Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 8.4s

Alternatives: 12

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. associate-+l+ N/A

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

   6. lower-+.f64 N/A

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

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

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

   8. *-commutative N/A

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

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

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

   10. lower-fma.f64 N/A

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

   11. lower-neg.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 340:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -5e+61)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 340.0) (- (* -0.5 (log y)) z) (fma (/ (- z) x) x x)))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double tmp;
	if (t_0 <= -5e+61) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 340.0) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = fma((-z / x), x, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	tmp = 0.0
	if (t_0 <= -5e+61)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 340.0)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = fma(Float64(Float64(-z) / x), x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+61], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 340.0], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -5.00000000000000018e61

   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 55.1

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

   5. Applied rewrites 55.1%

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

   if -5.00000000000000018e61 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 340

   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. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 95.7

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

   5. Applied rewrites 95.7%

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

   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 84.9%

      \[\leadsto -0.5 \cdot \log y - \color{blue}{z} \]

   if 340 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 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 99.4%

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

Alternative 3: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{if}\;x \leq -240000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-144}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -0.5 - y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ (- z) x) x x)))
   (if (<= x -240000.0)
     t_0
     (if (<= x -2.6e-144)
       (- (* -0.5 (log y)) z)
       (if (<= x 2.6e-24) (fma (log y) (- -0.5 y) y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-z / x), x, x);
	double tmp;
	if (x <= -240000.0) {
		tmp = t_0;
	} else if (x <= -2.6e-144) {
		tmp = (-0.5 * log(y)) - z;
	} else if (x <= 2.6e-24) {
		tmp = fma(log(y), (-0.5 - y), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(Float64(-z) / x), x, x)
	tmp = 0.0
	if (x <= -240000.0)
		tmp = t_0;
	elseif (x <= -2.6e-144)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	elseif (x <= 2.6e-24)
		tmp = fma(log(y), Float64(-0.5 - y), y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[x, -240000.0], t$95$0, If[LessEqual[x, -2.6e-144], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 2.6e-24], N[(N[Log[y], $MachinePrecision] * N[(-0.5 - y), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4e5 or 2.6e-24 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   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 79.0%

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

   if -2.4e5 < x < -2.6000000000000001e-144

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   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 75.6%

      \[\leadsto -0.5 \cdot \log y - \color{blue}{z} \]

   if -2.6000000000000001e-144 < x < 2.6e-24

   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. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.9%

      \[\leadsto y - \log y \cdot \color{blue}{y} \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 71.1%

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

Alternative 4: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y + x\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+27}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5 + y, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.8e+90)
   (fma (log y) (- y) (+ y x))
   (if (<= x 5.5e+27)
     (- y (fma (+ 0.5 y) (log y) z))
     (- (fma -0.5 (log y) x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.8e+90) {
		tmp = fma(log(y), -y, (y + x));
	} else if (x <= 5.5e+27) {
		tmp = y - fma((0.5 + y), log(y), z);
	} else {
		tmp = fma(-0.5, log(y), x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.8e+90)
		tmp = fma(log(y), Float64(-y), Float64(y + x));
	elseif (x <= 5.5e+27)
		tmp = Float64(y - fma(Float64(0.5 + y), log(y), z));
	else
		tmp = Float64(fma(-0.5, log(y), x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.8e+90], N[(N[Log[y], $MachinePrecision] * (-y) + N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+27], N[(y - N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8000000000000001e90

   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 y around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in z around 0

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

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

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. distribute-lft-in N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. mul-1-neg N/A

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

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

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

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

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

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

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

       13. lower-fma.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower--.f64 N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 89.0

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

   11. Applied rewrites 89.0%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 89.0%

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

   if -3.8000000000000001e90 < x < 5.49999999999999966e27

   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. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if 5.49999999999999966e27 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 5: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -240000 \lor \neg \left(x \leq 4.4 \cdot 10^{+16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -240000.0) (not (<= x 4.4e+16)))
   (fma (/ (- z) x) x x)
   (- (* -0.5 (log y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -240000.0) || !(x <= 4.4e+16)) {
		tmp = fma((-z / x), x, x);
	} else {
		tmp = (-0.5 * log(y)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -240000.0) || !(x <= 4.4e+16))
		tmp = fma(Float64(Float64(-z) / x), x, x);
	else
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -240000.0], N[Not[LessEqual[x, 4.4e+16]], $MachinePrecision]], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e5 or 4.4e16 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   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 79.6%

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

   if -2.4e5 < x < 4.4e16

   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. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   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 59.2%

      \[\leadsto -0.5 \cdot \log y - \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -240000 \lor \neg \left(x \leq 4.4 \cdot 10^{+16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + \left(1 - \log y\right) \cdot y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.038)
   (- (fma -0.5 (log y) x) z)
   (- (+ x (* (- 1.0 (log y)) y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.038) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (x + ((1.0 - log(y)) * y)) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.038)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(x + Float64(Float64(1.0 - log(y)) * y)) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.038], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0379999999999999991

   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 0.0379999999999999991 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-+l+ N/A

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

      6. lower-+.f64 N/A

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

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

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

      8. *-commutative N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 99.7

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

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

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 98.7

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

   7. Applied rewrites 98.7%

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

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.8e-7)
   (- (fma -0.5 (log y) x) z)
   (+ (fma (- -0.5 y) (log y) y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.8e-7) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma((-0.5 - y), log(y), y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.8e-7)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(fma(Float64(-0.5 - y), log(y), y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.8e-7], 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] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.79999999999999948e-7

   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 100.0

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

   5. Applied rewrites 100.0%

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

   if 6.79999999999999948e-7 < 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

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 21.3%

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

   9. Taylor expanded in z around 0

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

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

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. distribute-lft-in N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. mul-1-neg N/A

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

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

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

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

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

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

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

       13. lower-fma.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower--.f64 N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 81.2

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

   11. Applied rewrites 81.2%

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

   13. Applied rewrites 81.2%

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

Alternative 8: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, -y, y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e+38) (- (fma -0.5 (log y) x) z) (fma (log y) (- y) (+ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e+38) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = fma(log(y), -y, (y + x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e+38)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = fma(log(y), Float64(-y), Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.3e+38], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * (-y) + N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3000000000000001e38

   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 96.3

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

   5. Applied rewrites 96.3%

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

   if 2.3000000000000001e38 < 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

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 15.3%

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

   9. Taylor expanded in z around 0

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

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

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

       4. +-commutative N/A

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

       5. distribute-rgt-in N/A

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

       6. distribute-lft-in N/A

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. mul-1-neg N/A

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

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

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

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

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

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

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

       13. lower-fma.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower--.f64 N/A

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

       16. +-commutative N/A

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

       17. lower-+.f64 81.9

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

   11. Applied rewrites 81.9%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 81.9%

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

Alternative 9: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9.5e+127) (- (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 <= 9.5e+127) {
		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 <= 9.5e+127)
		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, 9.5e+127], 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 < 9.49999999999999975e127

   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 87.0

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

   5. Applied rewrites 87.0%

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

   if 9.49999999999999975e127 < y

   1. Initial program 99.4%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 10: 57.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-8} \lor \neg \left(x \leq 4.4 \cdot 10^{+16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.5e-8) (not (<= x 4.4e+16))) (fma (/ (- z) x) x x) (- z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e-8) || !(x <= 4.4e+16)) {
		tmp = fma((-z / x), x, x);
	} else {
		tmp = -z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.5e-8) || !(x <= 4.4e+16))
		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, -8.5e-8], N[Not[LessEqual[x, 4.4e+16]], $MachinePrecision]], N[(N[((-z) / x), $MachinePrecision] * x + x), $MachinePrecision], (-z)]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999935e-8 or 4.4e16 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. div-add-rev N/A

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

   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 79.0%

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

   if -8.49999999999999935e-8 < x < 4.4e16

   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 36.0

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

   5. Applied rewrites 36.0%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-8} \lor \neg \left(x \leq 4.4 \cdot 10^{+16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{x}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+58} \lor \neg \left(z \leq 2.25 \cdot 10^{+23}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, y, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.8e+58) (not (<= z 2.25e+23))) (- z) (fma (/ x y) y y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.8e+58) || !(z <= 2.25e+23)) {
		tmp = -z;
	} else {
		tmp = fma((x / y), y, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.8e+58) || !(z <= 2.25e+23))
		tmp = Float64(-z);
	else
		tmp = fma(Float64(x / y), y, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.8e+58], N[Not[LessEqual[z, 2.25e+23]], $MachinePrecision]], (-z), N[(N[(x / y), $MachinePrecision] * y + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.8000000000000003e58 or 2.2499999999999999e23 < z

   1. Initial program 99.8%

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

   3. 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 63.7

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

   5. Applied rewrites 63.7%

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

   if -8.8000000000000003e58 < z < 2.2499999999999999e23

   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

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 28.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, y, y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+58} \lor \neg \left(z \leq 2.25 \cdot 10^{+23}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, y, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.7% 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 29.5

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

5. Applied rewrites 29.5%

   \[\leadsto \color{blue}{-z} \]
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 2024343 
(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))