Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 8.1s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 73.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+58}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -5e+58)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 500.0) (- (* -0.5 (log y)) z) (fma (/ (- z) x) x x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double tmp;
	if (t_0 <= -5e+58) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 500.0) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = fma((-z / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	tmp = 0.0
	if (t_0 <= -5e+58)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = fma(Float64(Float64(-z) / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+58], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4.99999999999999986e58

   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 54.9

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

   5. Applied rewrites 54.9%

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

   if -4.99999999999999986e58 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.7%

      \[\leadsto -0.5 \cdot \log y - z \]

   if 500 < (+.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 x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{z}{x}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

Alternative 3: 88.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{-z}{x} \cdot x + x\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+104}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.5e+129)
   (+ (* (/ (- z) x) x) x)
   (if (<= x 8e+104)
     (- y (fma (+ 0.5 y) (log y) z))
     (- (fma -0.5 (log y) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.5e+129) {
		tmp = ((-z / x) * x) + x;
	} else if (x <= 8e+104) {
		tmp = y - fma((0.5 + y), log(y), z);
	} else {
		tmp = fma(-0.5, log(y), x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.5e+129)
		tmp = Float64(Float64(Float64(Float64(-z) / x) * x) + x);
	elseif (x <= 8e+104)
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	else
		tmp = Float64(fma(-0.5, log(y), x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.5e+129], N[(N[(N[((-z) / x), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 8e+104], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.5000000000000001e129

   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 inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{z}{x}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 90.9%

      \[\leadsto \mathsf{fma}\left(\frac{-z}{x}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 90.9%

       \[\leadsto \frac{-z}{x} \cdot x + \color{blue}{x} \]

   if -4.5000000000000001e129 < x < 8e104

   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 93.3

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

   5. Applied rewrites 93.3%

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

   if 8e104 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 85.3

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

   5. Applied rewrites 85.3%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e-9) (- (fma -0.5 (log y) x) z) (- (+ (- x (* (log y) y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e-9) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((x - (log(y) * y)) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e-9)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e-9], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999976e-9

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 4.49999999999999976e-9 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-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 98.1

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

   5. Applied rewrites 98.1%

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

Alternative 5: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 38:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, x + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 38.0) (- (fma -0.5 (log y) x) z) (fma (log y) (- -0.5 y) (+ x y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 38.0) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(log(y), (-0.5 - y), (x + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 38.0)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(log(y), Float64(-0.5 - y), Float64(x + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 38.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 38

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 38 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 75.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - \mathsf{fma}\left(0.5 + y, \log y, z\right)}{x}, x, x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 75.6%

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

   8. Taylor expanded in z around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. log-rec N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. log-rec N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. metadata-eval N/A

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

      10. log-rec N/A

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

      11. distribute-rgt-neg-out N/A

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

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

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-out-- N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower--.f64 N/A

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

      19. lower-+.f64 84.5

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

   10. Applied rewrites 84.5%

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

Alternative 6: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.4e+56) (- (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 <= 4.4e+56) {
		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 <= 4.4e+56)
		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, 4.4e+56], 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 < 4.40000000000000032e56

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if 4.40000000000000032e56 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 68.8

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

   5. Applied rewrites 68.8%

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

Alternative 7: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{-z}{x} \cdot x + x\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.2e+56) (+ (* (/ (- z) x) x) x) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.2e+56) {
		tmp = ((-z / x) * x) + x;
	} 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 <= 4.2d+56) then
        tmp = ((-z / x) * x) + x
    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 <= 4.2e+56) {
		tmp = ((-z / x) * x) + x;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.2e+56:
		tmp = ((-z / x) * x) + x
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.2e+56)
		tmp = Float64(Float64(Float64(Float64(-z) / x) * x) + x);
	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 <= 4.2e+56)
		tmp = ((-z / x) * x) + x;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.2e+56], N[(N[(N[((-z) / x), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.20000000000000034e56

   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 inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{z}{x}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 59.2%

      \[\leadsto \mathsf{fma}\left(\frac{-z}{x}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 59.2%

       \[\leadsto \frac{-z}{x} \cdot x + \color{blue}{x} \]

   if 4.20000000000000034e56 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(1 - -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   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 68.8

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

   5. Applied rewrites 68.8%

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

Alternative 8: 57.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-69} \lor \neg \left(x \leq 1.3 \cdot 10^{-50}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4e-69) (not (<= x 1.3e-50))) (fma (/ (- z) x) x x) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4e-69) || !(x <= 1.3e-50)) {
		tmp = fma((-z / x), x, x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4e-69) || !(x <= 1.3e-50))
		tmp = fma(Float64(Float64(-z) / x), x, x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4e-69], N[Not[LessEqual[x, 1.3e-50]], $MachinePrecision]], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999999e-69 or 1.3000000000000001e-50 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{z}{x}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 68.7%

      \[\leadsto \mathsf{fma}\left(\frac{-z}{x}, x, x\right) \]

   if -3.9999999999999999e-69 < x < 1.3000000000000001e-50

   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 36.4

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

   5. Applied rewrites 36.4%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-69} \lor \neg \left(x \leq 1.3 \cdot 10^{-50}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-z}{x}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-69}:\\ \;\;\;\;t\_0 \cdot x + x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- z) x)))
   (if (<= x -4e-69) (+ (* t_0 x) x) (if (<= x 1.3e-50) (- z) (fma t_0 x x)))))

C

double code(double x, double y, double z) {
	double t_0 = -z / x;
	double tmp;
	if (x <= -4e-69) {
		tmp = (t_0 * x) + x;
	} else if (x <= 1.3e-50) {
		tmp = -z;
	} else {
		tmp = fma(t_0, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-z) / x)
	tmp = 0.0
	if (x <= -4e-69)
		tmp = Float64(Float64(t_0 * x) + x);
	elseif (x <= 1.3e-50)
		tmp = Float64(-z);
	else
		tmp = fma(t_0, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[((-z) / x), $MachinePrecision]}, If[LessEqual[x, -4e-69], N[(N[(t$95$0 * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 1.3e-50], (-z), N[(t$95$0 * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999999e-69

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{z}{x}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.9%

      \[\leadsto \mathsf{fma}\left(\frac{-z}{x}, x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 64.9%

       \[\leadsto \frac{-z}{x} \cdot x + \color{blue}{x} \]

   if -3.9999999999999999e-69 < x < 1.3000000000000001e-50

   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 36.4

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

   5. Applied rewrites 36.4%

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

   if 1.3000000000000001e-50 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{z}{x}, x, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 74.0%

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

Alternative 10: 30.0% 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 29.3

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

5. Applied rewrites 29.3%

   \[\leadsto \color{blue}{-z} \]
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 2024329 
(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))