Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] + x), $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. associate-+l+ N/A

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

   5. lower-+.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. distribute-neg-in N/A

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

   12. unsub-neg N/A

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

   13. lower--.f64 N/A

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

   14. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 \cdot x + y\right) - z\\ t_1 := \left(\left(x - \left(0.5 + y\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_1 \leq -40000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (* 1.0 x) y) z))
        (t_1 (- (+ (- x (* (+ 0.5 y) (log y))) y) z)))
   (if (<= t_1 -40000000000.0) t_0 (if (<= t_1 500.0) (* (log y) -0.5) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 * x) + y) - z;
	double t_1 = ((x - ((0.5 + y) * log(y))) + y) - z;
	double tmp;
	if (t_1 <= -40000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = log(y) * -0.5;
	} 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 = ((1.0d0 * x) + y) - z
    t_1 = ((x - ((0.5d0 + y) * log(y))) + y) - z
    if (t_1 <= (-40000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 500.0d0) then
        tmp = log(y) * (-0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 * x) + y) - z;
	double t_1 = ((x - ((0.5 + y) * Math.log(y))) + y) - z;
	double tmp;
	if (t_1 <= -40000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = Math.log(y) * -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 * x) + y) - z
	t_1 = ((x - ((0.5 + y) * math.log(y))) + y) - z
	tmp = 0
	if t_1 <= -40000000000.0:
		tmp = t_0
	elif t_1 <= 500.0:
		tmp = math.log(y) * -0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 * x) + y) - z)
	t_1 = Float64(Float64(Float64(x - Float64(Float64(0.5 + y) * log(y))) + y) - z)
	tmp = 0.0
	if (t_1 <= -40000000000.0)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = Float64(log(y) * -0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 * x) + y) - z;
	t_1 = ((x - ((0.5 + y) * log(y))) + y) - z;
	tmp = 0.0;
	if (t_1 <= -40000000000.0)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = log(y) * -0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 * x), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -40000000000.0], t$95$0, If[LessEqual[t$95$1, 500.0], N[(N[Log[y], $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]

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) < -4e10 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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 87.5

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 65.7%

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

   if -4e10 < (-.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. 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 0

      \[\leadsto \color{blue}{y - \left(z + \log y \cdot \left(\frac{1}{2} + y\right)\right)} \]
   6. 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 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(x - \left(0.5 + y\right) \cdot \log y\right) + y\right) - z \leq -40000000000:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{elif}\;\left(\left(x - \left(0.5 + y\right) \cdot \log y\right) + y\right) - z \leq 500:\\ \;\;\;\;\log y \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.000225:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, \log y, y\right) + x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.000225)
   (- (fma -0.5 (log y) x) z)
   (- (+ (fma (- y) (log y) y) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.000225) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (fma(-y, log(y), y) + x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.000225)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(fma(Float64(-y), log(y), y) + x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.000225], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[((-y) * N[Log[y], $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2499999999999999e-4

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 2.2499999999999999e-4 < y

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      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. associate-+l+ N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.000225:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-y, \log y, y\right) + x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 \cdot x + y\right) - z\\ \mathbf{if}\;z \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 320:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (* 1.0 x) y) z)))
   (if (<= z -3800000000.0) t_0 (if (<= z 320.0) (fma -0.5 (log y) x) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 * x) + y) - z;
	double tmp;
	if (z <= -3800000000.0) {
		tmp = t_0;
	} else if (z <= 320.0) {
		tmp = fma(-0.5, log(y), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 * x) + y) - z)
	tmp = 0.0
	if (z <= -3800000000.0)
		tmp = t_0;
	elseif (z <= 320.0)
		tmp = fma(-0.5, log(y), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 * x), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -3800000000.0], t$95$0, If[LessEqual[z, 320.0], N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8e9 or 320 < 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

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

      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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 90.9

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

   7. Applied rewrites 90.9%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 78.9%

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

   if -3.8e9 < z < 320

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      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-rgt-neg-in N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. div-inv N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + -1 \cdot \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(x + y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(x + y\right) - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(y + x\right)} - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(y + x\right) - \color{blue}{\frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      8. *-commutative N/A

         \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} - \frac{1}{4}\right) \cdot \log y}}{y - \frac{1}{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} - \frac{1}{4}\right) \cdot \log y}}{y - \frac{1}{2}} \]

      10. sub-neg N/A

          \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot \log y}{y - \frac{1}{2}} \]

      11. unpow2 N/A

          \[\leadsto \left(y + x\right) - \frac{\left(\color{blue}{y \cdot y} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot \log y}{y - \frac{1}{2}} \]

      12. metadata-eval N/A

          \[\leadsto \left(y + x\right) - \frac{\left(y \cdot y + \color{blue}{\frac{-1}{4}}\right) \cdot \log y}{y - \frac{1}{2}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(y + x\right) - \frac{\color{blue}{\mathsf{fma}\left(y, y, \frac{-1}{4}\right)} \cdot \log y}{y - \frac{1}{2}} \]

      14. lower-log.f64 N/A

          \[\leadsto \left(y + x\right) - \frac{\mathsf{fma}\left(y, y, \frac{-1}{4}\right) \cdot \color{blue}{\log y}}{y - \frac{1}{2}} \]

      15. lower--.f64 75.9

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

   7. Applied rewrites 75.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto x - \color{blue}{\frac{1}{2} \cdot \log y} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.8%

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

Alternative 5: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e+50)
   (- (fma -0.5 (log y) x) z)
   (+ (fma (log y) (- -0.5 y) y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+50) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(log(y), (-0.5 - y), y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e+50)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(fma(log(y), Float64(-0.5 - y), y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e+50], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e50

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. 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 2.1e50 < 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-rgt-neg-in N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. div-inv N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + -1 \cdot \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(x + y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(x + y\right) - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(y + x\right)} - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(y + x\right) - \color{blue}{\frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      8. *-commutative N/A

         \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} - \frac{1}{4}\right) \cdot \log y}}{y - \frac{1}{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} - \frac{1}{4}\right) \cdot \log y}}{y - \frac{1}{2}} \]

      10. sub-neg N/A

          \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot \log y}{y - \frac{1}{2}} \]

      11. unpow2 N/A

          \[\leadsto \left(y + x\right) - \frac{\left(\color{blue}{y \cdot y} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot \log y}{y - \frac{1}{2}} \]

      12. metadata-eval N/A

          \[\leadsto \left(y + x\right) - \frac{\left(y \cdot y + \color{blue}{\frac{-1}{4}}\right) \cdot \log y}{y - \frac{1}{2}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(y + x\right) - \frac{\color{blue}{\mathsf{fma}\left(y, y, \frac{-1}{4}\right)} \cdot \log y}{y - \frac{1}{2}} \]

      14. lower-log.f64 N/A

          \[\leadsto \left(y + x\right) - \frac{\mathsf{fma}\left(y, y, \frac{-1}{4}\right) \cdot \color{blue}{\log y}}{y - \frac{1}{2}} \]

      15. lower--.f64 38.6

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

   7. Applied rewrites 38.6%

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

   9. Applied rewrites 86.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.1e+50) (- (fma -0.5 (log y) x) z) (+ (- y (* (log y) y)) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+50) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (y - (log(y) * y)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e+50)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(y - Float64(log(y) * y)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e+50], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e50

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. 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 2.1e50 < 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-rgt-neg-in N/A

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

      10. lift-+.f64 N/A

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

      11. flip-+ N/A

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

      12. div-inv N/A

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

      13. associate-*l* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 48.4%

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

   5. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + y\right) + -1 \cdot \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \left(x + y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{\left(x + y\right) - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x + y\right) - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(y + x\right)} - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(y + x\right)} - \frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(y + x\right) - \color{blue}{\frac{\log y \cdot \left({y}^{2} - \frac{1}{4}\right)}{y - \frac{1}{2}}} \]

      8. *-commutative N/A

         \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} - \frac{1}{4}\right) \cdot \log y}}{y - \frac{1}{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} - \frac{1}{4}\right) \cdot \log y}}{y - \frac{1}{2}} \]

      10. sub-neg N/A

          \[\leadsto \left(y + x\right) - \frac{\color{blue}{\left({y}^{2} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \cdot \log y}{y - \frac{1}{2}} \]

      11. unpow2 N/A

          \[\leadsto \left(y + x\right) - \frac{\left(\color{blue}{y \cdot y} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \cdot \log y}{y - \frac{1}{2}} \]

      12. metadata-eval N/A

          \[\leadsto \left(y + x\right) - \frac{\left(y \cdot y + \color{blue}{\frac{-1}{4}}\right) \cdot \log y}{y - \frac{1}{2}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(y + x\right) - \frac{\color{blue}{\mathsf{fma}\left(y, y, \frac{-1}{4}\right)} \cdot \log y}{y - \frac{1}{2}} \]

      14. lower-log.f64 N/A

          \[\leadsto \left(y + x\right) - \frac{\mathsf{fma}\left(y, y, \frac{-1}{4}\right) \cdot \color{blue}{\log y}}{y - \frac{1}{2}} \]

      15. lower--.f64 38.6

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

   7. Applied rewrites 38.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 86.5%

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

   12. Applied rewrites 86.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y \cdot y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+96}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.2e+96) (- (fma -0.5 (log y) x) z) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.2e+96) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.2e+96)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.2e+96], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.20000000000000026e96

   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

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-log.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 7.20000000000000026e96 < 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

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

      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 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y - \left(z + \log y \cdot \left(\frac{1}{2} + y\right)\right)} \]
   6. 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 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 73.7%

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

Alternative 8: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+67}:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.05e+67) (- (+ (* 1.0 x) y) z) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e+67) {
		tmp = ((1.0 * x) + y) - z;
	} else {
		tmp = y - (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 <= 1.05d+67) then
        tmp = ((1.0d0 * x) + y) - z
    else
        tmp = y - (log(y) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e+67) {
		tmp = ((1.0 * x) + y) - z;
	} else {
		tmp = y - (Math.log(y) * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.05e+67:
		tmp = ((1.0 * x) + y) - z
	else:
		tmp = y - (math.log(y) * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.05e+67)
		tmp = Float64(Float64(Float64(1.0 * x) + y) - z);
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.05e+67)
		tmp = ((1.0 * x) + y) - z;
	else
		tmp = y - (log(y) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.05e+67], N[(N[(N[(1.0 * x), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e67

   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.9%

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

   if 1.0500000000000001e67 < 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

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

      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 99.6

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

      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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y - \left(z + \log y \cdot \left(\frac{1}{2} + y\right)\right)} \]
   6. 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 83.0

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

   7. Applied rewrites 83.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 70.5%

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

Alternative 9: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+67}:\\ \;\;\;\;\left(1 \cdot x + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.05e+67) (- (+ (* 1.0 x) y) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.05e+67) {
		tmp = ((1.0 * x) + y) - z;
	} 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 <= 1.05d+67) then
        tmp = ((1.0d0 * x) + y) - z
    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 <= 1.05e+67) {
		tmp = ((1.0 * x) + y) - z;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.05e+67:
		tmp = ((1.0 * x) + y) - z
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.05e+67)
		tmp = Float64(Float64(Float64(1.0 * x) + y) - z);
	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 <= 1.05e+67)
		tmp = ((1.0 * x) + y) - z;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.05e+67], N[(N[(N[(1.0 * x), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e67

   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. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lower-/.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-log.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 79.9%

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

   if 1.0500000000000001e67 < 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

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 70.5

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

   5. Applied rewrites 70.5%

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

Alternative 10: 56.8% 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. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. associate-/l* N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-fma.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. lower-/.f64 N/A

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

   11. distribute-neg-in N/A

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

   12. metadata-eval N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-log.f64 88.4

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

7. Applied rewrites 88.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 59.3%

    \[\leadsto \left(1 \cdot x + y\right) - z \]
11. 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 29.8

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

5. Applied rewrites 29.8%

   \[\leadsto \color{blue}{-z} \]
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 2024279 
(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))