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

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 19

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return (((x + y) + z) - (z * log(t))) + ((a - 0.5) * b);
}

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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = (((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\\ \mathbf{if}\;x + y \leq -2 \cdot 10^{-155}:\\ \;\;\;\;x + t\_1\\ \mathbf{else}:\\ \;\;\;\;y + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (fma (- a 0.5) b z) (* (log t) z))))
   (if (<= (+ x y) -2e-155) (+ x t_1) (+ y t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a - 0.5), b, z) - (log(t) * z);
	double tmp;
	if ((x + y) <= -2e-155) {
		tmp = x + t_1;
	} else {
		tmp = y + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(a - 0.5), b, z) - Float64(log(t) * z))
	tmp = 0.0
	if (Float64(x + y) <= -2e-155)
		tmp = Float64(x + t_1);
	else
		tmp = Float64(y + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + y), $MachinePrecision], -2e-155], N[(x + t$95$1), $MachinePrecision], N[(y + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < -2.00000000000000003e-155

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 77.6

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 77.6%

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

   if -2.00000000000000003e-155 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 80.2

          \[\leadsto y + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 80.2%

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

Alternative 3: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 2 \cdot 10^{+21}:\\ \;\;\;\;x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 2e+21)
   (+ x (- (fma (- a 0.5) b z) (* (log t) z)))
   (+ (fma (- a 0.5) b y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 2e+21) {
		tmp = x + (fma((a - 0.5), b, z) - (log(t) * z));
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 2e+21)
		tmp = Float64(x + Float64(fma(Float64(a - 0.5), b, z) - Float64(log(t) * z)));
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], 2e+21], N[(x + N[(N[(N[(a - 0.5), $MachinePrecision] * b + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 2e21

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 82.1

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 82.1%

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

   if 2e21 < (+.f64 x y)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

      6. lift--.f64 84.1

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

   4. Applied rewrites 84.1%

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

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \log t \cdot z\\ t_3 := x + \left(\mathsf{fma}\left(-0.5, b, z\right) - t\_2\right)\\ t_4 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+121}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 10^{-93}:\\ \;\;\;\;\left(y + x\right) + \left(z - t\_2\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+49}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b))
        (t_2 (* (log t) z))
        (t_3 (+ x (- (fma -0.5 b z) t_2)))
        (t_4 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -2e+121)
     t_4
     (if (<= t_1 -2e+21)
       t_3
       (if (<= t_1 1e-93)
         (+ (+ y x) (- z t_2))
         (if (<= t_1 2e+49) t_3 t_4))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = log(t) * z;
	double t_3 = x + (fma(-0.5, b, z) - t_2);
	double t_4 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -2e+121) {
		tmp = t_4;
	} else if (t_1 <= -2e+21) {
		tmp = t_3;
	} else if (t_1 <= 1e-93) {
		tmp = (y + x) + (z - t_2);
	} else if (t_1 <= 2e+49) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(log(t) * z)
	t_3 = Float64(x + Float64(fma(-0.5, b, z) - t_2))
	t_4 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -2e+121)
		tmp = t_4;
	elseif (t_1 <= -2e+21)
		tmp = t_3;
	elseif (t_1 <= 1e-93)
		tmp = Float64(Float64(y + x) + Float64(z - t_2));
	elseif (t_1 <= 2e+49)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$3 = N[(x + N[(N[(-0.5 * b + z), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+121], t$95$4, If[LessEqual[t$95$1, -2e+21], t$95$3, If[LessEqual[t$95$1, 1e-93], N[(N[(y + x), $MachinePrecision] + N[(z - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+49], t$95$3, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.00000000000000007e121 or 1.99999999999999989e49 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

      6. lift--.f64 88.5

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

   4. Applied rewrites 88.5%

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

   if -2.00000000000000007e121 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2e21 or 9.999999999999999e-94 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999989e49

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 70.0

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in a around 0

      \[\leadsto x + \left(\mathsf{fma}\left(\frac{-1}{2}, b, z\right) - \log \color{blue}{t} \cdot z\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 62.0%

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

   if -2e21 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 9.999999999999999e-94

   1. Initial program 99.8%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-log.f64 N/A

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

      15. lift-*.f64 96.2

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

   6. Applied rewrites 96.2%

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

Alternative 5: 84.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+68}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+22}:\\ \;\;\;\;\left(y + x\right) + \left(z - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -5e+68)
     t_2
     (if (<= t_1 1e+22) (+ (+ y x) (- z (* (log t) z))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -5e+68) {
		tmp = t_2;
	} else if (t_1 <= 1e+22) {
		tmp = (y + x) + (z - (log(t) * z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -5e+68)
		tmp = t_2;
	elseif (t_1 <= 1e+22)
		tmp = Float64(Float64(y + x) + Float64(z - Float64(log(t) * z)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+68], t$95$2, If[LessEqual[t$95$1, 1e+22], N[(N[(y + x), $MachinePrecision] + N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -5.0000000000000004e68 or 1e22 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

      6. lift--.f64 87.0

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

   4. Applied rewrites 87.0%

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

   if -5.0000000000000004e68 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1e22

   1. Initial program 99.8%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 93.8

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

   4. Applied rewrites 93.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-log.f64 N/A

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

      15. lift-*.f64 93.8

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

   6. Applied rewrites 93.8%

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

Alternative 6: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot z\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, z\right) - t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log t) z)))
   (if (<= z -4.9e+222)
     (- (fma -0.5 b z) t_1)
     (if (<= z 6.6e+191) (+ (fma (- a 0.5) b y) x) (+ x (- z t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(t) * z;
	double tmp;
	if (z <= -4.9e+222) {
		tmp = fma(-0.5, b, z) - t_1;
	} else if (z <= 6.6e+191) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = x + (z - t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(t) * z)
	tmp = 0.0
	if (z <= -4.9e+222)
		tmp = Float64(fma(-0.5, b, z) - t_1);
	elseif (z <= 6.6e+191)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(x + Float64(z - t_1));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.9e+222], N[(N[(-0.5 * b + z), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[z, 6.6e+191], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(z - t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.89999999999999991e222

   1. Initial program 99.5%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 93.9

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 93.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-log.f64 N/A

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

      12. lift-*.f64 88.1

          \[\leadsto \mathsf{fma}\left(b, a - 0.5, z - \log t \cdot z\right) \]

   7. Applied rewrites 88.1%

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

   8. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 72.2

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

   10. Applied rewrites 72.2%

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

   if -4.89999999999999991e222 < z < 6.5999999999999997e191

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

      6. lift--.f64 87.7

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

   4. Applied rewrites 87.7%

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

   if 6.5999999999999997e191 < z

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 91.1

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 68.1%

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

Alternative 7: 82.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(z - \log t \cdot z\right)\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ x (- z (* (log t) z)))))
   (if (<= z -1.65e+173)
     t_1
     (if (<= z 6.6e+191) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (z - (log(t) * z));
	double tmp;
	if (z <= -1.65e+173) {
		tmp = t_1;
	} else if (z <= 6.6e+191) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(z - Float64(log(t) * z)))
	tmp = 0.0
	if (z <= -1.65e+173)
		tmp = t_1;
	elseif (z <= 6.6e+191)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.65e+173], t$95$1, If[LessEqual[z, 6.6e+191], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999998e173 or 6.5999999999999997e191 < z

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 90.9

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 70.2%

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

   if -1.64999999999999998e173 < z < 6.5999999999999997e191

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

      6. lift--.f64 89.6

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

   4. Applied rewrites 89.6%

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

Alternative 8: 79.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -1.05 \cdot 10^{+244}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+263}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -1.05e+244)
     t_1
     (if (<= z 4.1e+263) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -1.05e+244) {
		tmp = t_1;
	} else if (z <= 4.1e+263) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -1.05e+244)
		tmp = t_1;
	elseif (z <= 4.1e+263)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.05e+244], t$95$1, If[LessEqual[z, 4.1e+263], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e244 or 4.09999999999999986e263 < z

   1. Initial program 99.4%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 75.3

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

   4. Applied rewrites 75.3%

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

   if -1.05e244 < z < 4.09999999999999986e263

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

      6. lift--.f64 84.2

         \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

   4. Applied rewrites 84.2%

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

Alternative 9: 78.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a - 0.5, b, y\right) + x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (+ (fma (- a 0.5) b y) x))

C

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

Julia

function code(x, y, z, t, a, b)
	return Float64(fma(Float64(a - 0.5), b, y) + x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, b, y\right) + x \]

   6. lift--.f64 78.6

      \[\leadsto \mathsf{fma}\left(a - 0.5, b, y\right) + x \]

4. Applied rewrites 78.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a - 0.5, b, y\right) + x} \]
5. Add Preprocessing

Alternative 10: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -2 \cdot 10^{-199}:\\ \;\;\;\;x + t\_1\\ \mathbf{else}:\\ \;\;\;\;y + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= (- (+ (+ x y) z) (* z (log t))) -2e-199) (+ x t_1) (+ y t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((((x + y) + z) - (z * log(t))) <= -2e-199) {
		tmp = x + t_1;
	} else {
		tmp = y + 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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if ((((x + y) + z) - (z * log(t))) <= (-2d-199)) then
        tmp = x + t_1
    else
        tmp = y + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if ((((x + y) + z) - (z * Math.log(t))) <= -2e-199) {
		tmp = x + t_1;
	} else {
		tmp = y + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if (((x + y) + z) - (z * math.log(t))) <= -2e-199:
		tmp = x + t_1
	else:
		tmp = y + t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) <= -2e-199)
		tmp = Float64(x + t_1);
	else
		tmp = Float64(y + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if ((((x + y) + z) - (z * log(t))) <= -2e-199)
		tmp = x + t_1;
	else
		tmp = y + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-199], N[(x + t$95$1), $MachinePrecision], N[(y + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -1.99999999999999996e-199

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \left(a - \frac{1}{2}\right) \cdot b \]
   3. Step-by-step derivation

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{x} + \left(a - 0.5\right) \cdot b \]

   if -1.99999999999999996e-199 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 58.1%

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

Alternative 11: 60.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;x + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 5e-16) (+ x (* (- a 0.5) b)) (fma a b y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 5e-16) {
		tmp = x + ((a - 0.5) * b);
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 5e-16)
		tmp = Float64(x + Float64(Float64(a - 0.5) * b));
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], 5e-16], N[(x + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(a * b + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 5.0000000000000004e-16

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \left(a - \frac{1}{2}\right) \cdot b \]
   3. Step-by-step derivation

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{x} + \left(a - 0.5\right) \cdot b \]

   if 5.0000000000000004e-16 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around inf

      \[\leadsto y + \color{blue}{a} \cdot b \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto y + \color{blue}{a} \cdot b \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + a \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot b + y} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot b} + y \]

      4. lower-fma.f64 48.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]

   9. Applied rewrites 48.8%

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

Alternative 12: 58.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -1 \cdot 10^{+39}:\\ \;\;\;\;x + a \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -1e+39)
   (+ x (* a b))
   (if (<= (+ x y) 5e-16) (* (- a 0.5) b) (fma a b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -1e+39) {
		tmp = x + (a * b);
	} else if ((x + y) <= 5e-16) {
		tmp = (a - 0.5) * b;
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -1e+39)
		tmp = Float64(x + Float64(a * b));
	elseif (Float64(x + y) <= 5e-16)
		tmp = Float64(Float64(a - 0.5) * b);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -1e+39], N[(x + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 5e-16], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -9.9999999999999994e38

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \left(a - \frac{1}{2}\right) \cdot b \]
   3. Step-by-step derivation

   4. Applied rewrites 57.2%

      \[\leadsto \color{blue}{x} + \left(a - 0.5\right) \cdot b \]

   5. Taylor expanded in a around inf

      \[\leadsto x + \color{blue}{a} \cdot b \]
   6. Step-by-step derivation

   7. Applied rewrites 49.2%

      \[\leadsto x + \color{blue}{a} \cdot b \]

   if -9.9999999999999994e38 < (+.f64 x y) < 5.0000000000000004e-16

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot \color{blue}{b} \]

      2. lift--.f64 N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b \]

      3. lift-*.f64 55.2

         \[\leadsto \left(a - 0.5\right) \cdot \color{blue}{b} \]

   4. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot b} \]

   if 5.0000000000000004e-16 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around inf

      \[\leadsto y + \color{blue}{a} \cdot b \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto y + \color{blue}{a} \cdot b \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + a \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot b + y} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot b} + y \]

      4. lower-fma.f64 48.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]

   9. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -9 \cdot 10^{+51}:\\ \;\;\;\;x + -0.5 \cdot b\\ \mathbf{elif}\;x + y \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -9e+51)
   (+ x (* -0.5 b))
   (if (<= (+ x y) 5e-16) (* (- a 0.5) b) (fma a b y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -9e+51) {
		tmp = x + (-0.5 * b);
	} else if ((x + y) <= 5e-16) {
		tmp = (a - 0.5) * b;
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -9e+51)
		tmp = Float64(x + Float64(-0.5 * b));
	elseif (Float64(x + y) <= 5e-16)
		tmp = Float64(Float64(a - 0.5) * b);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -9e+51], N[(x + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 5e-16], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -8.9999999999999999e51

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} + \left(a - \frac{1}{2}\right) \cdot b \]
   3. Step-by-step derivation

   4. Applied rewrites 57.2%

      \[\leadsto \color{blue}{x} + \left(a - 0.5\right) \cdot b \]

   5. Taylor expanded in a around 0

      \[\leadsto x + \color{blue}{\frac{-1}{2}} \cdot b \]
   6. Step-by-step derivation

   7. Applied rewrites 37.0%

      \[\leadsto x + \color{blue}{-0.5} \cdot b \]

   if -8.9999999999999999e51 < (+.f64 x y) < 5.0000000000000004e-16

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot \color{blue}{b} \]

      2. lift--.f64 N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b \]

      3. lift-*.f64 54.8

         \[\leadsto \left(a - 0.5\right) \cdot \color{blue}{b} \]

   4. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot b} \]

   if 5.0000000000000004e-16 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in a around inf

      \[\leadsto y + \color{blue}{a} \cdot b \]
   6. Step-by-step derivation

   7. Applied rewrites 48.8%

      \[\leadsto y + \color{blue}{a} \cdot b \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + a \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot b + y} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot b} + y \]

      4. lower-fma.f64 48.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]

   9. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+180}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -2e+138) t_1 (if (<= t_1 4e+180) (+ y x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -2e+138) {
		tmp = t_1;
	} else if (t_1 <= 4e+180) {
		tmp = y + x;
	} 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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-2d+138)) then
        tmp = t_1
    else if (t_1 <= 4d+180) then
        tmp = y + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -2e+138) {
		tmp = t_1;
	} else if (t_1 <= 4e+180) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -2e+138:
		tmp = t_1
	elif t_1 <= 4e+180:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -2e+138)
		tmp = t_1;
	elseif (t_1 <= 4e+180)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -2e+138)
		tmp = t_1;
	elseif (t_1 <= 4e+180)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+138], t$95$1, If[LessEqual[t$95$1, 4e+180], N[(y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e138 or 4e180 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a - \frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot \color{blue}{b} \]

      2. lift--.f64 N/A

         \[\leadsto \left(a - \frac{1}{2}\right) \cdot b \]

      3. lift-*.f64 77.9

         \[\leadsto \left(a - 0.5\right) \cdot \color{blue}{b} \]

   4. Applied rewrites 77.9%

      \[\leadsto \color{blue}{\left(a - 0.5\right) \cdot b} \]

   if -2.0000000000000001e138 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4e180

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 70.2

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 70.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a - \frac{1}{2}, y + x\right) \]

      10. lift-+.f64 70.6

          \[\leadsto \mathsf{fma}\left(b, a - 0.5, y + x\right) \]

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in b around 0

      \[\leadsto x + \color{blue}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + y \]

      2. associate-+l+ N/A

         \[\leadsto x + y \]

      3. +-commutative N/A

         \[\leadsto y + x \]

      4. lift-+.f64 58.2

         \[\leadsto y + x \]

   10. Applied rewrites 58.2%

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

Alternative 15: 50.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+180}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -1e+138) (fma a b y) (if (<= t_1 4e+180) (+ y x) (fma a b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -1e+138) {
		tmp = fma(a, b, y);
	} else if (t_1 <= 4e+180) {
		tmp = y + x;
	} else {
		tmp = fma(a, b, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -1e+138)
		tmp = fma(a, b, y);
	elseif (t_1 <= 4e+180)
		tmp = Float64(y + x);
	else
		tmp = fma(a, b, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+138], N[(a * b + y), $MachinePrecision], If[LessEqual[t$95$1, 4e+180], N[(y + x), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e138 or 4e180 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in a around inf

      \[\leadsto y + \color{blue}{a} \cdot b \]
   6. Step-by-step derivation

   7. Applied rewrites 63.1%

      \[\leadsto y + \color{blue}{a} \cdot b \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{y + a \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot b + y} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot b} + y \]

      4. lower-fma.f64 63.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]

   9. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b, y\right)} \]

   if -1e138 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4e180

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 70.2

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 70.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a - \frac{1}{2}, y + x\right) \]

      10. lift-+.f64 70.6

          \[\leadsto \mathsf{fma}\left(b, a - 0.5, y + x\right) \]

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in b around 0

      \[\leadsto x + \color{blue}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + y \]

      2. associate-+l+ N/A

         \[\leadsto x + y \]

      3. +-commutative N/A

         \[\leadsto y + x \]

      4. lift-+.f64 58.2

         \[\leadsto y + x \]

   10. Applied rewrites 58.2%

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

Alternative 16: 46.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+138}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+180}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -1e+138) (* b a) (if (<= t_1 4e+180) (+ y x) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -1e+138) {
		tmp = b * a;
	} else if (t_1 <= 4e+180) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-1d+138)) then
        tmp = b * a
    else if (t_1 <= 4d+180) then
        tmp = y + x
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -1e+138) {
		tmp = b * a;
	} else if (t_1 <= 4e+180) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -1e+138:
		tmp = b * a
	elif t_1 <= 4e+180:
		tmp = y + x
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -1e+138)
		tmp = Float64(b * a);
	elseif (t_1 <= 4e+180)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -1e+138)
		tmp = b * a;
	elseif (t_1 <= 4e+180)
		tmp = y + x;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+138], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 4e+180], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e138 or 4e180 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.9%

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

   2. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto b \cdot \color{blue}{a} \]

      2. lower-*.f64 56.6

         \[\leadsto b \cdot \color{blue}{a} \]

   4. Applied rewrites 56.6%

      \[\leadsto \color{blue}{b \cdot a} \]

   if -1e138 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4e180

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 70.2

          \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

   4. Applied rewrites 70.2%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 N/A

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

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, a - \frac{1}{2}, y + x\right) \]

      10. lift-+.f64 70.6

          \[\leadsto \mathsf{fma}\left(b, a - 0.5, y + x\right) \]

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in b around 0

      \[\leadsto x + \color{blue}{y} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x + y \]

      2. associate-+l+ N/A

         \[\leadsto x + y \]

      3. +-commutative N/A

         \[\leadsto y + x \]

      4. lift-+.f64 58.2

         \[\leadsto y + x \]

   10. Applied rewrites 58.2%

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

Alternative 17: 42.6% accurate, 7.0× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 (+ y x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return y + x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = y + x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return y + x;
}

Python

def code(x, y, z, t, a, b):
	return y + x

Julia

function code(x, y, z, t, a, b)
	return Float64(y + x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = y + x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(y + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower--.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-log.f64 78.4

       \[\leadsto x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right) \]

4. Applied rewrites 78.4%

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

5. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. associate-+l+ N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift--.f64 N/A

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

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(b, a - \frac{1}{2}, y + x\right) \]

   10. lift-+.f64 78.6

       \[\leadsto \mathsf{fma}\left(b, a - 0.5, y + x\right) \]

7. Applied rewrites 78.6%

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

8. Taylor expanded in b around 0

   \[\leadsto x + \color{blue}{y} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto x + y \]

   2. associate-+l+ N/A

      \[\leadsto x + y \]

   3. +-commutative N/A

      \[\leadsto y + x \]

   4. lift-+.f64 42.6

      \[\leadsto y + x \]

10. Applied rewrites 42.6%

    \[\leadsto y + \color{blue}{x} \]
11. Add Preprocessing

Alternative 18: 22.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{-199}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -2e-199) x y))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((x + y) + z) - (z * log(t))) + ((a - 0.5) * b)) <= -2e-199) {
		tmp = x;
	} 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, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (((((x + y) + z) - (z * log(t))) + ((a - 0.5d0) * b)) <= (-2d-199)) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((((x + y) + z) - (z * Math.log(t))) + ((a - 0.5) * b)) <= -2e-199) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if ((((x + y) + z) - (z * math.log(t))) + ((a - 0.5) * b)) <= -2e-199:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b)) <= -2e-199)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (((((x + y) + z) - (z * log(t))) + ((a - 0.5) * b)) <= -2e-199)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], -2e-199], x, y]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)) < -1.99999999999999996e-199

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 22.6%

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

   if -1.99999999999999996e-199 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 22.5%

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

Alternative 19: 21.9% accurate, 26.1× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 21.9%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))