Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 89.5% → 99.6%

Time: 10.0s

Alternatives: 16

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (*
    (- z 1.0)
    (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

5. Applied rewrites 99.2%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]
6. Add Preprocessing

Alternative 2: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -1.25 \cdot 10^{+77} \lor \neg \left(t\_1 \leq 10^{+67}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 -1.25e+77) (not (<= t_1 1e+67)))
     (* (log y) x)
     (- (- y (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= -1.25e+77) || !(t_1 <= 1e+67)) {
		tmp = log(y) * x;
	} else {
		tmp = (y - log(y)) - t;
	}
	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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))
    if ((t_1 <= (-1.25d+77)) .or. (.not. (t_1 <= 1d+67))) then
        tmp = log(y) * x
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)));
	double tmp;
	if ((t_1 <= -1.25e+77) || !(t_1 <= 1e+67)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))
	tmp = 0
	if (t_1 <= -1.25e+77) or not (t_1 <= 1e+67):
		tmp = math.log(y) * x
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= -1.25e+77) || !(t_1 <= 1e+67))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	tmp = 0.0;
	if ((t_1 <= -1.25e+77) || ~((t_1 <= 1e+67)))
		tmp = log(y) * x;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1.25e+77], N[Not[LessEqual[t$95$1, 1e+67]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -1.25000000000000001e77 or 9.99999999999999983e66 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

         \[\leadsto \color{blue}{\log y} \cdot x \]

      9. lower-log.f64 79.4

         \[\leadsto \color{blue}{\log y} \cdot x \]

   5. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\log y \cdot x} \]

   if -1.25000000000000001e77 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 9.99999999999999983e66

   1. Initial program 84.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(y + -1 \cdot \log y\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 78.9%

       \[\leadsto \left(y - \log y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq -1.25 \cdot 10^{+77} \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 10^{+67}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq -1.25 \cdot 10^{+77} \lor \neg \left(t\_1 \leq 10^{+67}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y))))))
   (if (or (<= t_1 -1.25e+77) (not (<= t_1 1e+67)))
     (* (log y) x)
     (- (- (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	double tmp;
	if ((t_1 <= -1.25e+77) || !(t_1 <= 1e+67)) {
		tmp = log(y) * x;
	} else {
		tmp = -log(y) - t;
	}
	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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))
    if ((t_1 <= (-1.25d+77)) .or. (.not. (t_1 <= 1d+67))) then
        tmp = log(y) * x
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)));
	double tmp;
	if ((t_1 <= -1.25e+77) || !(t_1 <= 1e+67)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))
	tmp = 0
	if (t_1 <= -1.25e+77) or not (t_1 <= 1e+67):
		tmp = math.log(y) * x
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y))))
	tmp = 0.0
	if ((t_1 <= -1.25e+77) || !(t_1 <= 1e+67))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)));
	tmp = 0.0;
	if ((t_1 <= -1.25e+77) || ~((t_1 <= 1e+67)))
		tmp = log(y) * x;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1.25e+77], N[Not[LessEqual[t$95$1, 1e+67]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -1.25000000000000001e77 or 9.99999999999999983e66 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 95.8%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

         \[\leadsto \color{blue}{\log y} \cdot x \]

      9. lower-log.f64 79.4

         \[\leadsto \color{blue}{\log y} \cdot x \]

   5. Applied rewrites 79.4%

      \[\leadsto \color{blue}{\log y \cdot x} \]

   if -1.25000000000000001e77 < (+.f64 (*.f64 (-.f64 x #s(literal 1 binary64)) (log.f64 y)) (*.f64 (-.f64 z #s(literal 1 binary64)) (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 9.99999999999999983e66

   1. Initial program 84.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\log y} - t \]
   10. Step-by-step derivation

   11. Applied rewrites 78.8%

       \[\leadsto \left(-\log y\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq -1.25 \cdot 10^{+77} \lor \neg \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right) \leq 10^{+67}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (- x 1.0) (log y))
   (* (- z 1.0) (* (fma (fma -0.3333333333333333 y -0.5) y -1.0) y)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * (fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y))) - t;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   3. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - \color{blue}{-1 \cdot -1}\right) \cdot y\right)\right) - t \]

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

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right)} \cdot y\right)\right) - t \]

   5. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot y} + \left(\mathsf{neg}\left(-1\right)\right) \cdot -1\right) \cdot y\right)\right) - t \]

   6. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot y + \color{blue}{1} \cdot -1\right) \cdot y\right)\right) - t \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\left(\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot y + \color{blue}{-1}\right) \cdot y\right)\right) - t \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot y - \frac{1}{2}, y, -1\right)} \cdot y\right)\right) - t \]

   9. metadata-eval N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{3} \cdot y - \color{blue}{\frac{1}{2} \cdot 1}, y, -1\right) \cdot y\right)\right) - t \]

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

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}, y, -1\right) \cdot y\right)\right) - t \]

   11. metadata-eval N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{3} \cdot y + \color{blue}{\frac{-1}{2}} \cdot 1, y, -1\right) \cdot y\right)\right) - t \]

   12. metadata-eval N/A

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\frac{-1}{3} \cdot y + \color{blue}{\frac{-1}{2}}, y, -1\right) \cdot y\right)\right) - t \]

   13. lower-fma.f64 99.1

       \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right)}, y, -1\right) \cdot y\right)\right) - t \]

5. Applied rewrites 99.1%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]
6. Add Preprocessing

Alternative 5: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

5. Applied rewrites 99.2%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]

6. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

8. Applied rewrites 99.0%

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

Alternative 6: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

   5. lower-fma.f64 N/A

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. remove-double-neg N/A

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

   9. lower-log.f64 N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. distribute-lft-in N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. mul-1-neg N/A

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

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

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

   17. mul-1-neg N/A

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

   18. distribute-rgt-out N/A

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

5. Applied rewrites 99.0%

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

Alternative 7: 94.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-88}:\\ \;\;\;\;\left(-\log y\right) - \mathsf{fma}\left(z - 1, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.1e+16)
   (- (* (log y) (- x 1.0)) t)
   (if (<= x 2.3e-88)
     (- (- (log y)) (fma (- z 1.0) y t))
     (- (fma (log y) (- x 1.0) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.1e+16) {
		tmp = (log(y) * (x - 1.0)) - t;
	} else if (x <= 2.3e-88) {
		tmp = -log(y) - fma((z - 1.0), y, t);
	} else {
		tmp = fma(log(y), (x - 1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.1e+16)
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	elseif (x <= 2.3e-88)
		tmp = Float64(Float64(-log(y)) - fma(Float64(z - 1.0), y, t));
	else
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.1e+16], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 2.3e-88], N[((-N[Log[y], $MachinePrecision]) - N[(N[(z - 1.0), $MachinePrecision] * y + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1e16

   1. Initial program 95.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower--.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if -2.1e16 < x < 2.29999999999999986e-88

   1. Initial program 79.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.5%

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

   if 2.29999999999999986e-88 < x

   1. Initial program 98.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 97.2%

      \[\leadsto \mathsf{fma}\left(\log y, x - 1, y\right) - \color{blue}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 66.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+66} \lor \neg \left(x - 1 \leq 10^{+75}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -5e+66) (not (<= (- x 1.0) 1e+75)))
   (* (log y) x)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -5e+66) || !((x - 1.0) <= 1e+75)) {
		tmp = log(y) * x;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x - 1.0d0) <= (-5d+66)) .or. (.not. ((x - 1.0d0) <= 1d+75))) then
        tmp = log(y) * x
    else
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -5e+66) || !((x - 1.0) <= 1e+75)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -5e+66) or not ((x - 1.0) <= 1e+75):
		tmp = math.log(y) * x
	else:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -5e+66) || !(Float64(x - 1.0) <= 1e+75))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - 1.0) <= -5e+66) || ~(((x - 1.0) <= 1e+75)))
		tmp = log(y) * x;
	else
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -5e+66], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 1e+75]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.99999999999999991e66 or 9.99999999999999927e74 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

         \[\leadsto \color{blue}{\log y} \cdot x \]

      9. lower-log.f64 80.7

         \[\leadsto \color{blue}{\log y} \cdot x \]

   5. Applied rewrites 80.7%

      \[\leadsto \color{blue}{\log y \cdot x} \]

   if -4.99999999999999991e66 < (-.f64 x #s(literal 1 binary64)) < 9.99999999999999927e74

   1. Initial program 84.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   5. Applied rewrites 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 63.9%

       \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -5 \cdot 10^{+66} \lor \neg \left(x - 1 \leq 10^{+75}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower--.f64 N/A

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

   7. remove-double-neg N/A

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

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

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

   9. log-rec N/A

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

   10. mul-1-neg N/A

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

   11. mul-1-neg N/A

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

   12. log-rec N/A

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

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

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

   14. remove-double-neg N/A

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

   15. *-rgt-identity N/A

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

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

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

5. Applied rewrites 98.8%

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

Alternative 10: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.1 \cdot 10^{+287}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 6.1e+287)
   (- (fma (log y) (- x 1.0) y) t)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.1e+287) {
		tmp = fma(log(y), (x - 1.0), y) - t;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 6.1e+287)
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 6.1e+287], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.10000000000000056e287

   1. Initial program 90.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.5%

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

   if 6.10000000000000056e287 < z

   1. Initial program 33.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   5. Applied rewrites 94.8%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 90.6%

       \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.1 \cdot 10^{+287}:\\ \;\;\;\;\log y \cdot \left(x - 1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 6.1e+287)
   (- (* (log y) (- x 1.0)) t)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.1e+287) {
		tmp = (log(y) * (x - 1.0)) - t;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 6.1d+287) then
        tmp = (log(y) * (x - 1.0d0)) - t
    else
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 6.1e+287) {
		tmp = (Math.log(y) * (x - 1.0)) - t;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 6.1e+287:
		tmp = (math.log(y) * (x - 1.0)) - t
	else:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 6.1e+287)
		tmp = Float64(Float64(log(y) * Float64(x - 1.0)) - t);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 6.1e+287)
		tmp = (log(y) * (x - 1.0)) - t;
	else
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 6.1e+287], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.10000000000000056e287

   1. Initial program 90.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-log.f64 N/A

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

      9. lower--.f64 89.4

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

   5. Applied rewrites 89.4%

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

   if 6.10000000000000056e287 < z

   1. Initial program 33.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   5. Applied rewrites 94.8%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 90.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 90.6%

       \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 42.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+38} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;\left(\left(\left(z - 1\right) \cdot y\right) \cdot y\right) \cdot -0.5 - t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.5e+38) (not (<= t 4500000.0)))
   (- (* (* (* (- z 1.0) y) y) -0.5) t)
   (* (- y) (- z 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.5e+38) || !(t <= 4500000.0)) {
		tmp = ((((z - 1.0) * y) * y) * -0.5) - t;
	} else {
		tmp = -y * (z - 1.0);
	}
	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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.5d+38)) .or. (.not. (t <= 4500000.0d0))) then
        tmp = ((((z - 1.0d0) * y) * y) * (-0.5d0)) - t
    else
        tmp = -y * (z - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.5e+38) || !(t <= 4500000.0)) {
		tmp = ((((z - 1.0) * y) * y) * -0.5) - t;
	} else {
		tmp = -y * (z - 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.5e+38) or not (t <= 4500000.0):
		tmp = ((((z - 1.0) * y) * y) * -0.5) - t
	else:
		tmp = -y * (z - 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.5e+38) || !(t <= 4500000.0))
		tmp = Float64(Float64(Float64(Float64(Float64(z - 1.0) * y) * y) * -0.5) - t);
	else
		tmp = Float64(Float64(-y) * Float64(z - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.5e+38) || ~((t <= 4500000.0)))
		tmp = ((((z - 1.0) * y) * y) * -0.5) - t;
	else
		tmp = -y * (z - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.5e+38], N[Not[LessEqual[t, 4500000.0]], $MachinePrecision]], N[(N[(N[(N[(N[(z - 1.0), $MachinePrecision] * y), $MachinePrecision] * y), $MachinePrecision] * -0.5), $MachinePrecision] - t), $MachinePrecision], N[((-y) * N[(z - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5000000000000001e38 or 4.5e6 < t

   1. Initial program 96.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   5. Applied rewrites 99.9%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 73.6%

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

   if -1.5000000000000001e38 < t < 4.5e6

   1. Initial program 83.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 19.8%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+38} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;\left(\left(\left(z - 1\right) \cdot y\right) \cdot y\right) \cdot -0.5 - t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+38} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.5e+38) (not (<= t 4500000.0))) (- t) (* (- y) (- z 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.5e+38) || !(t <= 4500000.0)) {
		tmp = -t;
	} else {
		tmp = -y * (z - 1.0);
	}
	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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.5d+38)) .or. (.not. (t <= 4500000.0d0))) then
        tmp = -t
    else
        tmp = -y * (z - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.5e+38) || !(t <= 4500000.0)) {
		tmp = -t;
	} else {
		tmp = -y * (z - 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.5e+38) or not (t <= 4500000.0):
		tmp = -t
	else:
		tmp = -y * (z - 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.5e+38) || !(t <= 4500000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * Float64(z - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.5e+38) || ~((t <= 4500000.0)))
		tmp = -t;
	else
		tmp = -y * (z - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.5e+38], N[Not[LessEqual[t, 4500000.0]], $MachinePrecision]], (-t), N[((-y) * N[(z - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5000000000000001e38 or 4.5e6 < t

   1. Initial program 96.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.5000000000000001e38 < t < 4.5e6

   1. Initial program 83.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 81.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 19.8%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+38} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.0% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((((-0.5 * y) - 1.0) * z) * y) - t;
}

Python

def code(x, y, z, t):
	return ((((-0.5 * y) - 1.0) * z) * y) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

5. Applied rewrites 99.2%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]

6. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 45.6%

    \[\leadsto \left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot \color{blue}{y} - t \]
12. Add Preprocessing

Alternative 15: 42.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+38} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.5e+38) (not (<= t 4500000.0))) (- t) (* (- z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.5e+38) || !(t <= 4500000.0)) {
		tmp = -t;
	} else {
		tmp = -z * 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, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1.5d+38)) .or. (.not. (t <= 4500000.0d0))) then
        tmp = -t
    else
        tmp = -z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.5e+38) || !(t <= 4500000.0)) {
		tmp = -t;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.5e+38) or not (t <= 4500000.0):
		tmp = -t
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.5e+38) || !(t <= 4500000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.5e+38) || ~((t <= 4500000.0)))
		tmp = -t;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.5e+38], N[Not[LessEqual[t, 4500000.0]], $MachinePrecision]], (-t), N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5000000000000001e38 or 4.5e6 < t

   1. Initial program 96.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.5000000000000001e38 < t < 4.5e6

   1. Initial program 83.5%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. remove-double-neg N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. log-rec N/A

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

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

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

      14. remove-double-neg N/A

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

      15. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 19.2%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+38} \lor \neg \left(t \leq 4500000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 35.7% accurate, 75.3× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 89.1%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 34.4

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

5. Applied rewrites 34.4%

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

Reproduce

herbie shell --seed 2025017 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))