Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.4s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lift-+.f64 N/A

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

   8. +-commutative N/A

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

   9. distribute-neg-in N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 N/A

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

   12. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 75.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+24} \lor \neg \left(t\_0 \leq 500\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -1e+144)
     (* (- 1.0 (log y)) y)
     (if (or (<= t_0 -1e+24) (not (<= t_0 500.0)))
       (- (+ (pow (pow x -1.0) -1.0) y) z)
       (- (* -0.5 (log y)) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double tmp;
	if (t_0 <= -1e+144) {
		tmp = (1.0 - log(y)) * y;
	} else if ((t_0 <= -1e+24) || !(t_0 <= 500.0)) {
		tmp = (pow(pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = (-0.5 * 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) :: t_0
    real(8) :: tmp
    t_0 = (x - ((y + 0.5d0) * log(y))) + y
    if (t_0 <= (-1d+144)) then
        tmp = (1.0d0 - log(y)) * y
    else if ((t_0 <= (-1d+24)) .or. (.not. (t_0 <= 500.0d0))) then
        tmp = (((x ** (-1.0d0)) ** (-1.0d0)) + y) - z
    else
        tmp = ((-0.5d0) * log(y)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * Math.log(y))) + y;
	double tmp;
	if (t_0 <= -1e+144) {
		tmp = (1.0 - Math.log(y)) * y;
	} else if ((t_0 <= -1e+24) || !(t_0 <= 500.0)) {
		tmp = (Math.pow(Math.pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = (-0.5 * Math.log(y)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - ((y + 0.5) * math.log(y))) + y
	tmp = 0
	if t_0 <= -1e+144:
		tmp = (1.0 - math.log(y)) * y
	elif (t_0 <= -1e+24) or not (t_0 <= 500.0):
		tmp = (math.pow(math.pow(x, -1.0), -1.0) + y) - z
	else:
		tmp = (-0.5 * math.log(y)) - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	tmp = 0.0
	if (t_0 <= -1e+144)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif ((t_0 <= -1e+24) || !(t_0 <= 500.0))
		tmp = Float64(Float64(((x ^ -1.0) ^ -1.0) + y) - z);
	else
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - ((y + 0.5) * log(y))) + y;
	tmp = 0.0;
	if (t_0 <= -1e+144)
		tmp = (1.0 - log(y)) * y;
	elseif ((t_0 <= -1e+24) || ~((t_0 <= 500.0)))
		tmp = (((x ^ -1.0) ^ -1.0) + y) - z;
	else
		tmp = (-0.5 * log(y)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+144], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[Or[LessEqual[t$95$0, -1e+24], N[Not[LessEqual[t$95$0, 500.0]], $MachinePrecision]], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.00000000000000002e144

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 66.3

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

   5. Applied rewrites 66.3%

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

   if -1.00000000000000002e144 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.9999999999999998e23 or 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

   if -9.9999999999999998e23 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 94.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq -1 \cdot 10^{+24} \lor \neg \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y \leq 500\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -1000 \lor \neg \left(t\_0 \leq 500\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ y 0.5) (log y))) y) z)))
   (if (or (<= t_0 -1000.0) (not (<= t_0 500.0)))
     (- (+ (pow (pow x -1.0) -1.0) y) z)
     (fma -0.5 (log y) y))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 500.0)) {
		tmp = (pow(pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = fma(-0.5, log(y), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
	tmp = 0.0
	if ((t_0 <= -1000.0) || !(t_0 <= 500.0))
		tmp = Float64(Float64(((x ^ -1.0) ^ -1.0) + y) - z);
	else
		tmp = fma(-0.5, log(y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000.0], N[Not[LessEqual[t$95$0, 500.0]], $MachinePrecision]], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(-0.5 * N[Log[y], $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -1e3 or 500 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 59.9

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

   7. Applied rewrites 59.9%

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

   if -1e3 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \log y, y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \leq -1000 \lor \neg \left(\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \leq 500\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -1000 \lor \neg \left(t\_0 \leq 500\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ y 0.5) (log y))) y) z)))
   (if (or (<= t_0 -1000.0) (not (<= t_0 500.0)))
     (- (+ (pow (pow x -1.0) -1.0) y) z)
     (* -0.5 (log y)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 500.0)) {
		tmp = (pow(pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = -0.5 * 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) :: t_0
    real(8) :: tmp
    t_0 = ((x - ((y + 0.5d0) * log(y))) + y) - z
    if ((t_0 <= (-1000.0d0)) .or. (.not. (t_0 <= 500.0d0))) then
        tmp = (((x ** (-1.0d0)) ** (-1.0d0)) + y) - z
    else
        tmp = (-0.5d0) * log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * Math.log(y))) + y) - z;
	double tmp;
	if ((t_0 <= -1000.0) || !(t_0 <= 500.0)) {
		tmp = (Math.pow(Math.pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = -0.5 * Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x - ((y + 0.5) * math.log(y))) + y) - z
	tmp = 0
	if (t_0 <= -1000.0) or not (t_0 <= 500.0):
		tmp = (math.pow(math.pow(x, -1.0), -1.0) + y) - z
	else:
		tmp = -0.5 * math.log(y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
	tmp = 0.0
	if ((t_0 <= -1000.0) || !(t_0 <= 500.0))
		tmp = Float64(Float64(((x ^ -1.0) ^ -1.0) + y) - z);
	else
		tmp = Float64(-0.5 * log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	tmp = 0.0;
	if ((t_0 <= -1000.0) || ~((t_0 <= 500.0)))
		tmp = (((x ^ -1.0) ^ -1.0) + y) - z;
	else
		tmp = -0.5 * log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1000.0], N[Not[LessEqual[t$95$0, 500.0]], $MachinePrecision]], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -1e3 or 500 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 59.9

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

   7. Applied rewrites 59.9%

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

   if -1e3 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 92.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \leq -1000 \lor \neg \left(\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \leq 500\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+18} \lor \neg \left(x \leq 7900000\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+18) (not (<= x 7900000.0)))
   (- (+ (pow (pow x -1.0) -1.0) y) z)
   (- (* -0.5 (log y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+18) || !(x <= 7900000.0)) {
		tmp = (pow(pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = (-0.5 * 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 ((x <= (-2.4d+18)) .or. (.not. (x <= 7900000.0d0))) then
        tmp = (((x ** (-1.0d0)) ** (-1.0d0)) + y) - z
    else
        tmp = ((-0.5d0) * log(y)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+18) || !(x <= 7900000.0)) {
		tmp = (Math.pow(Math.pow(x, -1.0), -1.0) + y) - z;
	} else {
		tmp = (-0.5 * Math.log(y)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.4e+18) or not (x <= 7900000.0):
		tmp = (math.pow(math.pow(x, -1.0), -1.0) + y) - z
	else:
		tmp = (-0.5 * math.log(y)) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+18) || !(x <= 7900000.0))
		tmp = Float64(Float64(((x ^ -1.0) ^ -1.0) + y) - z);
	else
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.4e+18) || ~((x <= 7900000.0)))
		tmp = (((x ^ -1.0) ^ -1.0) + y) - z;
	else
		tmp = (-0.5 * log(y)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+18], N[Not[LessEqual[x, 7900000.0]], $MachinePrecision]], N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e18 or 7.9e6 < x

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -2.4e18 < x < 7.9e6

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.5%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+18} \lor \neg \left(x \leq 7900000\right):\\ \;\;\;\;\left({\left({x}^{-1}\right)}^{-1} + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z):
	return (math.pow(math.pow(x, -1.0), -1.0) + y) - z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[Power[N[Power[x, -1.0], $MachinePrecision], -1.0], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 52.4

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

7. Applied rewrites 52.4%

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

8. Final simplification 52.4%

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

Alternative 7: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+36} \lor \neg \left(x \leq 4.3 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.25e+36) (not (<= x 4.3e+73)))
   (- (fma -0.5 (log y) x) z)
   (- y (fma (+ 0.5 y) (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.25e+36) || !(x <= 4.3e+73)) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.25e+36) || !(x <= 4.3e+73))
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.25e+36], N[Not[LessEqual[x, 4.3e+73]], $MachinePrecision]], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.24999999999999998e36 or 4.30000000000000013e73 < 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

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -2.24999999999999998e36 < x < 4.30000000000000013e73

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+36} \lor \neg \left(x \leq 4.3 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, \log y, x\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28) (- (fma -0.5 (log y) x) z) (- (+ (fma (- y) (log y) x) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (fma(-y, log(y), x) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(fma(Float64(-y), log(y), x) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[((-y) * N[Log[y], $MachinePrecision] + x), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.28000000000000003 < y

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 9: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28) (- (fma -0.5 (log y) x) z) (- (+ (- x (* (log y) y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.28) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((x - (log(y) * y)) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.28)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.28], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.28000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 0.28000000000000003 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

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

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 10: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.45e+68)
   (- (fma -0.5 (log y) x) z)
   (fma (- -0.5 y) (log y) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.45e+68) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma((-0.5 - y), log(y), (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.45e+68)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.45e+68], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.45000000000000006e68

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 95.0

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

   5. Applied rewrites 95.0%

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

   if 1.45000000000000006e68 < 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 z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 11.8

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

   5. Applied rewrites 11.8%

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

   6. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. distribute-neg-in N/A

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

      9. metadata-eval N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-+.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 11: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.55e+138) (- (fma -0.5 (log y) x) z) (fma (- -0.5 y) (log y) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.55e+138) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma((-0.5 - y), log(y), y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.55e+138)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(Float64(-0.5 - y), log(y), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.55e+138], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5499999999999999e138

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if 1.5499999999999999e138 < 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 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 86.8

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 86.1%

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

Alternative 12: 29.9% accurate, 39.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 26.4

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

5. Applied rewrites 26.4%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024299 
(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))