Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 6.5s

Alternatives: 9

Speedup: N/A×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. lower-fma.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. associate-+l+ N/A

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

   5. +-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

   9. lift-neg.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. associate--l- N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-fma.f64 N/A

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

   16. lower--.f64 99.9

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.8e-35)
   (- (fma -0.5 (log y) x) z)
   (- (+ (- x (* (log y) y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.8e-35) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((x - (log(y) * y)) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.8e-35)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.8e-35], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000009e-35

   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 1.80000000000000009e-35 < y

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 3: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+29) (- (fma -0.5 (log y) x) z) (- (+ x y) (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+29) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (x + y) - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+29)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(x + y) - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+29], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999966e29

   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 96.9

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

   5. Applied rewrites 96.9%

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

   if 3.99999999999999966e29 < y

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. lower-fma.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. associate--l- N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lower--.f64 99.7

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 86.2

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

   9. Applied rewrites 86.2%

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

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+24) (- (fma -0.5 (log y) x) z) (- (* (- 1.0 (log y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+24) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = ((1.0 - log(y)) * y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+24)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+24], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999996e24

   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 97.5

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

   5. Applied rewrites 97.5%

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

   if 6.4999999999999996e24 < y

   1. Initial program 99.7%

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

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lower-+.f64 99.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

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

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

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

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

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

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

      11. associate-*r* N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-neg.f64 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 78.0%

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

Alternative 5: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4e+168) (- (fma -0.5 (log y) x) z) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4e+168) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4e+168)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4e+168], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999997e168

   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 86.7

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

   5. Applied rewrites 86.7%

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

   if 3.9999999999999997e168 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 90.9

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

   5. Applied rewrites 90.9%

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

   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 86.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e+57) (fma -0.5 (log y) (- z)) (- y (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e+57) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = y - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e+57)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = Float64(y - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e+57], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e57

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 66.6

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

   5. Applied rewrites 66.6%

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

   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 60.8%

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

   if 1.1e57 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 77.8

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

   5. Applied rewrites 77.8%

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

   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 67.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.1e+57) (fma -0.5 (log y) (- z)) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.1e+57) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.1e+57)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.1e+57], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e57

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-log.f64 66.6

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

   5. Applied rewrites 66.6%

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

   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 60.8%

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

   if 1.1e57 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 67.5

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

   5. Applied rewrites 67.5%

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

Alternative 8: 42.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-log.f64 70.9

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

5. Applied rewrites 70.9%

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

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 42.4%

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

Alternative 9: 29.8% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 99.9%

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
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 28.7

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

5. Applied rewrites 28.7%

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025008 
(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))