
   (+ x (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e-8) {
		tmp = (x + (log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - log(y))) - 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 (y <= 3.8d-8) then
        tmp = (x + (log(y) * (-0.5d0))) - z
    else
        tmp = x + ((y * (1.0d0 - log(y))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.8e-8) {
		tmp = (x + (Math.log(y) * -0.5)) - z;
	} else {
		tmp = x + ((y * (1.0 - Math.log(y))) - z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.8e-8:
		tmp = (x + (math.log(y) * -0.5)) - z
	else:
		tmp = x + ((y * (1.0 - math.log(y))) - z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.8e-8)
		tmp = Float64(Float64(x + Float64(log(y) * -0.5)) - z);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(1.0 - log(y))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.8e-8)
		tmp = (x + (log(y) * -0.5)) - z;
	else
		tmp = x + ((y * (1.0 - log(y))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.8e-8], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(x + N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000028e-8

   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+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.8%

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

   if 3.80000000000000028e-8 < 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+ 99.7%

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

      2. sub-neg 99.7%

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

      3. associate-+l+ 99.7%

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

      4. associate-+r- 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-define 99.8%

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

      8. +-commutative 99.8%

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

      9. distribute-neg-in 99.8%

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

      10. unsub-neg 99.8%

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

      11. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.4%

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

      1. log-rec 99.4%

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

      2. sub-neg 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot \left(1 - \log y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-281} \lor \neg \left(z \leq 3.2 \cdot 10^{-286}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e-281) (not (<= z 3.2e-286))) (- x z) (* (log y) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e-281) || !(z <= 3.2e-286)) {
		tmp = x - z;
	} else {
		tmp = log(y) * -0.5;
	}
	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 <= (-2d-281)) .or. (.not. (z <= 3.2d-286))) then
        tmp = x - z
    else
        tmp = log(y) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e-281) || !(z <= 3.2e-286)) {
		tmp = x - z;
	} else {
		tmp = Math.log(y) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2e-281) or not (z <= 3.2e-286):
		tmp = x - z
	else:
		tmp = math.log(y) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e-281) || !(z <= 3.2e-286))
		tmp = Float64(x - z);
	else
		tmp = Float64(log(y) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2e-281) || ~((z <= 3.2e-286)))
		tmp = x - z;
	else
		tmp = log(y) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e-281], N[Not[LessEqual[z, 3.2e-286]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2e-281 or 3.20000000000000007e-286 < 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+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 92.0%

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

   6. Taylor expanded in y around inf 82.0%

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

      1. log-rec 82.0%

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

      2. distribute-frac-neg 82.0%

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

      3. unsub-neg 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in y around inf 82.0%

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

       1. cancel-sign-sub-inv 82.0%

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

       2. metadata-eval 82.0%

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

       3. log-rec 82.0%

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

       4. distribute-neg-frac 82.0%

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

       5. *-lft-identity 82.0%

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

       6. sub-neg 82.0%

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

       7. div-sub 82.0%

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

   11. Simplified 82.0%

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

   12. Taylor expanded in y around 0 62.3%

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

       1. neg-mul-1 62.3%

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

       2. sub-neg 62.3%

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

   14. Simplified 62.3%

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

   if -2e-281 < z < 3.20000000000000007e-286

   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+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 99.7%

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

   6. Taylor expanded in x around inf 83.2%

      \[\leadsto \color{blue}{x \cdot \left(\left(1 + \frac{y}{x}\right) - \frac{\log y \cdot \left(0.5 + y\right)}{x}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 83.2%

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

      2. div-sub 84.2%

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

      3. cancel-sign-sub-inv 84.2%

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

      4. +-commutative 84.2%

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

      5. cancel-sign-sub-inv 84.2%

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

   8. Simplified 84.2%

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

   9. Taylor expanded in y around 0 76.3%

      \[\leadsto x \cdot \left(1 + \frac{\color{blue}{-0.5 \cdot \log y}}{x}\right) \]
   10. Step-by-step derivation

       1. *-commutative 76.3%

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

   11. Simplified 76.3%

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

   12. Taylor expanded in x around 0 69.8%

       \[\leadsto \color{blue}{-0.5 \cdot \log y} \]
   13. Step-by-step derivation

       1. *-commutative 69.8%

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

   14. Simplified 69.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-281} \lor \neg \left(z \leq 3.2 \cdot 10^{-286}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+94}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+94) (- (+ x (* (log y) -0.5)) z) (- (+ x y) (* y (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.5e+94) {
		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 <= 9.5d+94) 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 <= 9.5e+94) {
		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 <= 9.5e+94:
		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 <= 9.5e+94)
		tmp = Float64(Float64(x + 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 <= 9.5e+94)
		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, 9.5e+94], N[(N[(x + N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.4999999999999998e94

   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+ 100.0%

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

      2. sub-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. associate-+r- 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-define 100.0%

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

      8. +-commutative 100.0%

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

      9. distribute-neg-in 100.0%

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

      10. unsub-neg 100.0%

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

      11. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 95.0%

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

   if 9.4999999999999998e94 < 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+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around 0 86.4%

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

   6. Taylor expanded in y around inf 86.4%

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

      1. mul-1-neg 86.4%

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

      2. distribute-rgt-neg-in 86.4%

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

      3. log-rec 86.4%

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

      4. remove-double-neg 86.4%

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

   8. Simplified 86.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+94}:\\ \;\;\;\;\left(x + \log y \cdot -0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - y \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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+ 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 11: 47.4% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.95e+32) x (if (<= x 4.4e+140) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.95e+32) {
		tmp = x;
	} else if (x <= 4.4e+140) {
		tmp = -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 <= (-2.95d+32)) then
        tmp = x
    else if (x <= 4.4d+140) then
        tmp = -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 <= -2.95e+32) {
		tmp = x;
	} else if (x <= 4.4e+140) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.95e+32:
		tmp = x
	elif x <= 4.4e+140:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.95e+32)
		tmp = x;
	elseif (x <= 4.4e+140)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.95e+32)
		tmp = x;
	elseif (x <= 4.4e+140)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.95e+32], x, If[LessEqual[x, 4.4e+140], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.94999999999999983e32 or 4.3999999999999997e140 < x

   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+ 99.9%

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

      2. sub-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-+r- 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 68.5%

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

   if -2.94999999999999983e32 < x < 4.3999999999999997e140

   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+ 99.8%

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

      2. sub-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. associate-+r- 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-define 99.9%

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

      8. +-commutative 99.9%

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

      9. distribute-neg-in 99.9%

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

      10. unsub-neg 99.9%

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

      11. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 42.8%

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

      1. neg-mul-1 42.8%

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

   7. Simplified 42.8%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+32}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+140}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.0% 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 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 91.6%

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

6. Taylor expanded in y around inf 78.8%

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

   1. log-rec 78.8%

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

   2. distribute-frac-neg 78.8%

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

   3. unsub-neg 78.8%

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

8. Simplified 78.8%

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

9. Taylor expanded in y around inf 78.8%

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

    1. cancel-sign-sub-inv 78.8%

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

    2. metadata-eval 78.8%

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

    3. log-rec 78.8%

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

    4. distribute-neg-frac 78.8%

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

    5. *-lft-identity 78.8%

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

    6. sub-neg 78.8%

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

    7. div-sub 78.8%

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

11. Simplified 78.8%

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

12. Taylor expanded in y around 0 59.9%

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

    1. neg-mul-1 59.9%

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

    2. sub-neg 59.9%

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

14. Simplified 59.9%

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

15. Final simplification 59.9%

    \[\leadsto x - z \]
16. Add Preprocessing

Alternative 13: 29.9% 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 99.8%

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

   1. associate--l+ 99.8%

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

   2. sub-neg 99.8%

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

   3. associate-+l+ 99.8%

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

   4. associate-+r- 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. fma-define 99.9%

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

   8. +-commutative 99.9%

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

   9. distribute-neg-in 99.9%

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

   10. unsub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 31.6%

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

6. Final simplification 31.6%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 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 2024071 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

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

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