Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

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

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

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]

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. Add Preprocessing

Alternative 2: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -1e+141)
     (fma (- -0.5 y) (log y) (+ x y))
     (if (<= t_0 2.0)
       (- y (fma (+ 0.5 y) (log y) z))
       (- (fma -0.5 (log y) x) 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+141) {
		tmp = fma((-0.5 - y), log(y), (x + y));
	} else if (t_0 <= 2.0) {
		tmp = y - fma((0.5 + y), log(y), z);
	} else {
		tmp = fma(-0.5, log(y), x) - 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+141)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	elseif (t_0 <= 2.0)
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	else
		tmp = Float64(fma(-0.5, log(y), x) - z);
	end
	return 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+141], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $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) < -1.00000000000000002e141

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.6

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. associate--l+ N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 91.7%

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

   if -1.00000000000000002e141 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 2

   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 88.2

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

   5. Applied rewrites 88.2%

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

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

   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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 3: 67.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -10000 \lor \neg \left(t\_0 \leq 500\right):\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ y 0.5) (log y))) y) z)))
   (if (or (<= t_0 -10000.0) (not (<= t_0 500.0)))
     (- (+ (* 1.0 x) 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 <= -10000.0) || !(t_0 <= 500.0)) {
		tmp = ((1.0 * x) + 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 <= (-10000.0d0)) .or. (.not. (t_0 <= 500.0d0))) then
        tmp = ((1.0d0 * x) + 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 <= -10000.0) || !(t_0 <= 500.0)) {
		tmp = ((1.0 * x) + 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 <= -10000.0) or not (t_0 <= 500.0):
		tmp = ((1.0 * x) + 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 <= -10000.0) || !(t_0 <= 500.0))
		tmp = Float64(Float64(Float64(1.0 * x) + 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 <= -10000.0) || ~((t_0 <= 500.0)))
		tmp = ((1.0 * x) + 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, -10000.0], N[Not[LessEqual[t$95$0, 500.0]], $MachinePrecision]], N[(N[(N[(1.0 * x), $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) < -1e4 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

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 89.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 64.4%

       \[\leadsto \left(1 \cdot x + y\right) - z \]

   if -1e4 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 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 97.7

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 96.6%

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

   12. Taylor expanded in y around 0

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

   14. Applied rewrites 91.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \leq -10000 \lor \neg \left(\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \leq 500\right):\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right)\\ \mathbf{elif}\;t\_0 \leq 340:\\ \;\;\;\;y - \mathsf{fma}\left(0.5, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 2.0)
     (fma (- -0.5 y) (log y) y)
     (if (<= t_0 340.0) (- y (fma 0.5 (log y) z)) (- (+ (* 1.0 x) 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 <= 2.0) {
		tmp = fma((-0.5 - y), log(y), y);
	} else if (t_0 <= 340.0) {
		tmp = y - fma(0.5, log(y), z);
	} else {
		tmp = ((1.0 * x) + 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 <= 2.0)
		tmp = fma(Float64(-0.5 - y), log(y), y);
	elseif (t_0 <= 340.0)
		tmp = Float64(y - fma(0.5, log(y), z));
	else
		tmp = Float64(Float64(Float64(1.0 * x) + y) - z);
	end
	return 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, 2.0], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$0, 340.0], N[(y - N[(0.5 * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * x), $MachinePrecision] + y), $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) < 2

   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 58.0%

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

   if 2 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 340

   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 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.0%

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

   if 340 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \left(1 \cdot x + y\right) - z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right)\\ \mathbf{elif}\;t\_0 \leq 340:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 2.0)
     (fma (- -0.5 y) (log y) y)
     (if (<= t_0 340.0) (- (fma (log y) 0.5 z)) (- (+ (* 1.0 x) 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 <= 2.0) {
		tmp = fma((-0.5 - y), log(y), y);
	} else if (t_0 <= 340.0) {
		tmp = -fma(log(y), 0.5, z);
	} else {
		tmp = ((1.0 * x) + 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 <= 2.0)
		tmp = fma(Float64(-0.5 - y), log(y), y);
	elseif (t_0 <= 340.0)
		tmp = Float64(-fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(Float64(1.0 * x) + y) - z);
	end
	return 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, 2.0], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$0, 340.0], (-N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), N[(N[(N[(1.0 * x), $MachinePrecision] + y), $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) < 2

   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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 61.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 58.0%

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

   if 2 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 340

   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 99.0

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

   5. Applied rewrites 99.0%

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

   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 99.0%

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

   if 340 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \left(1 \cdot x + y\right) - z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+47}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 340:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -5e+47)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 340.0) (- (fma (log y) 0.5 z)) (- (+ (* 1.0 x) 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 <= -5e+47) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 340.0) {
		tmp = -fma(log(y), 0.5, z);
	} else {
		tmp = ((1.0 * x) + 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 <= -5e+47)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 340.0)
		tmp = Float64(-fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(Float64(1.0 * x) + y) - z);
	end
	return 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, -5e+47], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 340.0], (-N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), N[(N[(N[(1.0 * x), $MachinePrecision] + y), $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) < -5.00000000000000022e47

   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 60.8

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

   5. Applied rewrites 60.8%

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

   if -5.00000000000000022e47 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 340

   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 93.9

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

   5. Applied rewrites 93.9%

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

   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 85.8%

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

   if 340 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   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. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 100.0

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 99.2%

       \[\leadsto \left(1 \cdot x + y\right) - z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -230 \lor \neg \left(x \leq 0.00095\right):\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -230.0) (not (<= x 0.00095)))
   (- (+ (* 1.0 x) y) z)
   (- (fma (log y) 0.5 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -230.0) || !(x <= 0.00095)) {
		tmp = ((1.0 * x) + y) - z;
	} else {
		tmp = -fma(log(y), 0.5, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -230.0) || !(x <= 0.00095))
		tmp = Float64(Float64(Float64(1.0 * x) + y) - z);
	else
		tmp = Float64(-fma(log(y), 0.5, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -230.0], N[Not[LessEqual[x, 0.00095]], $MachinePrecision]], N[(N[(N[(1.0 * x), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -230 or 9.49999999999999998e-4 < 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

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. flip3-+ N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

      5. clear-num N/A

         \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      6. un-div-inv N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

      8. clear-num N/A

         \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

      9. flip3-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. neg-mul-1 N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \left(1 \cdot x + y\right) - z \]
   9. Step-by-step derivation

   10. Applied rewrites 78.1%

       \[\leadsto \left(1 \cdot x + y\right) - z \]

   if -230 < x < 9.49999999999999998e-4

   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 99.2

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

   5. Applied rewrites 99.2%

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

   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 62.2%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -230 \lor \neg \left(x \leq 0.00095\right):\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3) (- (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 <= 3.3) {
		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 <= 3.3)
		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, 3.3], 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 < 3.2999999999999998

   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 98.8

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

   5. Applied rewrites 98.8%

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

   if 3.2999999999999998 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \left(\left(x - \color{blue}{-1 \cdot \left(y \cdot \log \left(\frac{1}{y}\right)\right)}\right) + y\right) - z \]
   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 9: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 135000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 135000000000.0)
   (- (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 (y <= 135000000000.0) {
		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 (y <= 135000000000.0)
		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[LessEqual[y, 135000000000.0], 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 y < 1.35e11

   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 98.2

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

   5. Applied rewrites 98.2%

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

   if 1.35e11 < 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 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 10: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+132) (- (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 <= 2.8e+132) {
		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 <= 2.8e+132)
		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, 2.8e+132], 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 < 2.7999999999999999e132

   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 88.2

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

   5. Applied rewrites 88.2%

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

   if 2.7999999999999999e132 < y

   1. Initial program 99.5%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 93.1

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

   5. Applied rewrites 93.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 83.0%

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

Alternative 11: 56.6% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(1.0 * 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

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

   2. *-commutative N/A

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

   3. lift-+.f64 N/A

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

   4. flip3-+ N/A

      \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}\right) + y\right) - z \]

   5. clear-num N/A

      \[\leadsto \left(\left(x - \log y \cdot \color{blue}{\frac{1}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

   6. un-div-inv N/A

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

   7. lower-/.f64 N/A

      \[\leadsto \left(\left(x - \color{blue}{\frac{\log y}{\frac{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}{{y}^{3} + {\frac{1}{2}}^{3}}}}\right) + y\right) - z \]

   8. clear-num N/A

      \[\leadsto \left(\left(x - \frac{\log y}{\color{blue}{\frac{1}{\frac{{y}^{3} + {\frac{1}{2}}^{3}}{y \cdot y + \left(\frac{1}{2} \cdot \frac{1}{2} - y \cdot \frac{1}{2}\right)}}}}\right) + y\right) - z \]

   9. flip3-+ N/A

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

   10. lift-+.f64 N/A

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

   11. lower-/.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. +-commutative N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. neg-mul-1 N/A

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

   8. lower-*.f64 N/A

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

7. Applied rewrites 90.7%

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

8. Taylor expanded in x around inf

   \[\leadsto \left(1 \cdot x + y\right) - z \]
9. Step-by-step derivation

10. Applied rewrites 56.9%

    \[\leadsto \left(1 \cdot x + y\right) - z \]
11. Add Preprocessing

Alternative 12: 30.1% 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 30.5

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

5. Applied rewrites 30.5%

   \[\leadsto \color{blue}{-z} \]
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 2024315 
(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))