Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 9.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. distribute-neg-in N/A

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

   14. unsub-neg N/A

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

   15. lower--.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-+.f64 N/A

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

   18. lower--.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 75.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x\right) \cdot -1 + \left(-z\right)\\ t_1 := y - \left(\left(0.5 + y\right) \cdot \log y - x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq -10000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* (- x) -1.0) (- z))) (t_1 (- y (- (* (+ 0.5 y) (log y)) x))))
   (if (<= t_1 -5e+139)
     (* (- 1.0 (log y)) y)
     (if (<= t_1 -10000.0)
       t_0
       (if (<= t_1 500.0) (- (* (log y) -0.5) z) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-x * -1.0) + -z;
	double t_1 = y - (((0.5 + y) * log(y)) - x);
	double tmp;
	if (t_1 <= -5e+139) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_1 <= -10000.0) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = (log(y) * -0.5) - z;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (-x * (-1.0d0)) + -z
    t_1 = y - (((0.5d0 + y) * log(y)) - x)
    if (t_1 <= (-5d+139)) then
        tmp = (1.0d0 - log(y)) * y
    else if (t_1 <= (-10000.0d0)) then
        tmp = t_0
    else if (t_1 <= 500.0d0) then
        tmp = (log(y) * (-0.5d0)) - z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-x * -1.0) + -z;
	double t_1 = y - (((0.5 + y) * Math.log(y)) - x);
	double tmp;
	if (t_1 <= -5e+139) {
		tmp = (1.0 - Math.log(y)) * y;
	} else if (t_1 <= -10000.0) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = (Math.log(y) * -0.5) - z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-x * -1.0) + -z
	t_1 = y - (((0.5 + y) * math.log(y)) - x)
	tmp = 0
	if t_1 <= -5e+139:
		tmp = (1.0 - math.log(y)) * y
	elif t_1 <= -10000.0:
		tmp = t_0
	elif t_1 <= 500.0:
		tmp = (math.log(y) * -0.5) - z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(-x) * -1.0) + Float64(-z))
	t_1 = Float64(y - Float64(Float64(Float64(0.5 + y) * log(y)) - x))
	tmp = 0.0
	if (t_1 <= -5e+139)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_1 <= -10000.0)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = Float64(Float64(log(y) * -0.5) - z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-x * -1.0) + -z;
	t_1 = y - (((0.5 + y) * log(y)) - x);
	tmp = 0.0;
	if (t_1 <= -5e+139)
		tmp = (1.0 - log(y)) * y;
	elseif (t_1 <= -10000.0)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = (log(y) * -0.5) - z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-x) * -1.0), $MachinePrecision] + (-z)), $MachinePrecision]}, Block[{t$95$1 = N[(y - N[(N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+139], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -10000.0], t$95$0, If[LessEqual[t$95$1, 500.0], N[(N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5.0000000000000003e139

   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

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 58.0

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

   5. Applied rewrites 58.0%

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

   if -5.0000000000000003e139 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1e4 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. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 81.2

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

   10. Applied rewrites 81.2%

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

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 81.7

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

   13. Applied rewrites 81.7%

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

   if -1e4 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 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 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-sub-sign-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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y - \left(\left(0.5 + y\right) \cdot \log y - x\right) \leq -5 \cdot 10^{+139}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;y - \left(\left(0.5 + y\right) \cdot \log y - x\right) \leq -10000:\\ \;\;\;\;\left(-x\right) \cdot -1 + \left(-z\right)\\ \mathbf{elif}\;y - \left(\left(0.5 + y\right) \cdot \log y - x\right) \leq 500:\\ \;\;\;\;\log y \cdot -0.5 - z\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot -1 + \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (- -0.5 y) (log y) (- x z))))
   (if (<= z -1.9e+24)
     t_0
     (if (<= z 4.4e+33) (fma (- -0.5 y) (log y) (+ x y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-0.5 - y), log(y), (x - z));
	double tmp;
	if (z <= -1.9e+24) {
		tmp = t_0;
	} else if (z <= 4.4e+33) {
		tmp = fma((-0.5 - y), log(y), (x + y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-0.5 - y), log(y), Float64(x - z))
	tmp = 0.0
	if (z <= -1.9e+24)
		tmp = t_0;
	elseif (z <= 4.4e+33)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+24], t$95$0, If[LessEqual[z, 4.4e+33], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000008e24 or 4.39999999999999988e33 < z

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 86.9

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

   7. Applied rewrites 86.9%

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

   if -1.90000000000000008e24 < z < 4.39999999999999988e33

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 99.5

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

   7. Applied rewrites 99.5%

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

Alternative 4: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\left(-x\right) \cdot -1 + \left(-z\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.3e+24)
   (+ (* (- x) -1.0) (- z))
   (if (<= z 3.1e+33)
     (fma (- -0.5 y) (log y) (+ x y))
     (- (fma (- -0.5 y) (log y) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.3e+24) {
		tmp = (-x * -1.0) + -z;
	} else if (z <= 3.1e+33) {
		tmp = fma((-0.5 - y), log(y), (x + y));
	} else {
		tmp = fma((-0.5 - y), log(y), y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.3e+24)
		tmp = Float64(Float64(Float64(-x) * -1.0) + Float64(-z));
	elseif (z <= 3.1e+33)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x + y));
	else
		tmp = Float64(fma(Float64(-0.5 - y), log(y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.3e+24], N[(N[((-x) * -1.0), $MachinePrecision] + (-z)), $MachinePrecision], If[LessEqual[z, 3.1e+33], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2999999999999999e24

   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 inf

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

      1. distribute-rgt-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 84.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 84.6

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

   10. Applied rewrites 84.6%

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

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 84.8

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

   13. Applied rewrites 84.8%

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

   if -2.2999999999999999e24 < z < 3.1e33

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 3.1e33 < 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. 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 83.4

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

   8. Applied rewrites 83.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+24}:\\ \;\;\;\;\left(-x\right) \cdot -1 + \left(-z\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.43:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \log y \cdot y\right) - \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.43)
   (fma (- -0.5 y) (log y) (- x z))
   (- (- x (* (log y) y)) (- z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.43) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = (x - (log(y) * y)) - (z - y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.43)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x - z));
	else
		tmp = Float64(Float64(x - Float64(log(y) * y)) - Float64(z - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.43], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - N[(z - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.429999999999999993

   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

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 98.6

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

   7. Applied rewrites 98.6%

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

   if 0.429999999999999993 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg 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 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.43:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - \log y \cdot y\right) - \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \log y, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.2e+39) (- (fma -0.5 (log y) x) z) (fma (- y) (log y) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9.2e+39) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(-y, log(y), (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9.2e+39)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(Float64(-y), log(y), Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9.2e+39], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[((-y) * N[Log[y], $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.20000000000000047e39

   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-sub-sign-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 96.3

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

   5. Applied rewrites 96.3%

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

   if 9.20000000000000047e39 < 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. 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. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-+.f64 N/A

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

      18. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 82.4

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.4

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

   10. Applied rewrites 82.4%

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

Alternative 7: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.6 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.6e+136) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.6e+136) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.6e+136)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.6e+136], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.60000000000000006e136

   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-sub-sign-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 89.7

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

   5. Applied rewrites 89.7%

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

   if 3.60000000000000006e136 < 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 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 73.2

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

   5. Applied rewrites 73.2%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
2. Add Preprocessing
3. 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. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 9: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+136}:\\ \;\;\;\;\left(-x\right) \cdot -1 - \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e+136) (- (* (- x) -1.0) (- z y)) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+136) {
		tmp = (-x * -1.0) - (z - y);
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d+136) then
        tmp = (-x * (-1.0d0)) - (z - y)
    else
        tmp = (1.0d0 - log(y)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+136) {
		tmp = (-x * -1.0) - (z - y);
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e+136:
		tmp = (-x * -1.0) - (z - y)
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e+136)
		tmp = Float64(Float64(Float64(-x) * -1.0) - Float64(z - y));
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e+136)
		tmp = (-x * -1.0) - (z - y);
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e+136], N[(N[((-x) * -1.0), $MachinePrecision] - N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.50000000000000001e136

   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 inf

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

      1. distribute-rgt-in N/A

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

      2. metadata-eval N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-neg N/A

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

      10. lower-*.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. sub-neg N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. *-commutative N/A

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

      17. metadata-eval N/A

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

      18. lower-fma.f64 N/A

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

   5. Applied rewrites 95.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 3.50000000000000001e136 < 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 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 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+136}:\\ \;\;\;\;\left(-x\right) \cdot -1 - \left(z - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.9% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[((-x) * -1.0), $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. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. distribute-rgt-in N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

   10. lower-*.f64 N/A

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

   11. mul-1-neg N/A

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

   12. lower-neg.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. associate-/l* N/A

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

   16. *-commutative N/A

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

   17. metadata-eval N/A

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

   18. lower-fma.f64 N/A

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

5. Applied rewrites 88.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 60.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower--.f64 60.3

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

10. Applied rewrites 60.3%

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

11. Taylor expanded in y around 0

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

    1. mul-1-neg N/A

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

    2. lower-neg.f64 60.8

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

13. Applied rewrites 60.8%

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

14. Final simplification 60.8%

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

Alternative 11: 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 27.5

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

5. Applied rewrites 27.5%

   \[\leadsto \color{blue}{-z} \]
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 2024298 
(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))