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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 14

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 2: 69.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(x \cdot \log y - y\right) - z\right) + \log t\\ \mathbf{if}\;t\_1 \leq -20000000000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- (- (* x (log y)) y) z) (log t))))
   (if (<= t_1 -20000000000.0)
     (- (- y) z)
     (if (<= t_1 4e+20) (- (log t) y) (- z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (((x * log(y)) - y) - z) + log(t);
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -y - z;
	} else if (t_1 <= 4e+20) {
		tmp = log(t) - 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) :: t_1
    real(8) :: tmp
    t_1 = (((x * log(y)) - y) - z) + log(t)
    if (t_1 <= (-20000000000.0d0)) then
        tmp = -y - z
    else if (t_1 <= 4d+20) then
        tmp = log(t) - y
    else
        tmp = -z
    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) - z) + Math.log(t);
	double tmp;
	if (t_1 <= -20000000000.0) {
		tmp = -y - z;
	} else if (t_1 <= 4e+20) {
		tmp = Math.log(t) - y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (((x * math.log(y)) - y) - z) + math.log(t)
	tmp = 0
	if t_1 <= -20000000000.0:
		tmp = -y - z
	elif t_1 <= 4e+20:
		tmp = math.log(t) - y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
	tmp = 0.0
	if (t_1 <= -20000000000.0)
		tmp = Float64(Float64(-y) - z);
	elseif (t_1 <= 4e+20)
		tmp = Float64(log(t) - y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (((x * log(y)) - y) - z) + log(t);
	tmp = 0.0;
	if (t_1 <= -20000000000.0)
		tmp = -y - z;
	elseif (t_1 <= 4e+20)
		tmp = log(t) - y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000.0], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$1, 4e+20], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t)) < -2e10

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. 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 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot y - z \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 71.4

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

   8. Applied rewrites 71.4%

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

   if -2e10 < (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t)) < 4e20

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - y \]
   7. Step-by-step derivation

      1. lift-log.f64 93.5

         \[\leadsto \log t - y \]

   8. Applied rewrites 93.5%

      \[\leadsto \log t - y \]

   if 4e20 < (+.f64 (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) (log.f64 t))

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.9

         \[\leadsto -z \]

   5. Applied rewrites 52.9%

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

Alternative 3: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+22}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -1e+22)
     (- (- y) z)
     (if (<= t_1 2e+81) (- (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 <= -1e+22) {
		tmp = -y - z;
	} else if (t_1 <= 2e+81) {
		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 <= (-1d+22)) then
        tmp = -y - z
    else if (t_1 <= 2d+81) 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 <= -1e+22) {
		tmp = -y - z;
	} else if (t_1 <= 2e+81) {
		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 <= -1e+22:
		tmp = -y - z
	elif t_1 <= 2e+81:
		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 <= -1e+22)
		tmp = Float64(Float64(-y) - z);
	elseif (t_1 <= 2e+81)
		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 <= -1e+22)
		tmp = -y - z;
	elseif (t_1 <= 2e+81)
		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, -1e+22], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$1, 2e+81], 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) < -1e22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. 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 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot y - z \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if -1e22 < (-.f64 (*.f64 x (log.f64 y)) y) < 1.99999999999999984e81

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - z} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

      1. lift-log.f64 93.4

         \[\leadsto \log t - z \]

   8. Applied rewrites 93.4%

      \[\leadsto \log t - z \]

   if 1.99999999999999984e81 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 84.4

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

   5. Applied rewrites 84.4%

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

Alternative 4: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -1 \cdot 10^{+22}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) -1e+22) (- (- y) z) (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= -1e+22) {
		tmp = -y - z;
	} else {
		tmp = log(t) - 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 (((x * log(y)) - y) <= (-1d+22)) then
        tmp = -y - z
    else
        tmp = log(t) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * Math.log(y)) - y) <= -1e+22) {
		tmp = -y - z;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * math.log(y)) - y) <= -1e+22:
		tmp = -y - z
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * log(y)) - y) <= -1e+22)
		tmp = Float64(Float64(-y) - z);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * log(y)) - y) <= -1e+22)
		tmp = -y - z;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e22

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. 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 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto -1 \cdot y - z \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.8

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

   8. Applied rewrites 68.8%

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

   if -1e22 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

      1. lift-log.f64 71.2

         \[\leadsto \log t - z \]

   8. Applied rewrites 71.2%

      \[\leadsto \log t - z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.4e+66)
   (fma (log y) x (- (log t) z))
   (fma (log y) x (- (log t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.4e+66) {
		tmp = fma(log(y), x, (log(t) - z));
	} else {
		tmp = fma(log(y), x, (log(t) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.4e+66)
		tmp = fma(log(y), x, Float64(log(t) - z));
	else
		tmp = fma(log(y), x, Float64(log(t) - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.4e+66], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000002e66

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - z} \]
   4. 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 97.3

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

   5. Applied rewrites 97.3%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 2.4000000000000002e66 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 86.4

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

   7. Applied rewrites 86.4%

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

Alternative 6: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.4e+66)
   (- (fma (log y) x (log t)) z)
   (fma (log y) x (- (log t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.4e+66) {
		tmp = fma(log(y), x, log(t)) - z;
	} else {
		tmp = fma(log(y), x, (log(t) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.4e+66)
		tmp = Float64(fma(log(y), x, log(t)) - z);
	else
		tmp = fma(log(y), x, Float64(log(t) - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.4e+66], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000002e66

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - z} \]
   4. 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 97.3

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

   5. Applied rewrites 97.3%

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

   if 2.4000000000000002e66 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 86.3

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

   5. Applied rewrites 86.3%

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

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate--l+ N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 86.4

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

   7. Applied rewrites 86.4%

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

Alternative 7: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.4e+66) (- (fma (log y) x (log t)) z) (- (* (log y) x) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.4e+66) {
		tmp = fma(log(y), x, log(t)) - z;
	} else {
		tmp = (log(y) * x) - y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.4e+66)
		tmp = Float64(fma(log(y), x, log(t)) - z);
	else
		tmp = Float64(Float64(log(y) * x) - y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.4e+66], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000002e66

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - z} \]
   4. 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 97.3

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

   5. Applied rewrites 97.3%

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

   if 2.4000000000000002e66 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 86.3

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

   5. Applied rewrites 86.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y - y \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

Alternative 8: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 9: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+85} \lor \neg \left(x \leq 1.22 \cdot 10^{+67}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.8e+85) (not (<= x 1.22e+67)))
   (- (* (log y) x) y)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.8e+85) || !(x <= 1.22e+67)) {
		tmp = (log(y) * x) - y;
	} else {
		tmp = (log(t) - y) - 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 ((x <= (-4.8d+85)) .or. (.not. (x <= 1.22d+67))) then
        tmp = (log(y) * x) - y
    else
        tmp = (log(t) - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.8e+85) || !(x <= 1.22e+67)) {
		tmp = (Math.log(y) * x) - y;
	} else {
		tmp = (Math.log(t) - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.8e+85) or not (x <= 1.22e+67):
		tmp = (math.log(y) * x) - y
	else:
		tmp = (math.log(t) - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.8e+85) || !(x <= 1.22e+67))
		tmp = Float64(Float64(log(y) * x) - y);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.8e+85) || ~((x <= 1.22e+67)))
		tmp = (log(y) * x) - y;
	else
		tmp = (log(t) - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.8e+85], N[Not[LessEqual[x, 1.22e+67]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.79999999999999993e85 or 1.22000000000000004e67 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y - y \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 85.9

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

   8. Applied rewrites 85.9%

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

   if -4.79999999999999993e85 < x < 1.22000000000000004e67

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. 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 94.5

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

   5. Applied rewrites 94.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+85} \lor \neg \left(x \leq 1.22 \cdot 10^{+67}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+85}:\\ \;\;\;\;t\_1 - y\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+34}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -4.8e+85)
     (- t_1 y)
     (if (<= x 1.1e+34) (- (- (log t) y) z) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -4.8e+85) {
		tmp = t_1 - y;
	} else if (x <= 1.1e+34) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-4.8d+85)) then
        tmp = t_1 - y
    else if (x <= 1.1d+34) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1 - z
    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 <= -4.8e+85) {
		tmp = t_1 - y;
	} else if (x <= 1.1e+34) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -4.8e+85:
		tmp = t_1 - y
	elif x <= 1.1e+34:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -4.8e+85)
		tmp = Float64(t_1 - y);
	elseif (x <= 1.1e+34)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -4.8e+85)
		tmp = t_1 - y;
	elseif (x <= 1.1e+34)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1 - z;
	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, -4.8e+85], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 1.1e+34], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.79999999999999993e85

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y - y \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if -4.79999999999999993e85 < x < 1.1000000000000001e34

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. 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 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.1000000000000001e34 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - z} \]
   4. 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 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y - z \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 88.5

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

   8. Applied rewrites 88.5%

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

Alternative 11: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+155} \lor \neg \left(x \leq 2.25 \cdot 10^{+123}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.4e+155) (not (<= x 2.25e+123)))
   (* (log y) x)
   (- (- (log t) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+155) || !(x <= 2.25e+123)) {
		tmp = log(y) * x;
	} else {
		tmp = (log(t) - y) - 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 ((x <= (-1.4d+155)) .or. (.not. (x <= 2.25d+123))) then
        tmp = log(y) * x
    else
        tmp = (log(t) - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.4e+155) || !(x <= 2.25e+123)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = (Math.log(t) - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.4e+155) or not (x <= 2.25e+123):
		tmp = math.log(y) * x
	else:
		tmp = (math.log(t) - y) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.4e+155) || !(x <= 2.25e+123))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(log(t) - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.4e+155) || ~((x <= 2.25e+123)))
		tmp = log(y) * x;
	else
		tmp = (log(t) - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.4e+155], N[Not[LessEqual[x, 2.25e+123]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000008e155 or 2.24999999999999991e123 < x

   1. Initial program 99.6%

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

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if -1.40000000000000008e155 < x < 2.24999999999999991e123

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. 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

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

   5. Applied rewrites 88.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+155} \lor \neg \left(x \leq 2.25 \cdot 10^{+123}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.9% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+66}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.4e+66) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.4e+66) {
		tmp = -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 <= 2.4d+66) then
        tmp = -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 <= 2.4e+66) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.4e+66:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.4e+66)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.4e+66)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.4e+66], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000002e66

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.0

         \[\leadsto -z \]

   5. Applied rewrites 40.0%

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

   if 2.4000000000000002e66 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.2

         \[\leadsto -y \]

   5. Applied rewrites 69.2%

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

Alternative 13: 58.2% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
4. 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 69.7

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

5. Applied rewrites 69.7%

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

6. Taylor expanded in y around inf

   \[\leadsto -1 \cdot y - z \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 58.5

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

8. Applied rewrites 58.5%

   \[\leadsto \left(-y\right) - z \]
9. Add Preprocessing

Alternative 14: 29.7% 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 99.8%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 30.5

      \[\leadsto -y \]

5. Applied rewrites 30.5%

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

Reproduce

herbie shell --seed 2025037 
(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)))