Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 5.1s

Alternatives: 13

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate--l- N/A

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

   4. lower--.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-+.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 349.5:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -2e+27)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 349.5)
       (- y (fma (log y) 0.5 z))
       (- (+ (* (- x) -1.0) y) 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 <= -2e+27) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 349.5) {
		tmp = y - fma(log(y), 0.5, z);
	} else {
		tmp = ((-x * -1.0) + y) - 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 <= -2e+27)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 349.5)
		tmp = Float64(y - fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(Float64(Float64(-x) * -1.0) + y) - 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, -2e+27], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 349.5], N[(y - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-x) * -1.0), $MachinePrecision] + y), $MachinePrecision] - 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) < -2e27

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 57.8

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

   5. Applied rewrites 57.8%

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

   if -2e27 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 349.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 97.0%

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

   if 349.5 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
   7. Step-by-step derivation

   8. Applied rewrites 99.4%

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

Alternative 3: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 349.5:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (- x (* (+ y 0.5) (log y))) y)))
   (if (<= t_0 -2e+27)
     (* (- 1.0 (log y)) y)
     (if (<= t_0 349.5) (fma -0.5 (log y) (- z)) (- (+ (* (- x) -1.0) y) 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 <= -2e+27) {
		tmp = (1.0 - log(y)) * y;
	} else if (t_0 <= 349.5) {
		tmp = fma(-0.5, log(y), -z);
	} else {
		tmp = ((-x * -1.0) + y) - 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 <= -2e+27)
		tmp = Float64(Float64(1.0 - log(y)) * y);
	elseif (t_0 <= 349.5)
		tmp = fma(-0.5, log(y), Float64(-z));
	else
		tmp = Float64(Float64(Float64(Float64(-x) * -1.0) + y) - 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, -2e+27], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 349.5], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision], N[(N[(N[((-x) * -1.0), $MachinePrecision] + y), $MachinePrecision] - 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) < -2e27

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-log.f64 57.8

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

   5. Applied rewrites 57.8%

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

   if -2e27 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) < 349.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

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

   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 97.0%

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

   if 349.5 < (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
   7. Step-by-step derivation

   8. Applied rewrites 99.4%

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

Alternative 4: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+27}:\\ \;\;\;\;y - \left(\mathsf{fma}\left(\log y, 0.5, z\right) - x\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+130}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot y\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(y - -0.5, \log y, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+27)
   (- y (- (fma (log y) 0.5 z) x))
   (if (<= y 5.2e+130)
     (- (+ x y) (* (log y) y))
     (- y (fma (- y -0.5) (log y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+27) {
		tmp = y - (fma(log(y), 0.5, z) - x);
	} else if (y <= 5.2e+130) {
		tmp = (x + y) - (log(y) * y);
	} else {
		tmp = y - fma((y - -0.5), log(y), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+27)
		tmp = Float64(y - Float64(fma(log(y), 0.5, z) - x));
	elseif (y <= 5.2e+130)
		tmp = Float64(Float64(x + y) - Float64(log(y) * y));
	else
		tmp = Float64(y - fma(Float64(y - -0.5), log(y), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+27], N[(y - N[(N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+130], N[(N[(x + y), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[(y - -0.5), $MachinePrecision] * N[Log[y], $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.9999999999999998e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.3%

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

   if 8.9999999999999998e27 < y < 5.1999999999999996e130

   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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 84.8

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

   9. Applied rewrites 84.8%

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

   if 5.1999999999999996e130 < 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. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 91.1

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

   5. Applied rewrites 91.1%

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

Alternative 5: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4000000 \lor \neg \left(x \leq 1650\right):\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4000000.0) (not (<= x 1650.0)))
   (- (+ (* (- x) -1.0) y) z)
   (fma -0.5 (log y) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4000000.0) || !(x <= 1650.0)) {
		tmp = ((-x * -1.0) + y) - z;
	} else {
		tmp = fma(-0.5, log(y), -z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4000000.0) || !(x <= 1650.0))
		tmp = Float64(Float64(Float64(Float64(-x) * -1.0) + y) - z);
	else
		tmp = fma(-0.5, log(y), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4000000.0], N[Not[LessEqual[x, 1650.0]], $MachinePrecision]], N[(N[(N[((-x) * -1.0), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision], N[(-0.5 * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4e6 or 1650 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-out N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. metadata-eval N/A

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

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

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
   7. Step-by-step derivation

   8. Applied rewrites 74.9%

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

   if -4e6 < x < 1650

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   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 65.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4000000 \lor \neg \left(x \leq 1650\right):\\ \;\;\;\;\left(\left(-x\right) \cdot -1 + y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, -z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.18:\\ \;\;\;\;y - \left(\mathsf{fma}\left(\log y, 0.5, z\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x - \log y \cdot y\right) + y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 0.18)
   (- y (- (fma (log y) 0.5 z) x))
   (- (+ (- x (* (log y) y)) y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 0.18) {
		tmp = y - (fma(log(y), 0.5, z) - x);
	} else {
		tmp = ((x - (log(y) * y)) + y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 0.18)
		tmp = Float64(y - Float64(fma(log(y), 0.5, z) - x));
	else
		tmp = Float64(Float64(Float64(x - Float64(log(y) * y)) + y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 0.18], N[(y - N[(N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.17999999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.2%

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

   if 0.17999999999999999 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. remove-double-neg N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

Alternative 7: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+27}:\\ \;\;\;\;y - \left(\mathsf{fma}\left(\log y, 0.5, z\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+27) (- y (- (fma (log y) 0.5 z) x)) (- (+ x y) (* (log y) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+27) {
		tmp = y - (fma(log(y), 0.5, z) - x);
	} else {
		tmp = (x + y) - (log(y) * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+27)
		tmp = Float64(y - Float64(fma(log(y), 0.5, z) - x));
	else
		tmp = Float64(Float64(x + y) - Float64(log(y) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+27], N[(y - N[(N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.9999999999999998e27

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l- N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. *-lft-identity N/A

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

      8. *-lft-identity N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 N/A

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

      17. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.3%

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

   if 8.9999999999999998e27 < 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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 82.5

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

   9. Applied rewrites 82.5%

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

Alternative 8: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) - \log y \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+27) (- (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 <= 9e+27) {
		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 <= 9e+27)
		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, 9e+27], 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 < 8.9999999999999998e27

   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

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 8.9999999999999998e27 < 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 99.7

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

      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 99.7

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

      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 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate--l- N/A

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

      4. lower--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 99.7

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

   6. Applied rewrites 99.7%

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

   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. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 82.5

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

   9. Applied rewrites 82.5%

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

Alternative 9: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;y - \left(y - -0.5\right) \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e+132) (- (fma -0.5 (log y) x) z) (- y (* (- y -0.5) (log y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e+132) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = y - ((y - -0.5) * log(y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e+132)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(y - Float64(Float64(y - -0.5) * log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e+132], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(y - N[(N[(y - -0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.49999999999999969e132

   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

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if 8.49999999999999969e132 < 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. *-lft-identity N/A

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

      4. *-lft-identity N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 90.9

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

   5. Applied rewrites 90.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 74.9%

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

Alternative 10: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8.5e+132) (- (fma -0.5 (log y) x) z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8.5e+132) {
		tmp = fma(-0.5, log(y), x) - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8.5e+132)
		tmp = Float64(fma(-0.5, log(y), x) - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8.5e+132], N[(N[(-0.5 * N[Log[y], $MachinePrecision] + x), $MachinePrecision] - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.49999999999999969e132

   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

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

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

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if 8.49999999999999969e132 < 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 74.9

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

   5. Applied rewrites 74.9%

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

Alternative 11: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. associate-+l- N/A

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

   4. lower--.f64 N/A

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

   5. lower--.f64 N/A

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

   6. +-commutative N/A

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

   7. *-lft-identity N/A

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

   8. *-lft-identity N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. metadata-eval N/A

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

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

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 12: 56.9% accurate, 8.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((-x * -1.0) + 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 * (-1.0d0)) + y) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-out N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. metadata-eval N/A

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

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

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

5. Applied rewrites 88.8%

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

6. Taylor expanded in x around inf

   \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
7. Step-by-step derivation

8. Applied rewrites 57.5%

   \[\leadsto \left(\left(-x\right) \cdot -1 + y\right) - z \]
9. Add Preprocessing

Alternative 13: 29.4% 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 99.8%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.5

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

5. Applied rewrites 30.5%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 2025017 
(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))