Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.9%

Time: 3.9s

Alternatives: 13

Speedup: 0.5×

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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(1 - \log y\right) \cdot y + x\right) - \log y \cdot 0.5\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (- (+ (* (- 1.0 (log y)) y) x) (* (log y) 0.5)) z))

C

double code(double x, double y, double z) {
	return ((((1.0 - log(y)) * y) + x) - (log(y) * 0.5)) - 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 - log(y)) * y) + x) - (log(y) * 0.5d0)) - z
end function

Java

public static double code(double x, double y, double z) {
	return ((((1.0 - Math.log(y)) * y) + x) - (Math.log(y) * 0.5)) - z;
}

Python

def code(x, y, z):
	return ((((1.0 - math.log(y)) * y) + x) - (math.log(y) * 0.5)) - z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(Float64(1.0 - log(y)) * y) + x) - Float64(log(y) * 0.5)) - z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((((1.0 - log(y)) * y) + x) - (log(y) * 0.5)) - z;
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f64 N/A

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

   9. unpow1/2 N/A

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

   10. lower-sqrt.f64 99.9

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

5. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 99.9

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

7. Applied rewrites 99.9%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 99.9

      \[\leadsto \left(\left(\left(1 - \log y\right) \cdot y + x\right) - \log y \cdot 0.5\right) - z \]

9. Applied rewrites 99.9%

   \[\leadsto \left(\left(\left(1 - \log y\right) \cdot y + x\right) - \log y \cdot \color{blue}{0.5}\right) - z \]
10. Add Preprocessing

Alternative 2: 69.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\\ \mathbf{if}\;t\_0 \leq -10000000000000:\\ \;\;\;\;x - z\\ \mathbf{elif}\;t\_0 \leq 500:\\ \;\;\;\;-0.5 \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (- x (* (+ y 0.5) (log y))) y) z)))
   (if (<= t_0 -10000000000000.0)
     (- x z)
     (if (<= t_0 500.0) (* -0.5 (log y)) (- x z)))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = -0.5 * log(y);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x - ((y + 0.5d0) * log(y))) + y) - z
    if (t_0 <= (-10000000000000.0d0)) then
        tmp = x - z
    else if (t_0 <= 500.0d0) then
        tmp = (-0.5d0) * log(y)
    else
        tmp = x - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((x - ((y + 0.5) * Math.log(y))) + y) - z;
	double tmp;
	if (t_0 <= -10000000000000.0) {
		tmp = x - z;
	} else if (t_0 <= 500.0) {
		tmp = -0.5 * Math.log(y);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((x - ((y + 0.5) * math.log(y))) + y) - z
	tmp = 0
	if t_0 <= -10000000000000.0:
		tmp = x - z
	elif t_0 <= 500.0:
		tmp = -0.5 * math.log(y)
	else:
		tmp = x - z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y) - z)
	tmp = 0.0
	if (t_0 <= -10000000000000.0)
		tmp = Float64(x - z);
	elseif (t_0 <= 500.0)
		tmp = Float64(-0.5 * log(y));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((x - ((y + 0.5) * log(y))) + y) - z;
	tmp = 0.0;
	if (t_0 <= -10000000000000.0)
		tmp = x - z;
	elseif (t_0 <= 500.0)
		tmp = -0.5 * log(y);
	else
		tmp = x - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$0, -10000000000000.0], N[(x - z), $MachinePrecision], If[LessEqual[t$95$0, 500.0], N[(-0.5 * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(x - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < -1e13 or 500 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z)

   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

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

   5. Applied rewrites 66.5%

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

   if -1e13 < (-.f64 (+.f64 (-.f64 x (*.f64 (+.f64 y #s(literal 1/2 binary64)) (log.f64 y))) y) z) < 500

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-log.f64 90.1

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

   8. Applied rewrites 90.1%

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

   9. Taylor expanded in z around 0

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

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1}{2} \cdot \log y \]

       2. lift-log.f64 87.0

          \[\leadsto -0.5 \cdot \log y \]

   11. Applied rewrites 87.0%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(1 - \log y, y, x\right) - \log y \cdot 0.5\right) - z \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (- (fma (- 1.0 (log y)) y x) (* (log y) 0.5)) z))

C

double code(double x, double y, double z) {
	return (fma((1.0 - log(y)), y, x) - (log(y) * 0.5)) - z;
}

Julia

function code(x, y, z)
	return Float64(Float64(fma(Float64(1.0 - log(y)), y, x) - Float64(log(y) * 0.5)) - z)
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. log-pow-rev N/A

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

   8. lower-log.f64 N/A

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

   9. unpow1/2 N/A

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

   10. lower-sqrt.f64 99.9

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

5. Applied rewrites 99.9%

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

   1. lift-log.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. pow1/2 N/A

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

   4. log-pow-rev N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-log.f64 99.9

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

7. Applied rewrites 99.9%

   \[\leadsto \left(\mathsf{fma}\left(1 - \log y, y, x\right) - \log y \cdot \color{blue}{0.5}\right) - z \]
8. Add Preprocessing

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+51}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+114}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y - z\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot \log y\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5.4e+51)
   (- x (fma (log y) 0.5 z))
   (if (<= y 9e+114) (- (* (- 1.0 (log y)) y) z) (+ (- x (* y (log y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5.4e+51) {
		tmp = x - fma(log(y), 0.5, z);
	} else if (y <= 9e+114) {
		tmp = ((1.0 - log(y)) * y) - z;
	} else {
		tmp = (x - (y * log(y))) + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5.4e+51)
		tmp = Float64(x - fma(log(y), 0.5, z));
	elseif (y <= 9e+114)
		tmp = Float64(Float64(Float64(1.0 - log(y)) * y) - z);
	else
		tmp = Float64(Float64(x - Float64(y * log(y))) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5.4e+51], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e+114], N[(N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - z), $MachinePrecision], N[(N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.39999999999999983e51

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 5.39999999999999983e51 < y < 9.0000000000000001e114

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-pow-rev N/A

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

      8. lower-log.f64 N/A

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

      9. unpow1/2 N/A

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

      10. lower-sqrt.f64 99.8

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

   5. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-log.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. pow1/2 N/A

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

      4. log-pow-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 99.8

         \[\leadsto \left(\left(\left(1 - \log y\right) \cdot y + x\right) - \log y \cdot 0.5\right) - z \]

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in y around inf

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

       1. associate-+l- N/A

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

       2. flip3-+ N/A

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

       3. metadata-eval N/A

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

       4. metadata-eval N/A

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

       5. associate-+l- N/A

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

       6. *-commutative N/A

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

       7. mul-1-neg N/A

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

       8. neg-log N/A

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

       9. remove-double-neg N/A

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

       10. lift-log.f64 N/A

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

       11. lift--.f64 N/A

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

       12. lift-*.f64 70.1

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

   12. Applied rewrites 70.1%

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

   if 9.0000000000000001e114 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(x - y \cdot \log y\right) + \mathsf{Rewrite=>}\left(lower--.f64, \left(y - z\right)\right) \]

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 85.5%

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

Alternative 5: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;x - z\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+47}:\\ \;\;\;\;-\mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -210.0)
   (- x z)
   (if (<= x 3.2e+47) (- (fma (log y) 0.5 z)) (- x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -210.0) {
		tmp = x - z;
	} else if (x <= 3.2e+47) {
		tmp = -fma(log(y), 0.5, z);
	} else {
		tmp = x - z;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -210.0)
		tmp = Float64(x - z);
	elseif (x <= 3.2e+47)
		tmp = Float64(-fma(log(y), 0.5, z));
	else
		tmp = Float64(x - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -210.0], N[(x - z), $MachinePrecision], If[LessEqual[x, 3.2e+47], (-N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), N[(x - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -210 or 3.2e47 < 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

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

   5. Applied rewrites 79.5%

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

   if -210 < x < 3.2e47

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-log.f64 61.1

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

   8. Applied rewrites 61.1%

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

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{-14}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot \log y\right) + \left(y - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.6e-14) (- x (fma (log y) 0.5 z)) (+ (- x (* y (log y))) (- y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.6e-14) {
		tmp = x - fma(log(y), 0.5, z);
	} else {
		tmp = (x - (y * log(y))) + (y - z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.6e-14)
		tmp = Float64(x - fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(x - Float64(y * log(y))) + Float64(y - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.6e-14], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.6000000000000004e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if 7.6000000000000004e-14 < 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

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

   5. Applied rewrites 98.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(x - y \cdot \log y\right) + \mathsf{Rewrite=>}\left(lower--.f64, \left(y - z\right)\right) \]

   7. Applied rewrites 98.0%

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

Alternative 7: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+51}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y \cdot \log y\right) + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 8e+51) (- x (fma (log y) 0.5 z)) (+ (- x (* y (log y))) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 8e+51) {
		tmp = x - fma(log(y), 0.5, z);
	} else {
		tmp = (x - (y * log(y))) + y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 8e+51)
		tmp = Float64(x - fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(x - Float64(y * log(y))) + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 8e+51], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8e51

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if 8e51 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lower-+.f64 N/A

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

      5. flip3-+ N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

         \[\leadsto \left(x - y \cdot \log y\right) + \mathsf{Rewrite=>}\left(lower--.f64, \left(y - z\right)\right) \]

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 82.3%

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

Alternative 8: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (x - (log(y) * (y - -0.5))) + (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 - (log(y) * (y - (-0.5d0)))) + (y - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(x - N[(N[Log[y], $MachinePrecision] * N[(y - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y - 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. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. associate--l+ N/A

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

   8. lower-+.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 9: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+153}:\\ \;\;\;\;x - \mathsf{fma}\left(\log y, 0.5, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+153) (- x (fma (log y) 0.5 z)) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+153) {
		tmp = x - fma(log(y), 0.5, z);
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+153)
		tmp = Float64(x - fma(log(y), 0.5, z));
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+153], N[(x - N[(N[Log[y], $MachinePrecision] * 0.5 + z), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.0000000000000002e153

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if 9.0000000000000002e153 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-pow-rev N/A

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

      8. lower-log.f64 N/A

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

      9. unpow1/2 N/A

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

      10. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

      1. associate-+l- N/A

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

      2. flip3-+ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

   10. Applied rewrites 76.5%

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

Alternative 10: 70.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+153}:\\ \;\;\;\;x - z\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \log y\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 9e+153) (- x z) (* (- 1.0 (log y)) y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+153) {
		tmp = x - z;
	} else {
		tmp = (1.0 - log(y)) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 9d+153) then
        tmp = x - z
    else
        tmp = (1.0d0 - log(y)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 9e+153) {
		tmp = x - z;
	} else {
		tmp = (1.0 - Math.log(y)) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 9e+153:
		tmp = x - z
	else:
		tmp = (1.0 - math.log(y)) * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 9e+153)
		tmp = Float64(x - z);
	else
		tmp = Float64(Float64(1.0 - log(y)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 9e+153)
		tmp = x - z;
	else
		tmp = (1.0 - log(y)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 9e+153], N[(x - z), $MachinePrecision], N[(N[(1.0 - N[Log[y], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.0000000000000002e153

   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

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

   5. Applied rewrites 69.1%

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

   if 9.0000000000000002e153 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. log-pow-rev N/A

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

      8. lower-log.f64 N/A

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

      9. unpow1/2 N/A

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

      10. lower-sqrt.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around inf

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

      1. associate-+l- N/A

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

      2. flip3-+ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 76.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. *-commutative N/A

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

      7. remove-double-neg N/A

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

   10. Applied rewrites 76.5%

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

Alternative 11: 48.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+51}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.12e+21) x (if (<= x 7e+51) (- z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.12e+21) {
		tmp = x;
	} else if (x <= 7e+51) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.12d+21)) then
        tmp = x
    else if (x <= 7d+51) then
        tmp = -z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.12e+21) {
		tmp = x;
	} else if (x <= 7e+51) {
		tmp = -z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.12e+21:
		tmp = x
	elif x <= 7e+51:
		tmp = -z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.12e+21)
		tmp = x;
	elseif (x <= 7e+51)
		tmp = Float64(-z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.12e+21)
		tmp = x;
	elseif (x <= 7e+51)
		tmp = -z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.12e+21], x, If[LessEqual[x, 7e+51], (-z), x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.12e21 or 7e51 < 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

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

   5. Applied rewrites 61.7%

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

   if -1.12e21 < x < 7e51

   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

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

      2. lower-neg.f64 37.3

         \[\leadsto -z \]

   5. Applied rewrites 37.3%

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

Alternative 12: 57.8% accurate, 29.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 57.8%

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

Alternative 13: 30.2% accurate, 118.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 30.2%

   \[\leadsto \color{blue}{x} \]
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 2025088 
(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))