Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 7.6s

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(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 99.9

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

   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 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. distribute-lft-in N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. *-rgt-identity N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. *-commutative N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. associate-*r* N/A

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. associate-*r* N/A

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

   14. distribute-lft-in N/A

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

   15. distribute-rgt-in N/A

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

7. Applied rewrites 99.9%

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

Alternative 2: 68.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+91}:\\ \;\;\;\;y - \mathsf{fma}\left(0.5, \log y, z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -5e+37)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 2e+91) (- y (fma 0.5 (log y) z)) (- y (- 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+37) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 2e+91) {
		tmp = y - fma(0.5, log(y), z);
	} else {
		tmp = y - -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+37)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 2e+91)
		tmp = Float64(y - fma(0.5, log(y), z));
	else
		tmp = Float64(y - Float64(-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+37], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e+91], N[(y - N[(0.5 * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(y - (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4.99999999999999989e37

   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 55.4

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

   5. Applied rewrites 55.4%

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

   if -4.99999999999999989e37 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 2.00000000000000016e91

   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 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.3%

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

   if 2.00000000000000016e91 < (+.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 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto y - -1 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.1%

      \[\leadsto y - \left(-x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+37}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+91}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -5e+37)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 2e+91) (- (* -0.5 (log y)) z) (- y (- 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+37) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 2e+91) {
		tmp = (-0.5 * log(y)) - z;
	} else {
		tmp = y - -x;
	}
	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
    if (t_0 <= (-5d+37)) then
        tmp = (1.0d0 - log(y)) * y
    else if (t_0 <= 2d+91) then
        tmp = ((-0.5d0) * log(y)) - z
    else
        tmp = y - -x
    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;
	double tmp;
	if (t_0 <= -5e+37) {
		tmp = (1.0 - Math.log(y)) * y;
	} else if (t_0 <= 2e+91) {
		tmp = (-0.5 * Math.log(y)) - z;
	} else {
		tmp = y - -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x - ((y + 0.5) * math.log(y))) + y
	tmp = 0
	if t_0 <= -5e+37:
		tmp = (1.0 - math.log(y)) * y
	elif t_0 <= 2e+91:
		tmp = (-0.5 * math.log(y)) - z
	else:
		tmp = y - -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+37)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 2e+91)
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	else
		tmp = Float64(y - Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x - ((y + 0.5) * log(y))) + y;
	tmp = 0.0;
	if (t_0 <= -5e+37)
		tmp = (1.0 - log(y)) * y;
	elseif (t_0 <= 2e+91)
		tmp = (-0.5 * log(y)) - z;
	else
		tmp = y - -x;
	end
	tmp_2 = 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+37], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 2e+91], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(y - (-x)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -4.99999999999999989e37

   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 55.4

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

   5. Applied rewrites 55.4%

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

   if -4.99999999999999989e37 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 2.00000000000000016e91

   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 87.8

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

   5. Applied rewrites 87.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.2%

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

   if 2.00000000000000016e91 < (+.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 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto y - -1 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.1%

      \[\leadsto y - \left(-x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 47.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y - \left(-x\right)\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+16}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4 \cdot 10^{-252}:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- y (- x))))
   (if (<= z -2.3e+16)
     (- z)
     (if (<= z -7e-127)
       t_0
       (if (<= z -4e-252) (* -0.5 (log y)) (if (<= z 1.06e+46) t_0 (- z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y - -x;
	double tmp;
	if (z <= -2.3e+16) {
		tmp = -z;
	} else if (z <= -7e-127) {
		tmp = t_0;
	} else if (z <= -4e-252) {
		tmp = -0.5 * log(y);
	} else if (z <= 1.06e+46) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y - -x
    if (z <= (-2.3d+16)) then
        tmp = -z
    else if (z <= (-7d-127)) then
        tmp = t_0
    else if (z <= (-4d-252)) then
        tmp = (-0.5d0) * log(y)
    else if (z <= 1.06d+46) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y - -x;
	double tmp;
	if (z <= -2.3e+16) {
		tmp = -z;
	} else if (z <= -7e-127) {
		tmp = t_0;
	} else if (z <= -4e-252) {
		tmp = -0.5 * Math.log(y);
	} else if (z <= 1.06e+46) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y - -x
	tmp = 0
	if z <= -2.3e+16:
		tmp = -z
	elif z <= -7e-127:
		tmp = t_0
	elif z <= -4e-252:
		tmp = -0.5 * math.log(y)
	elif z <= 1.06e+46:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y - Float64(-x))
	tmp = 0.0
	if (z <= -2.3e+16)
		tmp = Float64(-z);
	elseif (z <= -7e-127)
		tmp = t_0;
	elseif (z <= -4e-252)
		tmp = Float64(-0.5 * log(y));
	elseif (z <= 1.06e+46)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y - -x;
	tmp = 0.0;
	if (z <= -2.3e+16)
		tmp = -z;
	elseif (z <= -7e-127)
		tmp = t_0;
	elseif (z <= -4e-252)
		tmp = -0.5 * log(y);
	elseif (z <= 1.06e+46)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y - (-x)), $MachinePrecision]}, If[LessEqual[z, -2.3e+16], (-z), If[LessEqual[z, -7e-127], t$95$0, If[LessEqual[z, -4e-252], N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.06e+46], t$95$0, (-z)]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.3e16 or 1.05999999999999998e46 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -2.3e16 < z < -6.99999999999999979e-127 or -3.99999999999999977e-252 < z < 1.05999999999999998e46

   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

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto y - -1 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.2%

      \[\leadsto y - \left(-x\right) \]

   if -6.99999999999999979e-127 < z < -3.99999999999999977e-252

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 85.2

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

   5. Applied rewrites 85.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 55.4%

       \[\leadsto -0.5 \cdot \log y \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+17} \lor \neg \left(x \leq 1.05 \cdot 10^{+100}\right):\\ \;\;\;\;y - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.5e+17) (not (<= x 1.05e+100)))
   (- y (- x))
   (- (* -0.5 (log y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e+17) || !(x <= 1.05e+100)) {
		tmp = y - -x;
	} else {
		tmp = (-0.5 * log(y)) - 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 <= (-9.5d+17)) .or. (.not. (x <= 1.05d+100))) then
        tmp = y - -x
    else
        tmp = ((-0.5d0) * log(y)) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e+17) || !(x <= 1.05e+100)) {
		tmp = y - -x;
	} else {
		tmp = (-0.5 * Math.log(y)) - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.5e+17) or not (x <= 1.05e+100):
		tmp = y - -x
	else:
		tmp = (-0.5 * math.log(y)) - z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.5e+17) || !(x <= 1.05e+100))
		tmp = Float64(y - Float64(-x));
	else
		tmp = Float64(Float64(-0.5 * log(y)) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.5e+17) || ~((x <= 1.05e+100)))
		tmp = y - -x;
	else
		tmp = (-0.5 * log(y)) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.5e+17], N[Not[LessEqual[x, 1.05e+100]], $MachinePrecision]], N[(y - (-x)), $MachinePrecision], N[(N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e17 or 1.0499999999999999e100 < 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 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto y - -1 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.4%

      \[\leadsto y - \left(-x\right) \]

   if -9.5e17 < x < 1.0499999999999999e100

   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 94.4

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

   5. Applied rewrites 94.4%

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

   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 62.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+17} \lor \neg \left(x \leq 1.05 \cdot 10^{+100}\right):\\ \;\;\;\;y - \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.5)
		tmp = Float64(Float64(fma(-0.5, log(y), x) + y) - 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, 0.5], N[(N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] + y), $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 < 0.5

   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

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 0.5 < y

   1. Initial program 99.6%

      \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
   2. Add Preprocessing
   3. 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 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 7: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(\left(y - -0.5\right) \cdot \log y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+35)
   (- (fma -0.5 (log y) x) z)
   (- y (- (* (- y -0.5) (log y)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+35) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - (((y - -0.5) * log(y)) - x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+35)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(Float64(Float64(y - -0.5) * log(y)) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+35], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(N[(y - -0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5000000000000003e35

   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 6.5000000000000003e35 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 83.8%

      \[\leadsto y - \left(\left(y - -0.5\right) \cdot \log y - x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(\log y \cdot y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+35) (- (fma -0.5 (log y) x) z) (- y (- (* (log y) y) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+35) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - ((log(y) * y) - x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+35)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(Float64(log(y) * y) - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+35], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.5000000000000003e35

   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 98.7

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

   5. Applied rewrites 98.7%

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

   if 6.5000000000000003e35 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.8%

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

Alternative 9: 85.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.4 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.4e+122) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.4e+122) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.4e+122)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.4e+122], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.40000000000000024e122

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

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

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-log.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 6.40000000000000024e122 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 73.0

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

   5. Applied rewrites 73.0%

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

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(y - N[(N[(N[(0.5 + y), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision] - x), $MachinePrecision]), $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 x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 11: 48.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+16} \lor \neg \left(z \leq 1.06 \cdot 10^{+46}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y - \left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.3e+16) (not (<= z 1.06e+46))) (- z) (- y (- x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+16) || !(z <= 1.06e+46)) {
		tmp = -z;
	} else {
		tmp = y - -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-2.3d+16)) .or. (.not. (z <= 1.06d+46))) then
        tmp = -z
    else
        tmp = y - -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.3e+16) || !(z <= 1.06e+46)) {
		tmp = -z;
	} else {
		tmp = y - -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.3e+16) or not (z <= 1.06e+46):
		tmp = -z
	else:
		tmp = y - -x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.3e+16) || !(z <= 1.06e+46))
		tmp = Float64(-z);
	else
		tmp = Float64(y - Float64(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2.3e+16) || ~((z <= 1.06e+46)))
		tmp = -z;
	else
		tmp = y - -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.3e+16], N[Not[LessEqual[z, 1.06e+46]], $MachinePrecision]], (-z), N[(y - (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3e16 or 1.05999999999999998e46 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.0

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

   5. Applied rewrites 66.0%

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

   if -2.3e16 < z < 1.05999999999999998e46

   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

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto y - -1 \cdot \color{blue}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.6%

      \[\leadsto y - \left(-x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+16} \lor \neg \left(z \leq 1.06 \cdot 10^{+46}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;y - \left(-x\right)\\ \end{array} \]
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 27.9

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

5. Applied rewrites 27.9%

   \[\leadsto \color{blue}{-z} \]
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 2024342 
(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))