Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

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: 64.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+28}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{z} - 1\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ y 0.5) (log y))) y) z)))
   (if (<= t_0 -5e+28)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 500.0) (- (* -0.5 (log y)) z) (* (- (/ x z) 1.0) z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	double tmp;
	if (t_0 <= -5e+28) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 500.0) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = ((x / z) - 1.0) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - ((y + 0.5d0) * log(y))) + y) - z
    if (t_0 <= (-5d+28)) then
        tmp = (1.0d0 - log(y)) * y
    else if (t_0 <= 500.0d0) then
        tmp = ((-0.5d0) * log(y)) - z
    else
        tmp = ((x / z) - 1.0d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * Math.log(y))) + y) - z;
	double tmp;
	if (t_0 <= -5e+28) {
		tmp = (1.0 - Math.log(y)) * y;
	} else if (t_0 <= 500.0) {
		tmp = (-0.5 * Math.log(y)) - z;
	} else {
		tmp = ((x / z) - 1.0) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x - ((y + 0.5) * math.log(y))) + y) - z
	tmp = 0
	if t_0 <= -5e+28:
		tmp = (1.0 - math.log(y)) * y
	elif t_0 <= 500.0:
		tmp = (-0.5 * math.log(y)) - z
	else:
		tmp = ((x / z) - 1.0) * z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
	tmp = 0.0
	if (t_0 <= -5e+28)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 500.0)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = Float64(Float64(Float64(x / z) - 1.0) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	tmp = 0.0;
	if (t_0 <= -5e+28)
		tmp = (1.0 - log(y)) * y;
	elseif (t_0 <= 500.0)
		tmp = (-0.5 * log(y)) - z;
	else
		tmp = ((x / z) - 1.0) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+28], 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[(N[(x / z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -4.99999999999999957e28

   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 57.7

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

   5. Applied rewrites 57.7%

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

   if -4.99999999999999957e28 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

      4. 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 96.1

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.8%

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

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

   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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 86.0%

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

   7. Applied rewrites 55.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 83.9%

       \[\leadsto \left(\frac{x}{z} - 1\right) \cdot z \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 920000000000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 920000000000.0)
   (- (fma -0.5 (log y) x) z)
   (if (<= y 2.75e+84)
     (fma (log y) (- -0.5 y) (+ x y))
     (- (fma (log y) (- y) y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 920000000000.0) {
		tmp = fma(-0.5, log(y), x) - z;
	} else if (y <= 2.75e+84) {
		tmp = fma(log(y), (-0.5 - y), (x + y));
	} else {
		tmp = fma(log(y), -y, y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 920000000000.0)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	elseif (y <= 2.75e+84)
		tmp = fma(log(y), Float64(-0.5 - y), Float64(x + y));
	else
		tmp = Float64(fma(log(y), Float64(-y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 920000000000.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 2.75e+84], N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.2e11

   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 99.4

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

   5. Applied rewrites 99.4%

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

   if 9.2e11 < y < 2.7500000000000002e84

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 36.8%

      \[\leadsto \frac{x}{z} \cdot z \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 82.7%

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

   if 2.7500000000000002e84 < 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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. log-rec N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 66.5%

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 90.3

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

   10. Applied rewrites 90.3%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 90.3%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.28:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.28) (- (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 <= 0.28) {
		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 <= 0.28)
		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, 0.28], 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 < 0.28000000000000003

   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.28000000000000003 < 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 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 5: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.4e+107) (- (fma -0.5 (log y) x) z) (- (fma (log y) (- y) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.4e+107) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(log(y), -y, y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.4e+107)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(fma(log(y), Float64(-y), y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.4e+107], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * (-y) + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.39999999999999992e107

   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 92.7

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

   5. Applied rewrites 92.7%

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

   if 1.39999999999999992e107 < 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

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. mul-1-neg N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

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

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

      11. log-rec N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 65.3%

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

   7. Applied rewrites 79.1%

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

   8. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-in N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

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

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

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

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 92.5

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

   10. Applied rewrites 92.5%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 92.5%

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

Alternative 6: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.2e+108) (- (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 <= 8.2e+108) {
		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 <= 8.2e+108)
		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, 8.2e+108], 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 < 8.1999999999999998e108

   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 92.7

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

   5. Applied rewrites 92.7%

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

   if 8.1999999999999998e108 < 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 73.9

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

   5. Applied rewrites 73.9%

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

Alternative 7: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.5 \cdot 10^{+104}:\\ \;\;\;\;\left(\frac{x}{z} - 1\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.5e+104) (* (- (/ x z) 1.0) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+104) {
		tmp = ((x / z) - 1.0) * z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7.5d+104) then
        tmp = ((x / z) - 1.0d0) * z
    else
        tmp = (1.0d0 - log(y)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.5e+104) {
		tmp = ((x / z) - 1.0) * z;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.5e+104:
		tmp = ((x / z) - 1.0) * z
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.5e+104)
		tmp = Float64(Float64(Float64(x / z) - 1.0) * z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.5e+104)
		tmp = ((x / z) - 1.0) * z;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.5e+104], N[(N[(N[(x / z), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5000000000000002e104

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.7%

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 61.0%

       \[\leadsto \left(\frac{x}{z} - 1\right) \cdot z \]

   if 7.5000000000000002e104 < 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 72.8

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

   5. Applied rewrites 72.8%

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

Alternative 8: 38.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+157} \lor \neg \left(x \leq 4 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.55e+157) (not (<= x 4e+36))) (* (/ x z) z) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55e+157) || !(x <= 4e+36)) {
		tmp = (x / z) * z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.55d+157)) .or. (.not. (x <= 4d+36))) then
        tmp = (x / z) * z
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.55e+157) || !(x <= 4e+36)) {
		tmp = (x / z) * z;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.55e+157) or not (x <= 4e+36):
		tmp = (x / z) * z
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.55e+157) || !(x <= 4e+36))
		tmp = Float64(Float64(x / z) * z);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.55e+157) || ~((x <= 4e+36)))
		tmp = (x / z) * z;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.55e+157], N[Not[LessEqual[x, 4e+36]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * z), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -1.5499999999999999e157 or 4.00000000000000017e36 < 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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{x}{z} \cdot z \]
   7. Step-by-step derivation

   8. Applied rewrites 45.7%

      \[\leadsto \frac{x}{z} \cdot z \]

   if -1.5499999999999999e157 < x < 4.00000000000000017e36

   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 37.5

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

   5. Applied rewrites 37.5%

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

4. Final simplification 40.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+157} \lor \neg \left(x \leq 4 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x}{z} \cdot z\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((x / z) - 1.0) * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x / z) - 1.0d0) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 81.5%

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

7. Applied rewrites 51.6%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 46.3%

    \[\leadsto \left(\frac{x}{z} - 1\right) \cdot z \]
11. Add Preprocessing

Alternative 10: 30.6% 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 29.0

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

5. Applied rewrites 29.0%

   \[\leadsto \color{blue}{-z} \]
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 2024337 
(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))