Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 3.8s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. +-commutative N/A

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

   8. associate--r+ N/A

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

   9. associate--l+ N/A

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

   10. lower--.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. lift-log.f64 N/A

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

   15. lift-log.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.9

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

3. Applied rewrites 99.9%

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

Alternative 2: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - \left(z + y\right)\\ t_2 := \left(\left(x \cdot \log y - y\right) - z\right) + \log t\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) (+ z y)))
        (t_2 (+ (- (- (* x (log y)) y) z) (log t))))
   (if (<= t_2 -2e+26) t_1 (if (<= t_2 1000.0) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - (z + y);
	double t_2 = (((x * log(y)) - y) - z) + log(t);
	double tmp;
	if (t_2 <= -2e+26) {
		tmp = t_1;
	} else if (t_2 <= 1000.0) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (log(y) * x) - (z + y)
    t_2 = (((x * log(y)) - y) - z) + log(t)
    if (t_2 <= (-2d+26)) then
        tmp = t_1
    else if (t_2 <= 1000.0d0) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - (z + y);
	double t_2 = (((x * Math.log(y)) - y) - z) + Math.log(t);
	double tmp;
	if (t_2 <= -2e+26) {
		tmp = t_1;
	} else if (t_2 <= 1000.0) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - (z + y)
	t_2 = (((x * math.log(y)) - y) - z) + math.log(t)
	tmp = 0
	if t_2 <= -2e+26:
		tmp = t_1
	elif t_2 <= 1000.0:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - Float64(z + y))
	t_2 = Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
	tmp = 0.0
	if (t_2 <= -2e+26)
		tmp = t_1;
	elseif (t_2 <= 1000.0)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - (z + y);
	t_2 = (((x * log(y)) - y) - z) + log(t);
	tmp = 0.0;
	if (t_2 <= -2e+26)
		tmp = t_1;
	elseif (t_2 <= 1000.0)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - N[(z + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+26], t$95$1, If[LessEqual[t$95$2, 1000.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t)) < -2.0000000000000001e26 or 1e3 < (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t))

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if -2.0000000000000001e26 < (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t)) < 1e3

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 94.9

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

   4. Applied rewrites 94.9%

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

Alternative 3: 90.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -1e+164)
     (fma (log y) x (- y))
     (if (<= t_1 5e-7) (- (- (log t) y) z) (- (* (log y) x) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -1e+164) {
		tmp = fma(log(y), x, -y);
	} else if (t_1 <= 5e-7) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = (log(y) * x) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -1e+164)
		tmp = fma(log(y), x, Float64(-y));
	elseif (t_1 <= 5e-7)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(Float64(log(y) * x) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+164], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[t$95$1, 5e-7], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e164

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lift-log.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.0

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

   8. Applied rewrites 88.0%

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

   if -1e164 < (-.f64 (*.f64 x (log.f64 y)) y) < 4.99999999999999977e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 88.7

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

   4. Applied rewrites 88.7%

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

   if 4.99999999999999977e-7 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 98.1

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

   6. Applied rewrites 98.1%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 96.8%

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

Alternative 4: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+65}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -5e+65) (- y) (if (<= t_1 5e+14) (- (log t) z) (* (log y) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -5e+65) {
		tmp = -y;
	} else if (t_1 <= 5e+14) {
		tmp = log(t) - z;
	} else {
		tmp = log(y) * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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 * log(y)) - y
    if (t_1 <= (-5d+65)) then
        tmp = -y
    else if (t_1 <= 5d+14) then
        tmp = log(t) - z
    else
        tmp = log(y) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double tmp;
	if (t_1 <= -5e+65) {
		tmp = -y;
	} else if (t_1 <= 5e+14) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if t_1 <= -5e+65:
		tmp = -y
	elif t_1 <= 5e+14:
		tmp = math.log(t) - z
	else:
		tmp = math.log(y) * x
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -5e+65)
		tmp = Float64(-y);
	elseif (t_1 <= 5e+14)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (t_1 <= -5e+65)
		tmp = -y;
	elseif (t_1 <= 5e+14)
		tmp = log(t) - z;
	else
		tmp = log(y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+65], (-y), If[LessEqual[t$95$1, 5e+14], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -4.99999999999999973e65

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 54.5

         \[\leadsto -y \]

   4. Applied rewrites 54.5%

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

   if -4.99999999999999973e65 < (-.f64 (*.f64 x (log.f64 y)) y) < 5e14

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 63.7

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

   6. Applied rewrites 63.7%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 55.3%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\log t} - z \]
   11. Step-by-step derivation

       1. lift-log.f64 84.4

          \[\leadsto \log t - z \]

   12. Applied rewrites 84.4%

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

   if 5e14 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 77.9

         \[\leadsto \log y \cdot x \]

   4. Applied rewrites 77.9%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 270:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - \left(z + y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 270.0) (- (fma (log y) x (log t)) z) (- (* (log y) x) (+ z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 270.0) {
		tmp = fma(log(y), x, log(t)) - z;
	} else {
		tmp = (log(y) * x) - (z + y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 270.0)
		tmp = Float64(fma(log(y), x, log(t)) - z);
	else
		tmp = Float64(Float64(log(y) * x) - Float64(z + y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 270.0], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - N[(z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 270

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 270 < y

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 6: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - z\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 35000000:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) z)))
   (if (<= x -2.1e+22) t_1 (if (<= x 35000000.0) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - z;
	double tmp;
	if (x <= -2.1e+22) {
		tmp = t_1;
	} else if (x <= 35000000.0) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	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 = (log(y) * x) - z
    if (x <= (-2.1d+22)) then
        tmp = t_1
    else if (x <= 35000000.0d0) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - z;
	double tmp;
	if (x <= -2.1e+22) {
		tmp = t_1;
	} else if (x <= 35000000.0) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - z
	tmp = 0
	if x <= -2.1e+22:
		tmp = t_1
	elif x <= 35000000.0:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - z)
	tmp = 0.0
	if (x <= -2.1e+22)
		tmp = t_1;
	elseif (x <= 35000000.0)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - z;
	tmp = 0.0;
	if (x <= -2.1e+22)
		tmp = t_1;
	elseif (x <= 35000000.0)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -2.1e+22], t$95$1, If[LessEqual[x, 35000000.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0999999999999998e22 or 3.5e7 < x

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 80.5%

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

   if -2.0999999999999998e22 < x < 3.5e7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 7: 83.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+69}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+149}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -3.5e+69) t_1 (if (<= x 1.65e+149) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -3.5e+69) {
		tmp = t_1;
	} else if (x <= 1.65e+149) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	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 = log(y) * x
    if (x <= (-3.5d+69)) then
        tmp = t_1
    else if (x <= 1.65d+149) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -3.5e+69) {
		tmp = t_1;
	} else if (x <= 1.65e+149) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -3.5e+69:
		tmp = t_1
	elif x <= 1.65e+149:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -3.5e+69)
		tmp = t_1;
	elseif (x <= 1.65e+149)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -3.5e+69)
		tmp = t_1;
	elseif (x <= 1.65e+149)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.5e+69], t$95$1, If[LessEqual[x, 1.65e+149], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999987e69 or 1.65e149 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 71.3

         \[\leadsto \log y \cdot x \]

   4. Applied rewrites 71.3%

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

   if -3.49999999999999987e69 < x < 1.65e149

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 89.9

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

   4. Applied rewrites 89.9%

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

Alternative 8: 60.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+62}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 5.5e+62) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.5e+62) {
		tmp = log(t) - z;
	} else {
		tmp = -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 (y <= 5.5d+62) then
        tmp = log(t) - z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.5e+62) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5.5e+62:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.5e+62)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.5e+62)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.5e+62], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 5.4999999999999997e62

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

      9. associate--l+ N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. lift-log.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 79.0

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 74.4%

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

   10. Taylor expanded in x around 0

       \[\leadsto \color{blue}{\log t} - z \]
   11. Step-by-step derivation

       1. lift-log.f64 57.7

          \[\leadsto \log t - z \]

   12. Applied rewrites 57.7%

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

   if 5.4999999999999997e62 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.8

         \[\leadsto -y \]

   4. Applied rewrites 63.8%

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

Alternative 9: 48.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+27}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 1.66 \cdot 10^{+56}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+27) (- z) (if (<= z 1.66e+56) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+27) {
		tmp = -z;
	} else if (z <= 1.66e+56) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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 <= (-3.5d+27)) then
        tmp = -z
    else if (z <= 1.66d+56) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+27) {
		tmp = -z;
	} else if (z <= 1.66e+56) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.5e+27:
		tmp = -z
	elif z <= 1.66e+56:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+27)
		tmp = Float64(-z);
	elseif (z <= 1.66e+56)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.5e+27)
		tmp = -z;
	elseif (z <= 1.66e+56)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+27], (-z), If[LessEqual[z, 1.66e+56], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000002e27 or 1.6600000000000001e56 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 62.4

         \[\leadsto -z \]

   4. Applied rewrites 62.4%

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

   if -3.5000000000000002e27 < z < 1.6600000000000001e56

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.6

         \[\leadsto -y \]

   4. Applied rewrites 37.6%

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

Alternative 10: 30.1% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, 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 = -y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.1

      \[\leadsto -y \]

4. Applied rewrites 30.1%

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

Reproduce

herbie shell --seed 2025105 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))