Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 4.2s

Alternatives: 12

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

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

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower--.f64 N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. metadata-eval N/A

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

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

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 72.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ t_1 := \left(1 - \log y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+111}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq -20000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 335:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)) (t_1 (* (- 1.0 (log y)) y)))
   (if (<= t_0 -5e+186)
     t_1
     (if (<= t_0 -1e+111)
       (- x z)
       (if (<= t_0 -20000000000.0)
         t_1
         (if (<= t_0 335.0) (- (fma (log y) 0.5 z)) (- x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double t_1 = (1.0 - log(y)) * y;
	double tmp;
	if (t_0 <= -5e+186) {
		tmp = t_1;
	} else if (t_0 <= -1e+111) {
		tmp = x - z;
	} else if (t_0 <= -20000000000.0) {
		tmp = t_1;
	} else if (t_0 <= 335.0) {
		tmp = -fma(log(y), 0.5, z);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	t_1 = Float64(Float64(1.0 - log(y)) * y)
	tmp = 0.0
	if (t_0 <= -5e+186)
		tmp = t_1;
	elseif (t_0 <= -1e+111)
		tmp = Float64(x - z);
	elseif (t_0 <= -20000000000.0)
		tmp = t_1;
	elseif (t_0 <= 335.0)
		tmp = Float64(-fma(log(y), 0.5, z));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+186], t$95$1, If[LessEqual[t$95$0, -1e+111], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, -20000000000.0], t$95$1, If[LessEqual[t$95$0, 335.0], (-N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), N[(x - z), $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) < -4.99999999999999954e186 or -9.99999999999999957e110 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e10

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{\frac{-1}{4}}\right)}{y - \frac{1}{2}} \cdot \log y\right) + y\right) - z \]

      8. lower--.f64 56.8

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y - 0.5}} \cdot \log y\right) + y\right) - z \]

   3. Applied rewrites 56.8%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y - 0.5}} \cdot \log y\right) + y\right) - z \]

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

         \[\leadsto \left(1 - \log y\right) \cdot y \]

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

         \[\leadsto \left(1 - \log y\right) \cdot y \]

      9. lift-log.f64 53.3

         \[\leadsto \left(1 - \log y\right) \cdot y \]

   6. Applied rewrites 53.3%

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

   if -4.99999999999999954e186 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -9.99999999999999957e110 or 335 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \]
   3. Step-by-step derivation

   4. Applied rewrites 76.9%

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

   if -2e10 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 335

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-log.f64 95.6

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

   7. Applied rewrites 95.6%

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

Alternative 3: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+176}:\\ \;\;\;\;x + \left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 18.12:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y - -0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -5e+176)
     (+ x (* (- 1.0 (log y)) y))
     (if (<= t_0 18.12)
       (- y (fma (log y) (- y -0.5) z))
       (- x (fma (log y) 0.5 z))))))

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+176) {
		tmp = x + ((1.0 - log(y)) * y);
	} else if (t_0 <= 18.12) {
		tmp = y - fma(log(y), (y - -0.5), z);
	} else {
		tmp = x - fma(log(y), 0.5, z);
	}
	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+176)
		tmp = Float64(x + Float64(Float64(1.0 - log(y)) * y));
	elseif (t_0 <= 18.12)
		tmp = Float64(y - fma(log(y), Float64(y - -0.5), z));
	else
		tmp = Float64(x - fma(log(y), 0.5, z));
	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+176], N[(x + N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 18.12], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y - -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $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) < -5e176

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      8. lower--.f64 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      9. lift-log.f64 89.2

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

   7. Applied rewrites 89.2%

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

   if -5e176 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 18.120000000000001

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if 18.120000000000001 < (+.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. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 4: 90.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ t_1 := \left(1 - \log y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+176}:\\ \;\;\;\;x + t\_1\\ \mathbf{elif}\;t\_0 \leq -2000000000:\\ \;\;\;\;t\_1 - z\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)) (t_1 (* (- 1.0 (log y)) y)))
   (if (<= t_0 -5e+176)
     (+ x t_1)
     (if (<= t_0 -2000000000.0) (- t_1 z) (- x (fma (log y) 0.5 z))))))

C

double code(double x, double y, double z) {
	double t_0 = (x - ((y + 0.5) * log(y))) + y;
	double t_1 = (1.0 - log(y)) * y;
	double tmp;
	if (t_0 <= -5e+176) {
		tmp = x + t_1;
	} else if (t_0 <= -2000000000.0) {
		tmp = t_1 - z;
	} else {
		tmp = x - fma(log(y), 0.5, z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y)
	t_1 = Float64(Float64(1.0 - log(y)) * y)
	tmp = 0.0
	if (t_0 <= -5e+176)
		tmp = Float64(x + t_1);
	elseif (t_0 <= -2000000000.0)
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(x - fma(log(y), 0.5, z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+176], N[(x + t$95$1), $MachinePrecision], If[LessEqual[t$95$0, -2000000000.0], N[(t$95$1 - z), $MachinePrecision], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $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) < -5e176

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      8. lower--.f64 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      9. lift-log.f64 89.2

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

   7. Applied rewrites 89.2%

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

   if -5e176 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < -2e9

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{\frac{-1}{4}}\right)}{y - \frac{1}{2}} \cdot \log y\right) + y\right) - z \]

      8. lower--.f64 89.8

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y - 0.5}} \cdot \log y\right) + y\right) - z \]

   3. Applied rewrites 89.8%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y - 0.5}} \cdot \log y\right) + y\right) - z \]

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

         \[\leadsto \left(1 - \log \left({\left({y}^{-1}\right)}^{-1}\right)\right) \cdot y - z \]

      4. pow-pow N/A

         \[\leadsto \left(1 - \log \left({y}^{\left(-1 \cdot -1\right)}\right)\right) \cdot y - z \]

      5. metadata-eval N/A

         \[\leadsto \left(1 - \log \left({y}^{1}\right)\right) \cdot y - z \]

      6. unpow1 N/A

         \[\leadsto \left(1 - \log y\right) \cdot y - z \]

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

         \[\leadsto \left(1 - \log y\right) \cdot y - z \]

      9. lift-log.f64 77.1

         \[\leadsto \left(1 - \log y\right) \cdot y - z \]

   6. Applied rewrites 77.1%

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

   if -2e9 < (+.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. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 98.3

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

   4. Applied rewrites 98.3%

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

Alternative 5: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.28 \cdot 10^{+36}:\\ \;\;\;\;x - z\\ \mathbf{elif}\;z \leq 750000:\\ \;\;\;\;\left(y + x\right) - \log y \cdot \left(y - -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y - -0.5, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.28e+36)
   (- x z)
   (if (<= z 750000.0)
     (- (+ y x) (* (log y) (- y -0.5)))
     (- y (fma (log y) (- y -0.5) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.28e+36) {
		tmp = x - z;
	} else if (z <= 750000.0) {
		tmp = (y + x) - (log(y) * (y - -0.5));
	} else {
		tmp = y - fma(log(y), (y - -0.5), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.28e+36)
		tmp = Float64(x - z);
	elseif (z <= 750000.0)
		tmp = Float64(Float64(y + x) - Float64(log(y) * Float64(y - -0.5)));
	else
		tmp = Float64(y - fma(log(y), Float64(y - -0.5), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.28e+36], N[(x - z), $MachinePrecision], If[LessEqual[z, 750000.0], N[(N[(y + x), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * N[(y - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y - -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.27999999999999993e36

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \]
   3. Step-by-step derivation

   4. Applied rewrites 79.1%

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

   if -1.27999999999999993e36 < z < 7.5e5

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower--.f64 97.3

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

   4. Applied rewrites 97.3%

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

   if 7.5e5 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 80.3

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

   4. Applied rewrites 80.3%

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

Alternative 6: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -255000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+49}:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -255000000000.0)
   (- x z)
   (if (<= x 1.7e+49) (- (fma (log y) 0.5 z)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -255000000000.0) {
		tmp = x - z;
	} else if (x <= 1.7e+49) {
		tmp = -fma(log(y), 0.5, z);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -255000000000.0)
		tmp = Float64(x - z);
	elseif (x <= 1.7e+49)
		tmp = Float64(-fma(log(y), 0.5, z));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -255000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 1.7e+49], (-N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55e11 or 1.7e49 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} - z \]
   3. Step-by-step derivation

   4. Applied rewrites 79.6%

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

   if -2.55e11 < x < 1.7e49

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 63.9

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

   4. Applied rewrites 63.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-log.f64 61.0

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

   7. Applied rewrites 61.0%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.32 \cdot 10^{-7}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - y \cdot \log y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.32e-7) (- x (fma (log y) 0.5 z)) (- (+ (- x (* y (log y))) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.32e-7) {
		tmp = x - fma(log(y), 0.5, z);
	} else {
		tmp = ((x - (y * log(y))) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.32e-7)
		tmp = Float64(x - fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(Float64(x - Float64(y * log(y))) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.32e-7], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.3200000000000001e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 2.3200000000000001e-7 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 98.6%

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

Alternative 8: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 55000:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 55000.0) (- x (fma (log y) 0.5 z)) (+ x (* (- 1.0 (log y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 55000.0) {
		tmp = x - fma(log(y), 0.5, z);
	} else {
		tmp = x + ((1.0 - log(y)) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 55000.0)
		tmp = Float64(x - fma(log(y), 0.5, z));
	else
		tmp = Float64(x + Float64(Float64(1.0 - log(y)) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 55000.0], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 55000

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if 55000 < y

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      7. lower-*.f64 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      8. lower--.f64 N/A

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

      9. lift-log.f64 79.0

         \[\leadsto x + \left(1 - \log y\right) \cdot y \]

   7. Applied rewrites 79.0%

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

Alternative 9: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+183}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.8e+183) (- x (fma (log y) 0.5 z)) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.8e+183) {
		tmp = x - fma(log(y), 0.5, z);
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.8e+183)
		tmp = Float64(x - fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.8e+183], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.80000000000000018e183

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if 2.80000000000000018e183 < y

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, \color{blue}{\frac{-1}{4}}\right)}{y - \frac{1}{2}} \cdot \log y\right) + y\right) - z \]

      8. lower--.f64 9.1

         \[\leadsto \left(\left(x - \frac{\mathsf{fma}\left(y, y, -0.25\right)}{\color{blue}{y - 0.5}} \cdot \log y\right) + y\right) - z \]

   3. Applied rewrites 9.1%

      \[\leadsto \left(\left(x - \color{blue}{\frac{\mathsf{fma}\left(y, y, -0.25\right)}{y - 0.5}} \cdot \log y\right) + y\right) - z \]

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. log-pow-rev N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

      5. metadata-eval N/A

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

      6. unpow1 N/A

         \[\leadsto \left(1 - \log y\right) \cdot y \]

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

         \[\leadsto \left(1 - \log y\right) \cdot y \]

      9. lift-log.f64 80.3

         \[\leadsto \left(1 - \log y\right) \cdot y \]

   6. Applied rewrites 80.3%

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

Alternative 10: 47.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-6}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+19}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.2e-6) (- z) (if (<= z 5.2e+19) x (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e-6) {
		tmp = -z;
	} else if (z <= 5.2e+19) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-1.2d-6)) then
        tmp = -z
    else if (z <= 5.2d+19) then
        tmp = x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.2e-6) {
		tmp = -z;
	} else if (z <= 5.2e+19) {
		tmp = x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.2e-6:
		tmp = -z
	elif z <= 5.2e+19:
		tmp = x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.2e-6)
		tmp = Float64(-z);
	elseif (z <= 5.2e+19)
		tmp = x;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.2e-6)
		tmp = -z;
	elseif (z <= 5.2e+19)
		tmp = x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.2e-6], (-z), If[LessEqual[z, 5.2e+19], x, (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1999999999999999e-6 or 5.2e19 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 57.9

         \[\leadsto -z \]

   4. Applied rewrites 57.9%

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

   if -1.1999999999999999e-6 < z < 5.2e19

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 38.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.8% accurate, 29.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return x - 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 - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} - z \]
3. Step-by-step derivation

4. Applied rewrites 57.8%

   \[\leadsto \color{blue}{x} - z \]
5. Add Preprocessing

Alternative 12: 29.8% accurate, 118.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 29.8%

   \[\leadsto \color{blue}{x} \]
5. 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 2025099 
(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))