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

Percentage Accurate: 99.9% → 99.9%

Time: 7.3s

Alternatives: 17

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 99.8%

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

Alternative 2: 89.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+158} \lor \neg \left(t\_1 \leq 4\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) + z\right) - \log t \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -2e+158) (not (<= t_1 4.0)))
     (+ (fma (- a 0.5) b y) x)
     (- (+ (+ y x) z) (* (log t) z)))))

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+158) || !(t_1 <= 4.0)) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = ((y + x) + z) - (log(t) * z);
	}
	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+158) || !(t_1 <= 4.0))
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(Float64(y + x) + z) - Float64(log(t) * z));
	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[Or[LessEqual[t$95$1, -2e+158], N[Not[LessEqual[t$95$1, 4.0]], $MachinePrecision]], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1.99999999999999991e158 or 4 < (*.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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 88.9

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

   5. Applied rewrites 88.9%

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

   if -1.99999999999999991e158 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   4. 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 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -2 \cdot 10^{+158} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 4\right):\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) + z\right) - \log t \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+118} \lor \neg \left(z \leq 2.4 \cdot 10^{+121}\right):\\ \;\;\;\;\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -7.8e+118) (not (<= z 2.4e+121)))
   (+ (- z (* z (log t))) (* (- a 0.5) b))
   (+ (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 ((z <= -7.8e+118) || !(z <= 2.4e+121)) {
		tmp = (z - (z * log(t))) + ((a - 0.5) * b);
	} else {
		tmp = fma((a - 0.5), b, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -7.8e+118) || !(z <= 2.4e+121))
		tmp = Float64(Float64(z - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b));
	else
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -7.8e+118], N[Not[LessEqual[z, 2.4e+121]], $MachinePrecision]], N[(N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.8e118 or 2.4e121 < 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. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 89.5%

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

   if -7.8e118 < z < 2.4e121

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+118} \lor \neg \left(z \leq 2.4 \cdot 10^{+121}\right):\\ \;\;\;\;\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= (- (+ (+ x y) z) (* z (log t))) -5e-68) (+ 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))) <= -5e-68) {
		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))) <= (-5d-68)) 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))) <= -5e-68) {
		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))) <= -5e-68:
		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))) <= -5e-68)
		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))) <= -5e-68)
		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], -5e-68], 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))) < -4.99999999999999971e-68

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 58.9%

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

   if -4.99999999999999971e-68 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 59.6%

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

Alternative 5: 21.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{-68}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -5e-68) 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)) <= -5e-68) {
		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)) <= (-5d-68)) 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)) <= -5e-68) {
		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)) <= -5e-68:
		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)) <= -5e-68)
		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)) <= -5e-68)
		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], -5e-68], 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)) < -4.99999999999999971e-68

   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. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 20.2%

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

   if -4.99999999999999971e-68 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 21.7%

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

Alternative 6: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot z\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(a, b, z - t\_1\right)\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y + x\right) + z\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log t) z)))
   (if (<= z -8.5e+118)
     (fma a b (- z t_1))
     (if (<= z 4.5e+138) (+ (fma (- a 0.5) b y) x) (- (+ (+ y 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 <= -8.5e+118) {
		tmp = fma(a, b, (z - t_1));
	} else if (z <= 4.5e+138) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = ((y + 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 <= -8.5e+118)
		tmp = fma(a, b, Float64(z - t_1));
	elseif (z <= 4.5e+138)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(Float64(y + x) + 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, -8.5e+118], N[(a * b + N[(z - t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.5e+138], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.50000000000000033e118

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 85.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 85.1

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 85.1

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

   10. Applied rewrites 85.1%

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

   if -8.50000000000000033e118 < z < 4.49999999999999982e138

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 94.1

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

   5. Applied rewrites 94.1%

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

   if 4.49999999999999982e138 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\left(x + \left(y + z\right)\right) - z \cdot \log t} \]
   4. 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 80.2

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

   5. Applied rewrites 80.2%

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

Alternative 7: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+197} \lor \neg \left(z \leq 5.2 \cdot 10^{+138}\right):\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.5e+197) (not (<= z 5.2e+138)))
   (* (- 1.0 (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 ((z <= -3.5e+197) || !(z <= 5.2e+138)) {
		tmp = (1.0 - 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 ((z <= -3.5e+197) || !(z <= 5.2e+138))
		tmp = Float64(Float64(1.0 - 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[Or[LessEqual[z, -3.5e+197], N[Not[LessEqual[z, 5.2e+138]], $MachinePrecision]], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.49999999999999999e197 or 5.2000000000000002e138 < 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. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(1 - \log t\right)} \]
   4. 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 70.7

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

   5. Applied rewrites 70.7%

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

   if -3.49999999999999999e197 < z < 5.2000000000000002e138

   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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 90.6

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

   5. Applied rewrites 90.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+197} \lor \neg \left(z \leq 5.2 \cdot 10^{+138}\right):\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+220}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+122}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+255}:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -1e+220)
     (* b a)
     (if (<= t_1 1e+122) (+ y x) (if (<= t_1 5e+255) (* -0.5 b) (* 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+220) {
		tmp = b * a;
	} else if (t_1 <= 1e+122) {
		tmp = y + x;
	} else if (t_1 <= 5e+255) {
		tmp = -0.5 * b;
	} 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+220)) then
        tmp = b * a
    else if (t_1 <= 1d+122) then
        tmp = y + x
    else if (t_1 <= 5d+255) then
        tmp = (-0.5d0) * b
    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+220) {
		tmp = b * a;
	} else if (t_1 <= 1e+122) {
		tmp = y + x;
	} else if (t_1 <= 5e+255) {
		tmp = -0.5 * b;
	} 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+220:
		tmp = b * a
	elif t_1 <= 1e+122:
		tmp = y + x
	elif t_1 <= 5e+255:
		tmp = -0.5 * b
	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+220)
		tmp = Float64(b * a);
	elseif (t_1 <= 1e+122)
		tmp = Float64(y + x);
	elseif (t_1 <= 5e+255)
		tmp = Float64(-0.5 * b);
	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+220)
		tmp = b * a;
	elseif (t_1 <= 1e+122)
		tmp = y + x;
	elseif (t_1 <= 5e+255)
		tmp = -0.5 * b;
	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+220], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+122], N[(y + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+255], N[(-0.5 * b), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e220 or 5.0000000000000002e255 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.8

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

   5. Applied rewrites 71.8%

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

   if -1e220 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.00000000000000001e122

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 51.2%

      \[\leadsto y + x \]

   if 1.00000000000000001e122 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.0000000000000002e255

   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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 62.2

         \[\leadsto \left(a - 0.5\right) \cdot b \]

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 48.4%

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

Alternative 9: 70.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;a - 0.5 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- a 0.5) -1e+130)
   (fma a b x)
   (if (<= (- a 0.5) -5e+18)
     (fma a b y)
     (if (<= (- a 0.5) 2e+19) (+ (fma -0.5 b y) x) (fma a b x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a - 0.5) <= -1e+130) {
		tmp = fma(a, b, x);
	} else if ((a - 0.5) <= -5e+18) {
		tmp = fma(a, b, y);
	} else if ((a - 0.5) <= 2e+19) {
		tmp = fma(-0.5, b, y) + x;
	} else {
		tmp = fma(a, b, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(a - 0.5) <= -1e+130)
		tmp = fma(a, b, x);
	elseif (Float64(a - 0.5) <= -5e+18)
		tmp = fma(a, b, y);
	elseif (Float64(a - 0.5) <= 2e+19)
		tmp = Float64(fma(-0.5, b, y) + x);
	else
		tmp = fma(a, b, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+130], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+18], N[(a * b + y), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], 2e+19], N[(N[(-0.5 * b + y), $MachinePrecision] + x), $MachinePrecision], N[(a * b + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1.0000000000000001e130 or 2e19 < (-.f64 a #s(literal 1/2 binary64))

   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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 74.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 74.0

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 74.0

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

   10. Applied rewrites 74.0%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 69.7%

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

   if -1.0000000000000001e130 < (-.f64 a #s(literal 1/2 binary64)) < -5e18

   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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 74.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 74.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 74.1

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 74.1

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

   10. Applied rewrites 74.1%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{y}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 59.2%

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

   if -5e18 < (-.f64 a #s(literal 1/2 binary64)) < 2e19

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 71.3

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

   5. Applied rewrites 71.3%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, b, y\right) + x \]
   7. Step-by-step derivation

   8. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(-0.5, b, y\right) + x \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;a - 0.5 \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \mathbf{elif}\;a - 0.5 \leq 2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+220} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+219}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (or (<= t_1 -1e+220) (not (<= t_1 2e+219))) (* b a) (+ y x))))

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+220) || !(t_1 <= 2e+219)) {
		tmp = b * a;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, 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+220)) .or. (.not. (t_1 <= 2d+219))) then
        tmp = b * a
    else
        tmp = y + x
    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+220) || !(t_1 <= 2e+219)) {
		tmp = b * a;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if (t_1 <= -1e+220) or not (t_1 <= 2e+219):
		tmp = b * a
	else:
		tmp = y + x
	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+220) || !(t_1 <= 2e+219))
		tmp = Float64(b * a);
	else
		tmp = Float64(y + x);
	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+220) || ~((t_1 <= 2e+219)))
		tmp = b * a;
	else
		tmp = y + x;
	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[Or[LessEqual[t$95$1, -1e+220], N[Not[LessEqual[t$95$1, 2e+219]], $MachinePrecision]], N[(b * a), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -1e220 or 1.99999999999999993e219 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 100.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -1e220 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 1.99999999999999993e219

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 48.1%

      \[\leadsto y + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a - 0.5\right) \cdot b \leq -1 \cdot 10^{+220} \lor \neg \left(\left(a - 0.5\right) \cdot b \leq 2 \cdot 10^{+219}\right):\\ \;\;\;\;b \cdot a\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.500000005 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= (- a 0.5) -0.500000005) (not (<= (- a 0.5) -0.4)))
   (+ (fma a b y) x)
   (+ (fma -0.5 b y) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((a - 0.5) <= -0.500000005) || !((a - 0.5) <= -0.4)) {
		tmp = fma(a, b, y) + x;
	} else {
		tmp = fma(-0.5, b, y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -0.500000005) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(fma(a, b, y) + x);
	else
		tmp = Float64(fma(-0.5, b, y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -0.500000005], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(-0.5 * b + y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -0.50000000499999997 or -0.40000000000000002 < (-.f64 a #s(literal 1/2 binary64))

   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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(a, b, y\right) + x \]
   7. Step-by-step derivation

   8. Applied rewrites 81.8%

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

   if -0.50000000499999997 < (-.f64 a #s(literal 1/2 binary64)) < -0.40000000000000002

   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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, b, y\right) + x \]
   7. Step-by-step derivation

   8. Applied rewrites 70.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.500000005 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, y\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.5% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -6 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 10^{+62}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -6e+35)
   (fma a b x)
   (if (<= (+ x y) 1e+62) (* (- 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) <= -6e+35) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= 1e+62) {
		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) <= -6e+35)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= 1e+62)
		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], -6e+35], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+62], 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) < -5.99999999999999981e35

   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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 42.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 42.6

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 42.6

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

   10. Applied rewrites 42.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 46.4%

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

   if -5.99999999999999981e35 < (+.f64 x y) < 1.00000000000000004e62

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 60.7

         \[\leadsto \left(a - 0.5\right) \cdot b \]

   5. Applied rewrites 60.7%

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

   if 1.00000000000000004e62 < (+.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. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 36.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 36.5

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 36.5

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

   10. Applied rewrites 36.5%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{y}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 52.2%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -6 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 10^{+62}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 0.5:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -5e-90)
   (fma a b x)
   (if (<= (+ x y) 0.5) (* -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-90) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= 0.5) {
		tmp = -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-90)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= 0.5)
		tmp = Float64(-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-90], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.5], N[(-0.5 * b), $MachinePrecision], N[(a * b + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -5.00000000000000019e-90

   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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 60.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 47.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 47.6

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 47.6

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

   10. Applied rewrites 47.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.8%

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

   if -5.00000000000000019e-90 < (+.f64 x y) < 0.5

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 63.1

         \[\leadsto \left(a - 0.5\right) \cdot b \]

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{-1}{2} \cdot b \]
   7. Step-by-step derivation

   8. Applied rewrites 38.4%

      \[\leadsto -0.5 \cdot b \]

   if 0.5 < (+.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. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 42.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 42.0

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 42.0

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

   10. Applied rewrites 42.0%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{y}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 48.7%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 0.5:\\ \;\;\;\;-0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, b, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 5e+80) (fma a b x) (+ y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 5e+80) {
		tmp = fma(a, b, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 5e+80)
		tmp = fma(a, b, x);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], 5e+80], N[(a * b + x), $MachinePrecision], N[(y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x y) < 4.99999999999999961e80

   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. Add Preprocessing

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 54.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 54.6

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

      5. lift-*.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-log.f64 N/A

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

      9. lift-*.f64 54.6

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

   10. Applied rewrites 54.6%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(a, b, \color{blue}{x}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 40.3%

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

   if 4.99999999999999961e80 < (+.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. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
   4. 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 83.0

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 60.3%

      \[\leadsto y + x \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + y \leq 5 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 78.4% accurate, 9.7× 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 99.8%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
4. 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 75.6

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

5. Applied rewrites 75.6%

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

Alternative 16: 41.9% accurate, 31.5× 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 99.8%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
4. 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 75.6

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

5. Applied rewrites 75.6%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 36.8%

   \[\leadsto y + x \]
9. Add Preprocessing

Alternative 17: 21.8% accurate, 126.0× 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 99.8%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 18.3%

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

Developer Target 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(x + y\right) + \frac{\left(1 - {\log t}^{2}\right) \cdot z}{1 + \log t}\right) + \left(a - 0.5\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  (+ (+ x y) (/ (* (- 1.0 (pow (log t) 2.0)) z) (+ 1.0 (log t))))
  (* (- a 0.5) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x + y) + (((1.0 - pow(log(t), 2.0)) * z) / (1.0 + 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) + (((1.0d0 - (log(t) ** 2.0d0)) * z) / (1.0d0 + 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) + (((1.0 - Math.pow(Math.log(t), 2.0)) * z) / (1.0 + Math.log(t)))) + ((a - 0.5) * b);
}

Python

def code(x, y, z, t, a, b):
	return ((x + y) + (((1.0 - math.pow(math.log(t), 2.0)) * z) / (1.0 + math.log(t)))) + ((a - 0.5) * b)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(x + y) + Float64(Float64(Float64(1.0 - (log(t) ^ 2.0)) * z) / Float64(1.0 + log(t)))) + Float64(Float64(a - 0.5) * b))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((x + y) + (((1.0 - (log(t) ^ 2.0)) * z) / (1.0 + log(t)))) + ((a - 0.5) * b);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(x + y), $MachinePrecision] + N[(N[(N[(1.0 - N[Power[N[Log[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] / N[(1.0 + N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025032 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 1/2) b)))

  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))