Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 6.4s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

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

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

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - N[(z - 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. 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. sub-negate-rev N/A

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

   3. lift-+.f64 N/A

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

   4. add-flip N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. associate--l+ N/A

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

   9. associate--r+ N/A

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

   10. sub-negate N/A

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

   11. lower--.f64 N/A

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

   12. add-flip-rev N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. add-flip N/A

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

   16. sub-negate N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval N/A

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

   19. lower--.f64 99.9%

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

3. Applied rewrites 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(N[(y + x), $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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. add-flip N/A

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

   13. sub-negate N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval N/A

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

   16. sub-flip N/A

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

   17. +-commutative N/A

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

   18. associate-+l+ N/A

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

   19. +-commutative N/A

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

   20. associate-+r+ N/A

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

   21. remove-double-neg N/A

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

   22. distribute-neg-in N/A

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

   23. sub-flip N/A

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

   24. sub-negate-rev N/A

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

   25. associate--r- N/A

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

3. Applied rewrites 99.9%

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

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, y - z\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y 5.2e+34)
  (fma (- -0.5 y) (log y) (- x z))
  (fma (- -0.5 y) (log y) (- y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e+34) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = fma((-0.5 - y), log(y), (y - z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e+34)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x - z));
	else
		tmp = fma(Float64(-0.5 - y), log(y), Float64(y - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e+34], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999999e34

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 79.5%

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

   6. Applied rewrites 79.5%

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

   if 5.1999999999999999e34 < y

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 70.6%

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

Alternative 4: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 - y, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y - -0.5, z\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y 5.2e+34)
  (fma (- -0.5 y) (log y) (- x z))
  (- y (fma (log y) (- y -0.5) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e+34) {
		tmp = fma((-0.5 - y), log(y), (x - z));
	} else {
		tmp = y - fma(log(y), (y - -0.5), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e+34)
		tmp = fma(Float64(-0.5 - y), log(y), Float64(x - z));
	else
		tmp = Float64(y - fma(log(y), Float64(y - -0.5), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e+34], N[(N[(-0.5 - y), $MachinePrecision] * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y - -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999999e34

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 79.5%

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

   6. Applied rewrites 79.5%

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

   if 5.1999999999999999e34 < y

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. add-flip-rev N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 70.5%

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

Alternative 5: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \mathsf{fma}\left(\log y, y - -0.5, z\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y 5.2e+34)
  (fma -0.5 (log y) (- x z))
  (- y (fma (log y) (- y -0.5) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.2e+34) {
		tmp = fma(-0.5, log(y), (x - z));
	} else {
		tmp = y - fma(log(y), (y - -0.5), z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.2e+34)
		tmp = fma(-0.5, log(y), Float64(x - z));
	else
		tmp = Float64(y - fma(log(y), Float64(y - -0.5), z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.2e+34], N[(-0.5 * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(y - -0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999999e34

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 79.5%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 71.0%

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

   if 5.1999999999999999e34 < y

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. add-flip-rev N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 70.5%

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

Alternative 6: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y 2.1e+124)
  (fma -0.5 (log y) (- x z))
  (* y (+ 1.0 (log (/ 1.0 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+124) {
		tmp = fma(-0.5, log(y), (x - z));
	} else {
		tmp = y * (1.0 + log((1.0 / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e+124)
		tmp = fma(-0.5, log(y), Float64(x - z));
	else
		tmp = Float64(y * Float64(1.0 + log(Float64(1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e+124], N[(-0.5 * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(y * N[(1.0 + N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1000000000000001e124

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 79.5%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 71.0%

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

   if 2.1000000000000001e124 < y

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 79.5%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 30.2%

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

   9. Applied rewrites 30.2%

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

Alternative 7: 84.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log y, x - z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \log y \cdot \left(0.5 + y\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  (if (<= y 2.1e+124)
  (fma -0.5 (log y) (- x z))
  (- y (* (log y) (+ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.1e+124) {
		tmp = fma(-0.5, log(y), (x - z));
	} else {
		tmp = y - (log(y) * (0.5 + y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.1e+124)
		tmp = fma(-0.5, log(y), Float64(x - z));
	else
		tmp = Float64(y - Float64(log(y) * Float64(0.5 + y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.1e+124], N[(-0.5 * N[Log[y], $MachinePrecision] + N[(x - z), $MachinePrecision]), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] * N[(0.5 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1000000000000001e124

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. add-flip N/A

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

      13. sub-negate N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. sub-flip N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

      20. associate-+r+ N/A

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

      21. remove-double-neg N/A

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

      22. distribute-neg-in N/A

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

      23. sub-flip N/A

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

      24. sub-negate-rev N/A

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

      25. associate--r- N/A

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. lower--.f64 79.5%

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 71.0%

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

   if 2.1000000000000001e124 < y

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. associate-+l- N/A

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

      8. lower--.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. add-flip-rev N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 99.8%

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

      14. lift-+.f64 N/A

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

      15. add-flip N/A

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

      16. lower--.f64 N/A

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

      17. metadata-eval 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 70.5%

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

   7. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 43.2%

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

   9. Applied rewrites 43.2%

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

Alternative 8: 71.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_] := N[(-0.5 * N[Log[y], $MachinePrecision] + N[(x - 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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. add-flip N/A

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

   13. sub-negate N/A

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

   14. lower--.f64 N/A

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

   15. metadata-eval N/A

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

   16. sub-flip N/A

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

   17. +-commutative N/A

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

   18. associate-+l+ N/A

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

   19. +-commutative N/A

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

   20. associate-+r+ N/A

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

   21. remove-double-neg N/A

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

   22. distribute-neg-in N/A

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

   23. sub-flip N/A

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

   24. sub-negate-rev N/A

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

   25. associate--r- N/A

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

3. Applied rewrites 99.9%

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

4. Taylor expanded in y around 0

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

   1. lower--.f64 79.5%

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

6. Applied rewrites 79.5%

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

7. Taylor expanded in y around 0

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

9. Applied rewrites 71.0%

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

Alternative 9: 57.3% accurate, 3.1× speedup

Math

\[1 \cdot x - z \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 * x) - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(1.0 * x), $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. 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. add-flip N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. associate--l+ N/A

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

   7. sum-to-mult N/A

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

   8. lower-unsound-*.f64 N/A

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

   9. lower-unsound-+.f64 N/A

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

   10. lower-unsound-/.f64 N/A

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

   11. add-flip-rev N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. add-flip N/A

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

   15. sub-negate N/A

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

   16. lower--.f64 N/A

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

   17. metadata-eval 88.1%

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

3. Applied rewrites 88.1%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 57.3%

   \[\leadsto \color{blue}{1} \cdot x - z \]
7. Add Preprocessing

Alternative 10: 29.1% accurate, 10.2× speedup

Math

\[-z \]

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. Taylor expanded in z around inf

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

   1. lower-*.f64 29.1%

      \[\leadsto -1 \cdot \color{blue}{z} \]

4. Applied rewrites 29.1%

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

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{z} \]

   2. mul-1-neg N/A

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

   3. lower-neg.f64 29.1%

      \[\leadsto -z \]

6. Applied rewrites 29.1%

   \[\leadsto -z \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (- x (* (+ y 0.5) (log y))) y) z))