Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.4s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 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- 99.8%

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

   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. Add Preprocessing

Alternative 2: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+117}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+62}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e+117)
   (- x z)
   (if (<= x 1.3e+62)
     (- (- y (* (log y) (+ y 0.5))) z)
     (- (- x (* (log y) 0.5)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+117) {
		tmp = x - z;
	} else if (x <= 1.3e+62) {
		tmp = (y - (log(y) * (y + 0.5))) - z;
	} 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 (x <= (-1.55d+117)) then
        tmp = x - z
    else if (x <= 1.3d+62) then
        tmp = (y - (log(y) * (y + 0.5d0))) - z
    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 (x <= -1.55e+117) {
		tmp = x - z;
	} else if (x <= 1.3e+62) {
		tmp = (y - (Math.log(y) * (y + 0.5))) - z;
	} else {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.55e+117:
		tmp = x - z
	elif x <= 1.3e+62:
		tmp = (y - (math.log(y) * (y + 0.5))) - z
	else:
		tmp = (x - (math.log(y) * 0.5)) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e+117)
		tmp = Float64(x - z);
	elseif (x <= 1.3e+62)
		tmp = Float64(Float64(y - Float64(log(y) * Float64(y + 0.5))) - z);
	else
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.55e+117)
		tmp = x - z;
	elseif (x <= 1.3e+62)
		tmp = (y - (log(y) * (y + 0.5))) - z;
	else
		tmp = (x - (log(y) * 0.5)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e+117], N[(x - z), $MachinePrecision], If[LessEqual[x, 1.3e+62], N[(N[(y - N[(N[Log[y], $MachinePrecision] * N[(y + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999988e117

   1. Initial program 100.0%

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

      1. flip-+ 88.6%

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

      2. div-inv 88.6%

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

      3. fmm-def 88.6%

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

      4. metadata-eval 88.6%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      5. metadata-eval 88.6%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      6. sub-neg 88.6%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{\color{blue}{y + \left(-0.5\right)}}\right) \cdot \log y\right) + y\right) - z \]

      7. metadata-eval 88.6%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + \color{blue}{-0.5}}\right) \cdot \log y\right) + y\right) - z \]

   4. Applied egg-rr 88.6%

      \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + -0.5}\right)} \cdot \log y\right) + y\right) - z \]
   5. Step-by-step derivation

      1. *-commutative 88.6%

         \[\leadsto \left(\left(x - \color{blue}{\left(\frac{1}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. associate-/r/ 88.6%

         \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]

   6. Simplified 88.6%

      \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-+l- 88.6%

         \[\leadsto \color{blue}{\left(x - \left(\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y - y\right)\right)} - z \]

      2. associate-*l/ 88.6%

         \[\leadsto \left(x - \left(\color{blue}{\frac{1 \cdot \log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} - y\right)\right) - z \]

      3. *-un-lft-identity 88.6%

         \[\leadsto \left(x - \left(\frac{\color{blue}{\log y}}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} - y\right)\right) - z \]

      4. clear-num 88.6%

         \[\leadsto \left(x - \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}} - y\right)\right) - z \]

      5. metadata-eval 88.6%

         \[\leadsto \left(x - \left(\frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}} - y\right)\right) - z \]

      6. metadata-eval 88.6%

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

      7. fmm-def 88.6%

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

      8. *-un-lft-identity 88.6%

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

      9. fma-define 88.6%

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

      10. metadata-eval 88.6%

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

      11. fmm-def 88.6%

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

      12. *-un-lft-identity 88.6%

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

      13. flip-+ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 90.7%

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

   if -1.54999999999999988e117 < x < 1.29999999999999992e62

   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 99.8%

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

      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 x around 0 96.6%

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

      1. associate-*r* 96.6%

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

      2. neg-mul-1 96.6%

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

      3. +-commutative 96.6%

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

      4. cancel-sign-sub-inv 96.6%

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

   7. Simplified 96.6%

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

   if 1.29999999999999992e62 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 87.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+117}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+62}:\\ \;\;\;\;\left(y - \log y \cdot \left(y + 0.5\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-147}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3e-147)
   (- (* (log y) -0.5) z)
   (if (<= y 1.5e+78) (- x z) (- (* y (- 1.0 (log y))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e-147) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 1.5e+78) {
		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 <= 3.3d-147) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 1.5d+78) 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 <= 3.3e-147) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 1.5e+78) {
		tmp = x - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3e-147:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 1.5e+78:
		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 <= 3.3e-147)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 1.5e+78)
		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 <= 3.3e-147)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 1.5e+78)
		tmp = x - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3e-147], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.5e+78], 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 < 3.29999999999999987e-147

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

      2. distribute-neg-in 76.5%

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

      3. sub-neg 76.5%

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

      4. neg-sub0 76.5%

         \[\leadsto \color{blue}{\left(0 - z\right)} - 0.5 \cdot \log y \]

      5. associate--r+ 76.5%

         \[\leadsto \color{blue}{0 - \left(z + 0.5 \cdot \log y\right)} \]

      6. +-commutative 76.5%

         \[\leadsto 0 - \color{blue}{\left(0.5 \cdot \log y + z\right)} \]

      7. associate--r+ 76.5%

         \[\leadsto \color{blue}{\left(0 - 0.5 \cdot \log y\right) - z} \]

      8. neg-sub0 76.5%

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

      9. distribute-lft-neg-in 76.5%

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

      10. metadata-eval 76.5%

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

      11. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   if 3.29999999999999987e-147 < y < 1.49999999999999991e78

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. div-inv 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      5. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      6. sub-neg 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{\color{blue}{y + \left(-0.5\right)}}\right) \cdot \log y\right) + y\right) - z \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + \color{blue}{-0.5}}\right) \cdot \log y\right) + y\right) - z \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + -0.5}\right)} \cdot \log y\right) + y\right) - z \]
   5. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(\frac{1}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. associate-/r/ 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]

   6. Simplified 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x - \left(\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y - y\right)\right)} - z \]

      2. associate-*l/ 100.0%

         \[\leadsto \left(x - \left(\color{blue}{\frac{1 \cdot \log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} - y\right)\right) - z \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(x - \left(\frac{\color{blue}{\log y}}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} - y\right)\right) - z \]

      4. clear-num 100.0%

         \[\leadsto \left(x - \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}} - y\right)\right) - z \]

      5. metadata-eval 100.0%

         \[\leadsto \left(x - \left(\frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}} - y\right)\right) - z \]

      6. metadata-eval 100.0%

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

      7. fmm-def 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. fma-define 100.0%

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

      10. metadata-eval 100.0%

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

      11. fmm-def 100.0%

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

      12. *-un-lft-identity 100.0%

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

      13. flip-+ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 79.6%

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

   if 1.49999999999999991e78 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around -inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. sub-neg 75.2%

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

      3. associate-/l* 75.1%

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

      4. +-commutative 75.1%

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

      5. metadata-eval 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in y around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. log-rec 82.5%

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

      3. remove-double-neg 82.5%

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

   8. Simplified 82.5%

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

Alternative 4: 67.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-149}:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{+78}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e-149)
   (- (* (log y) -0.5) z)
   (if (<= y 2.85e+78) (- x z) (* y (- 1.0 (log y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.2e-149) {
		tmp = (log(y) * -0.5) - z;
	} else if (y <= 2.85e+78) {
		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 <= 3.2d-149) then
        tmp = (log(y) * (-0.5d0)) - z
    else if (y <= 2.85d+78) 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 <= 3.2e-149) {
		tmp = (Math.log(y) * -0.5) - z;
	} else if (y <= 2.85e+78) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.2e-149:
		tmp = (math.log(y) * -0.5) - z
	elif y <= 2.85e+78:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.2e-149)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	elseif (y <= 2.85e+78)
		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 <= 3.2e-149)
		tmp = (log(y) * -0.5) - z;
	elseif (y <= 2.85e+78)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.2e-149], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.85e+78], N[(x - z), $MachinePrecision], N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.20000000000000002e-149

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in x around 0 76.5%

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

      1. mul-1-neg 76.5%

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

      2. distribute-neg-in 76.5%

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

      3. sub-neg 76.5%

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

      4. neg-sub0 76.5%

         \[\leadsto \color{blue}{\left(0 - z\right)} - 0.5 \cdot \log y \]

      5. associate--r+ 76.5%

         \[\leadsto \color{blue}{0 - \left(z + 0.5 \cdot \log y\right)} \]

      6. +-commutative 76.5%

         \[\leadsto 0 - \color{blue}{\left(0.5 \cdot \log y + z\right)} \]

      7. associate--r+ 76.5%

         \[\leadsto \color{blue}{\left(0 - 0.5 \cdot \log y\right) - z} \]

      8. neg-sub0 76.5%

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

      9. distribute-lft-neg-in 76.5%

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

      10. metadata-eval 76.5%

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

      11. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   if 3.20000000000000002e-149 < y < 2.84999999999999993e78

   1. Initial program 99.9%

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

      1. flip-+ 99.9%

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

      2. div-inv 99.9%

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

      3. fmm-def 99.9%

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

      4. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      5. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      6. sub-neg 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{\color{blue}{y + \left(-0.5\right)}}\right) \cdot \log y\right) + y\right) - z \]

      7. metadata-eval 99.9%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + \color{blue}{-0.5}}\right) \cdot \log y\right) + y\right) - z \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + -0.5}\right)} \cdot \log y\right) + y\right) - z \]
   5. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(\left(x - \color{blue}{\left(\frac{1}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. associate-/r/ 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]

   6. Simplified 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x - \left(\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y - y\right)\right)} - z \]

      2. associate-*l/ 100.0%

         \[\leadsto \left(x - \left(\color{blue}{\frac{1 \cdot \log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} - y\right)\right) - z \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(x - \left(\frac{\color{blue}{\log y}}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} - y\right)\right) - z \]

      4. clear-num 100.0%

         \[\leadsto \left(x - \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}} - y\right)\right) - z \]

      5. metadata-eval 100.0%

         \[\leadsto \left(x - \left(\frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}} - y\right)\right) - z \]

      6. metadata-eval 100.0%

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

      7. fmm-def 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. fma-define 100.0%

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

      10. metadata-eval 100.0%

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

      11. fmm-def 100.0%

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

      12. *-un-lft-identity 100.0%

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

      13. flip-+ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 79.6%

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

   if 2.84999999999999993e78 < 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%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 69.9%

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

      1. log-rec 69.9%

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

      2. sub-neg 69.9%

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

   7. Simplified 69.9%

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

Alternative 5: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -105 \lor \neg \left(z \leq -7.2 \cdot 10^{-122}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -105.0) (not (<= z -7.2e-122))) (- x z) (* (log y) -0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -105.0) || !(z <= -7.2e-122)) {
		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 <= (-105.0d0)) .or. (.not. (z <= (-7.2d-122)))) 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 <= -105.0) || !(z <= -7.2e-122)) {
		tmp = x - z;
	} else {
		tmp = Math.log(y) * -0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -105.0) or not (z <= -7.2e-122):
		tmp = x - z
	else:
		tmp = math.log(y) * -0.5
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -105.0) || !(z <= -7.2e-122))
		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 <= -105.0) || ~((z <= -7.2e-122)))
		tmp = x - z;
	else
		tmp = log(y) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -105.0], N[Not[LessEqual[z, -7.2e-122]], $MachinePrecision]], N[(x - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -105 or -7.19999999999999989e-122 < z

   1. Initial program 99.9%

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

      1. flip-+ 80.2%

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

      2. div-inv 80.2%

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

      3. fmm-def 80.2%

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

      4. metadata-eval 80.2%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      5. metadata-eval 80.2%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      6. sub-neg 80.2%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{\color{blue}{y + \left(-0.5\right)}}\right) \cdot \log y\right) + y\right) - z \]

      7. metadata-eval 80.2%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + \color{blue}{-0.5}}\right) \cdot \log y\right) + y\right) - z \]

   4. Applied egg-rr 80.2%

      \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + -0.5}\right)} \cdot \log y\right) + y\right) - z \]
   5. Step-by-step derivation

      1. *-commutative 80.2%

         \[\leadsto \left(\left(x - \color{blue}{\left(\frac{1}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. associate-/r/ 80.2%

         \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]

   6. Simplified 80.2%

      \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-+l- 80.2%

         \[\leadsto \color{blue}{\left(x - \left(\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y - y\right)\right)} - z \]

      2. associate-*l/ 80.2%

         \[\leadsto \left(x - \left(\color{blue}{\frac{1 \cdot \log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} - y\right)\right) - z \]

      3. *-un-lft-identity 80.2%

         \[\leadsto \left(x - \left(\frac{\color{blue}{\log y}}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} - y\right)\right) - z \]

      4. clear-num 80.2%

         \[\leadsto \left(x - \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}} - y\right)\right) - z \]

      5. metadata-eval 80.2%

         \[\leadsto \left(x - \left(\frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}} - y\right)\right) - z \]

      6. metadata-eval 80.2%

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

      7. fmm-def 80.2%

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

      8. *-un-lft-identity 80.2%

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

      9. fma-define 80.2%

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

      10. metadata-eval 80.2%

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

      11. fmm-def 80.2%

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

      12. *-un-lft-identity 80.2%

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

      13. flip-+ 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in x around inf 61.1%

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

   if -105 < z < -7.19999999999999989e-122

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0 66.4%

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

   4. Taylor expanded in z around 0 56.3%

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

   5. Taylor expanded in x around 0 46.4%

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

      1. *-commutative 46.4%

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

   7. Simplified 46.4%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -105 \lor \neg \left(z \leq -7.2 \cdot 10^{-122}\right):\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+78}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.85e+78) (- (- x (* (log y) 0.5)) z) (- (* y (- 1.0 (log y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.85e+78) {
		tmp = (x - (log(y) * 0.5)) - 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 <= 1.85d+78) then
        tmp = (x - (log(y) * 0.5d0)) - 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 <= 1.85e+78) {
		tmp = (x - (Math.log(y) * 0.5)) - z;
	} else {
		tmp = (y * (1.0 - Math.log(y))) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.85e+78:
		tmp = (x - (math.log(y) * 0.5)) - z
	else:
		tmp = (y * (1.0 - math.log(y))) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.85e+78)
		tmp = Float64(Float64(x - Float64(log(y) * 0.5)) - 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 <= 1.85e+78)
		tmp = (x - (log(y) * 0.5)) - z;
	else
		tmp = (y * (1.0 - log(y))) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.85e+78], N[(N[(x - N[(N[Log[y], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(y * N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.84999999999999992e78

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.6%

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

   if 1.84999999999999992e78 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around -inf 75.2%

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

      1. mul-1-neg 75.2%

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

      2. sub-neg 75.2%

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

      3. associate-/l* 75.1%

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

      4. +-commutative 75.1%

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

      5. metadata-eval 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in y around inf 82.5%

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

      1. mul-1-neg 82.5%

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

      2. log-rec 82.5%

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

      3. remove-double-neg 82.5%

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

   8. Simplified 82.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{+78}:\\ \;\;\;\;\left(x - \log y \cdot 0.5\right) - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 8: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.68 \cdot 10^{+78}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.68e+78) (- x z) (* y (- 1.0 (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.68e+78) {
		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 <= 1.68d+78) 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 <= 1.68e+78) {
		tmp = x - z;
	} else {
		tmp = y * (1.0 - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.68e+78:
		tmp = x - z
	else:
		tmp = y * (1.0 - math.log(y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.68e+78)
		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 <= 1.68e+78)
		tmp = x - z;
	else
		tmp = y * (1.0 - log(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.68e+78], 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 < 1.6799999999999999e78

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. div-inv 100.0%

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

      3. fmm-def 100.0%

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

      4. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      5. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

      6. sub-neg 100.0%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{\color{blue}{y + \left(-0.5\right)}}\right) \cdot \log y\right) + y\right) - z \]

      7. metadata-eval 100.0%

         \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + \color{blue}{-0.5}}\right) \cdot \log y\right) + y\right) - z \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + -0.5}\right)} \cdot \log y\right) + y\right) - z \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\left(\frac{1}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right)} \cdot \log y\right) + y\right) - z \]

      2. associate-/r/ 100.0%

         \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]

   6. Simplified 100.0%

      \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]
   7. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(x - \left(\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y - y\right)\right)} - z \]

      2. associate-*l/ 100.0%

         \[\leadsto \left(x - \left(\color{blue}{\frac{1 \cdot \log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} - y\right)\right) - z \]

      3. *-un-lft-identity 100.0%

         \[\leadsto \left(x - \left(\frac{\color{blue}{\log y}}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} - y\right)\right) - z \]

      4. clear-num 100.0%

         \[\leadsto \left(x - \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}} - y\right)\right) - z \]

      5. metadata-eval 100.0%

         \[\leadsto \left(x - \left(\frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}} - y\right)\right) - z \]

      6. metadata-eval 100.0%

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

      7. fmm-def 100.0%

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

      8. *-un-lft-identity 100.0%

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

      9. fma-define 100.0%

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

      10. metadata-eval 100.0%

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

      11. fmm-def 100.0%

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

      12. *-un-lft-identity 100.0%

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

      13. flip-+ 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around inf 72.1%

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

   if 1.6799999999999999e78 < 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%

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

      2. sub-neg 99.6%

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

      3. associate-+l+ 99.6%

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

      4. associate-+r- 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-define 99.7%

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

      8. +-commutative 99.7%

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

      9. distribute-neg-in 99.7%

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

      10. unsub-neg 99.7%

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

      11. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 69.9%

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

      1. log-rec 69.9%

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

      2. sub-neg 69.9%

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

   7. Simplified 69.9%

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

Alternative 9: 49.1% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+81}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.85e+81) x (if (<= x 1.35e+59) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.85e+81) {
		tmp = x;
	} else if (x <= 1.35e+59) {
		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 <= (-1.85d+81)) then
        tmp = x
    else if (x <= 1.35d+59) 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 <= -1.85e+81) {
		tmp = x;
	} else if (x <= 1.35e+59) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.85e+81:
		tmp = x
	elif x <= 1.35e+59:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.85e+81)
		tmp = x;
	elseif (x <= 1.35e+59)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.85e+81)
		tmp = x;
	elseif (x <= 1.35e+59)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.85e+81], x, If[LessEqual[x, 1.35e+59], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e81 or 1.3500000000000001e59 < 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 69.8%

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

   if -1.85e81 < x < 1.3500000000000001e59

   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 99.8%

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

      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 z around inf 39.8%

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

      1. neg-mul-1 39.8%

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

   7. Simplified 39.8%

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

Alternative 10: 58.9% 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. Add Preprocessing
3. Step-by-step derivation

   1. flip-+ 79.7%

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

   2. div-inv 79.7%

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

   3. fmm-def 79.7%

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

   4. metadata-eval 79.7%

      \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -\color{blue}{0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

   5. metadata-eval 79.7%

      \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right) \cdot \frac{1}{y - 0.5}\right) \cdot \log y\right) + y\right) - z \]

   6. sub-neg 79.7%

      \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{\color{blue}{y + \left(-0.5\right)}}\right) \cdot \log y\right) + y\right) - z \]

   7. metadata-eval 79.7%

      \[\leadsto \left(\left(x - \left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + \color{blue}{-0.5}}\right) \cdot \log y\right) + y\right) - z \]

4. Applied egg-rr 79.7%

   \[\leadsto \left(\left(x - \color{blue}{\left(\mathsf{fma}\left(y, y, -0.25\right) \cdot \frac{1}{y + -0.5}\right)} \cdot \log y\right) + y\right) - z \]
5. Step-by-step derivation

   1. *-commutative 79.7%

      \[\leadsto \left(\left(x - \color{blue}{\left(\frac{1}{y + -0.5} \cdot \mathsf{fma}\left(y, y, -0.25\right)\right)} \cdot \log y\right) + y\right) - z \]

   2. associate-/r/ 79.7%

      \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]

6. Simplified 79.7%

   \[\leadsto \left(\left(x - \color{blue}{\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} \cdot \log y\right) + y\right) - z \]
7. Step-by-step derivation

   1. associate-+l- 79.7%

      \[\leadsto \color{blue}{\left(x - \left(\frac{1}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} \cdot \log y - y\right)\right)} - z \]

   2. associate-*l/ 79.8%

      \[\leadsto \left(x - \left(\color{blue}{\frac{1 \cdot \log y}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}}} - y\right)\right) - z \]

   3. *-un-lft-identity 79.8%

      \[\leadsto \left(x - \left(\frac{\color{blue}{\log y}}{\frac{y + -0.5}{\mathsf{fma}\left(y, y, -0.25\right)}} - y\right)\right) - z \]

   4. clear-num 79.7%

      \[\leadsto \left(x - \left(\frac{\log y}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y + -0.5}}}} - y\right)\right) - z \]

   5. metadata-eval 79.7%

      \[\leadsto \left(x - \left(\frac{\log y}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \color{blue}{-0.25}\right)}{y + -0.5}}} - y\right)\right) - z \]

   6. metadata-eval 79.7%

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

   7. fmm-def 79.7%

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

   8. *-un-lft-identity 79.7%

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

   9. fma-define 79.7%

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

   10. metadata-eval 79.7%

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

   11. fmm-def 79.7%

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

   12. *-un-lft-identity 79.7%

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

   13. flip-+ 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in x around inf 56.5%

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

Alternative 11: 30.3% 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- 99.8%

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

   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 x around inf 27.1%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024185 
(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))