Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ t_1 := \frac{-z}{x}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+142}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+19}:\\ \;\;\;\;t\_1 \cdot x + x\\ \mathbf{elif}\;t\_0 \leq 351.5:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)) (t_1 (/ (- z) x)))
   (if (<= t_0 -1e+142)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 -1e+19)
       (+ (* t_1 x) x)
       (if (<= t_0 351.5) (fma -0.5 (log y) (- z)) (fma t_1 x x))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double t_1 = -z / x;
	double tmp;
	if (t_0 <= -1e+142) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= -1e+19) {
		tmp = (t_1 * x) + x;
	} else if (t_0 <= 351.5) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = fma(t_1, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	t_1 = Float64(Float64(-z) / x)
	tmp = 0.0
	if (t_0 <= -1e+142)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= -1e+19)
		tmp = Float64(Float64(t_1 * x) + x);
	elseif (t_0 <= 351.5)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = fma(t_1, 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]}, Block[{t$95$1 = N[((-z) / x), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+142], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, -1e+19], N[(N[(t$95$1 * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$0, 351.5], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(t$95$1 * x + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1.00000000000000005e142

   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 64.8

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

   5. Applied rewrites 64.8%

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

   if -1.00000000000000005e142 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -1e19

   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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 82.9%

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

   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 66.1%

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

   10. Applied rewrites 66.2%

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

   if -1e19 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 351.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 98.7

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

   5. Applied rewrites 98.7%

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

   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 94.5%

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

   if 351.5 < (+.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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 98.0%

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

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-z}{x}\\ \mathbf{if}\;x \leq -215:\\ \;\;\;\;t\_0 \cdot x + x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- z) x)))
   (if (<= x -215.0)
     (+ (* t_0 x) x)
     (if (<= x 2.65e+20) (fma -0.5 (log y) (- z)) (fma t_0 x x)))))

C

double code(double x, double y, double z) {
	double t_0 = -z / x;
	double tmp;
	if (x <= -215.0) {
		tmp = (t_0 * x) + x;
	} else if (x <= 2.65e+20) {
		tmp = fma(-0.5, log(y), -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 <= -215.0)
		tmp = Float64(Float64(t_0 * x) + x);
	elseif (x <= 2.65e+20)
		tmp = fma(-0.5, log(y), 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, -215.0], N[(N[(t$95$0 * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 2.65e+20], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(t$95$0 * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -215

   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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 99.8%

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

   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 77.8%

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

   10. Applied rewrites 77.9%

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

   if -215 < x < 2.65e20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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 61.6%

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

   if 2.65e20 < 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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 99.8%

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

   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 73.2%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e-8) (- (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 <= 9.5e-8) {
		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 <= 9.5e-8)
		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, 9.5e-8], 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 < 9.50000000000000036e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 9.50000000000000036e-8 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 5: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e+137)
   (- (fma -0.5 (log y) x) z)
   (- (fma (- -0.5 y) (log y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.25e+137) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma((-0.5 - y), log(y), y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.25e+137)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(fma(Float64(-0.5 - y), log(y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.25e+137], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25e137

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 1.25e137 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(y - \log y \cdot \left(\frac{1}{2} + y\right)\right)} - z \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

Alternative 6: 87.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.25e+137)
   (- (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 <= 1.25e+137) {
		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 <= 1.25e+137)
		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, 1.25e+137], 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 < 1.25e137

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 1.25e137 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 93.4

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

   5. Applied rewrites 93.4%

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

Alternative 7: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e+137) (- (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 <= 6.8e+137) {
		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 <= 6.8e+137)
		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, 6.8e+137], 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 < 6.79999999999999973e137

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-log.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 6.79999999999999973e137 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 82.5

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

   5. Applied rewrites 82.5%

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

Alternative 8: 57.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -250 \lor \neg \left(x \leq 2.65 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -250.0) (not (<= x 2.65e+20))) (fma (/ (- z) x) x x) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -250.0) || !(x <= 2.65e+20)) {
		tmp = fma((-z / x), x, x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -250.0) || !(x <= 2.65e+20))
		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, -250.0], N[Not[LessEqual[x, 2.65e+20]], $MachinePrecision]], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -250 or 2.65e20 < 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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 99.8%

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

   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 76.5%

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

   if -250 < x < 2.65e20

   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 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 40.7

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

   5. Applied rewrites 40.7%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250 \lor \neg \left(x \leq 2.65 \cdot 10^{+20}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-z}{x}\\ \mathbf{if}\;x \leq -250:\\ \;\;\;\;t\_0 \cdot x + x\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{+20}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- z) x)))
   (if (<= x -250.0)
     (+ (* t_0 x) x)
     (if (<= x 2.65e+20) (- z) (fma t_0 x x)))))

C

double code(double x, double y, double z) {
	double t_0 = -z / x;
	double tmp;
	if (x <= -250.0) {
		tmp = (t_0 * x) + x;
	} else if (x <= 2.65e+20) {
		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 <= -250.0)
		tmp = Float64(Float64(t_0 * x) + x);
	elseif (x <= 2.65e+20)
		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, -250.0], N[(N[(t$95$0 * x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 2.65e+20], (-z), N[(t$95$0 * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -250

   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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 99.8%

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

   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 78.9%

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

   10. Applied rewrites 79.0%

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

   if -250 < x < 2.65e20

   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 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 40.7

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

   5. Applied rewrites 40.7%

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

   if 2.65e20 < 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. distribute-rgt-in N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate--r+ N/A

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

      6. div-sub N/A

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

      7. div-sub N/A

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

      8. associate--r+ N/A

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

      9. 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 99.8%

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

   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 73.2%

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

Alternative 10: 30.1% accurate, 39.3× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- z))

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return -z

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-1 \cdot z} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 31.0

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

5. Applied rewrites 31.0%

   \[\leadsto \color{blue}{-z} \]
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 2024322 
(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))