Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

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

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((x - (log(y) * (0.5 + 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 - (log(y) * (0.5d0 + y))) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 75.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{\frac{1}{x}} + y\right) - z\\ t_1 := \left(x - \log y \cdot \left(0.5 + y\right)\right) + y\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 500:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (/ 1.0 (/ 1.0 x)) y) z))
        (t_1 (+ (- x (* (log y) (+ 0.5 y))) y)))
   (if (<= t_1 -2e+146)
     (* (- 1.0 (log y)) y)
     (if (<= t_1 -2e+19)
       t_0
       (if (<= t_1 500.0) (fma -0.5 (log y) (- z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 / (1.0 / x)) + y) - z;
	double t_1 = (x - (log(y) * (0.5 + y))) + y;
	double tmp;
	if (t_1 <= -2e+146) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_1 <= -2e+19) {
		tmp = t_0;
	} else if (t_1 <= 500.0) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 / x)) + y) - z)
	t_1 = Float64(Float64(x - Float64(log(y) * Float64(0.5 + y))) + y)
	tmp = 0.0
	if (t_1 <= -2e+146)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_1 <= -2e+19)
		tmp = t_0;
	elseif (t_1 <= 500.0)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - N[(N[Log[y], $MachinePrecision] * N[(0.5 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+146], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$1, -2e+19], t$95$0, If[LessEqual[t$95$1, 500.0], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.99999999999999987e146

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -1.99999999999999987e146 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e19 or 500 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.8

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if -2e19 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.4%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \log y \cdot \left(0.5 + y\right)\right) + y \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;\left(x - \log y \cdot \left(0.5 + y\right)\right) + y \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \mathbf{elif}\;\left(x - \log y \cdot \left(0.5 + y\right)\right) + y \leq 500:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(x - \log y \cdot \left(0.5 + y\right)\right) + y \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (+ (- x (* (log y) (+ 0.5 y))) y) -2e+14)
   (- (+ (- x (* (log y) y)) y) z)
   (- (fma -0.5 (log y) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x - (log(y) * (0.5 + y))) + y) <= -2e+14) {
		tmp = ((x - (log(y) * y)) + y) - z;
	} else {
		tmp = fma(-0.5, log(y), x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(x - Float64(log(y) * Float64(0.5 + y))) + y) <= -2e+14)
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	else
		tmp = Float64(fma(-0.5, log(y), x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(x - N[(N[Log[y], $MachinePrecision] * N[(0.5 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], -2e+14], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e14

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -2e14 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - \log y \cdot \left(0.5 + y\right)\right) + y \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;x \leq 102000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e+60)
   (- (fma -0.5 (log y) x) z)
   (if (<= x 102000000.0)
     (- (fma (- -0.5 y) (log y) y) z)
     (fma (log y) (- -0.5 y) (+ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+60) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (x <= 102000000.0) {
		tmp = fma((-0.5 - y), log(y), y) - z;
	} else {
		tmp = fma(log(y), (-0.5 - y), (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e+60)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (x <= 102000000.0)
		tmp = Float64(fma(Float64(-0.5 - y), log(y), y) - z);
	else
		tmp = fma(log(y), Float64(-0.5 - y), Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e+60], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 102000000.0], N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e60

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.55e60 < x < 1.02e8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-log.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.02e8 < x

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. associate--l+ N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. lower--.f64 N/A

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

      21. lower-+.f64 94.6

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

   7. Applied rewrites 94.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;x \leq 102000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;x \leq 102000000:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.55e+60)
   (- (fma -0.5 (log y) x) z)
   (if (<= x 102000000.0)
     (- y (fma (+ 0.5 y) (log y) z))
     (fma (log y) (- -0.5 y) (+ y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.55e+60) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (x <= 102000000.0) {
		tmp = y - fma((0.5 + y), log(y), z);
	} else {
		tmp = fma(log(y), (-0.5 - y), (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.55e+60)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (x <= 102000000.0)
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	else
		tmp = fma(log(y), Float64(-0.5 - y), Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.55e+60], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 102000000.0], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e60

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -1.55e60 < x < 1.02e8

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.02e8 < x

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. associate--l+ N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-rgt-in N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. associate-*r* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. mul-1-neg N/A

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

      19. unsub-neg N/A

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

      20. lower--.f64 N/A

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

      21. lower-+.f64 94.6

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

   7. Applied rewrites 94.6%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;x \leq 102000000:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{\frac{1}{x}} + y\right) - z\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (/ 1.0 (/ 1.0 x)) y) z)))
   (if (<= x -1.5e+19) t_0 (if (<= x 1.55e+16) (fma -0.5 (log y) (- z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 / (1.0 / x)) + y) - z;
	double tmp;
	if (x <= -1.5e+19) {
		tmp = t_0;
	} else if (x <= 1.55e+16) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 / Float64(1.0 / x)) + y) - z)
	tmp = 0.0
	if (x <= -1.5e+19)
		tmp = t_0;
	elseif (x <= 1.55e+16)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -1.5e+19], t$95$0, If[LessEqual[x, 1.55e+16], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e19 or 1.55e16 < x

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

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

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

      14. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -1.5e19 < x < 1.55e16

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 54.4%

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

Alternative 7: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.35e+57)
   (- (fma -0.5 (log y) x) z)
   (- y (fma (+ 0.5 y) (log y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.35e+57) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - fma((0.5 + y), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.35e+57)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.35e+57], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3499999999999999e57

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 1.3499999999999999e57 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 85.3

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

   5. Applied rewrites 85.3%

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

Alternative 8: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+115}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.4e+115) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+115) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e+115)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.4e+115], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4e115

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 6.4e115 < 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 82.7

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

   5. Applied rewrites 82.7%

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

Alternative 9: 47.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+79}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.8e+79) (- z) (if (<= z 2.1e+65) (/ 1.0 (/ 1.0 x)) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e+79) {
		tmp = -z;
	} else if (z <= 2.1e+65) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-3.8d+79)) then
        tmp = -z
    else if (z <= 2.1d+65) then
        tmp = 1.0d0 / (1.0d0 / x)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.8e+79) {
		tmp = -z;
	} else if (z <= 2.1e+65) {
		tmp = 1.0 / (1.0 / x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.8e+79:
		tmp = -z
	elif z <= 2.1e+65:
		tmp = 1.0 / (1.0 / x)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.8e+79)
		tmp = Float64(-z);
	elseif (z <= 2.1e+65)
		tmp = Float64(1.0 / Float64(1.0 / x));
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.8e+79)
		tmp = -z;
	elseif (z <= 2.1e+65)
		tmp = 1.0 / (1.0 / x);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.8e+79], (-z), If[LessEqual[z, 2.1e+65], N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000002e79 or 2.09999999999999991e65 < 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. 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 62.3

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

   5. Applied rewrites 62.3%

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

   if -3.8000000000000002e79 < z < 2.09999999999999991e65

   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. flip-- N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip-- N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 99.7

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

      9. lift--.f64 N/A

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

      10. lift-+.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 38.1

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 38.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 99.7

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 51.1

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

7. Applied rewrites 51.1%

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

Alternative 11: 29.8% 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 22.7

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

5. Applied rewrites 22.7%

   \[\leadsto \color{blue}{-z} \]
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 2024331 
(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))