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

Percentage Accurate: 99.8% → 99.9%

Time: 7.0s

Alternatives: 21

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: 99.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] + 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. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. *-commutative N/A

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(b, a - 0.5, y\right)\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- 1.0 (log t)) z (fma b (- a 0.5) y))))
   (if (<= z -1.25e+21)
     t_1
     (if (<= z 9.4e+157) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((1.0 - log(t)), z, fma(b, (a - 0.5), y));
	double tmp;
	if (z <= -1.25e+21) {
		tmp = t_1;
	} else if (z <= 9.4e+157) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(1.0 - log(t)), z, fma(b, Float64(a - 0.5), y))
	tmp = 0.0
	if (z <= -1.25e+21)
		tmp = t_1;
	elseif (z <= 9.4e+157)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.25e+21], t$95$1, If[LessEqual[z, 9.4e+157], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e21 or 9.40000000000000061e157 < z

   1. Initial program 99.5%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lift--.f64 86.8

          \[\leadsto \mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(b, a - 0.5, y\right)\right) \]

   7. Applied rewrites 86.8%

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

   if -1.25e21 < z < 9.40000000000000061e157

   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. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 92.9

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

   4. Applied rewrites 92.9%

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

Alternative 3: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(z + y\right) - \log t \cdot z\right) + a \cdot b\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (- (+ z y) (* (log t) z)) (* a b))))
   (if (<= z -3.8e+138)
     t_1
     (if (<= z 1.05e+158) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((z + y) - (log(t) * z)) + (a * b);
	double tmp;
	if (z <= -3.8e+138) {
		tmp = t_1;
	} else if (z <= 1.05e+158) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(z + y) - Float64(log(t) * z)) + Float64(a * b))
	tmp = 0.0
	if (z <= -3.8e+138)
		tmp = t_1;
	elseif (z <= 1.05e+158)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(z + y), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e+138], t$95$1, If[LessEqual[z, 1.05e+158], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000012e138 or 1.0499999999999999e158 < z

   1. Initial program 99.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 32.8%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-log.f64 N/A

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

      10. lift-*.f64 84.2

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

   10. Applied rewrites 84.2%

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

   if -3.80000000000000012e138 < z < 1.0499999999999999e158

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 91.1

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

   4. Applied rewrites 91.1%

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

Alternative 4: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+148}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z + a \cdot b\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(z - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.2e+148)
   (+ (* (- 1.0 (log t)) z) (* a b))
   (if (<= z 7.8e+162)
     (+ (fma (- a 0.5) b y) x)
     (+ (- z (* z (log t))) (* (- a 0.5) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.2e+148) {
		tmp = ((1.0 - log(t)) * z) + (a * b);
	} else if (z <= 7.8e+162) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = (z - (z * log(t))) + ((a - 0.5) * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.2e+148)
		tmp = Float64(Float64(Float64(1.0 - log(t)) * z) + Float64(a * b));
	elseif (z <= 7.8e+162)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(z - Float64(z * log(t))) + Float64(Float64(a - 0.5) * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.2e+148], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.8e+162], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(z - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.20000000000000013e148

   1. Initial program 99.6%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 35.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 29.5%

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

   8. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 74.3

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

   10. Applied rewrites 74.3%

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

   if -7.20000000000000013e148 < z < 7.80000000000000079e162

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.7

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

   4. Applied rewrites 90.7%

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

   if 7.80000000000000079e162 < z

   1. Initial program 99.1%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 80.6%

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

Alternative 5: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z + a \cdot b\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (* (- 1.0 (log t)) z) (* a b))))
   (if (<= z -7.2e+148) t_1 (if (<= z 5e+164) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((1.0 - log(t)) * z) + (a * b);
	double tmp;
	if (z <= -7.2e+148) {
		tmp = t_1;
	} else if (z <= 5e+164) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(1.0 - log(t)) * z) + Float64(a * b))
	tmp = 0.0
	if (z <= -7.2e+148)
		tmp = t_1;
	elseif (z <= 5e+164)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+148], t$95$1, If[LessEqual[z, 5e+164], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.20000000000000013e148 or 4.9999999999999995e164 < z

   1. Initial program 99.4%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 31.9%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 27.5%

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

   8. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 75.9

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

   10. Applied rewrites 75.9%

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

   if -7.20000000000000013e148 < z < 4.9999999999999995e164

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 6: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-67}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000000000:\\ \;\;\;\;\left(\left(y + x\right) + z\right) - \log t \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (+ (fma (- a 0.5) b y) x)))
   (if (<= t_1 -5e-67)
     t_2
     (if (<= t_1 2000000000.0) (- (+ (+ y x) z) (* (log t) z)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double t_2 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (t_1 <= -5e-67) {
		tmp = t_2;
	} else if (t_1 <= 2000000000.0) {
		tmp = ((y + x) + z) - (log(t) * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	t_2 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (t_1 <= -5e-67)
		tmp = t_2;
	elseif (t_1 <= 2000000000.0)
		tmp = Float64(Float64(Float64(y + x) + z) - Float64(log(t) * z));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-67], t$95$2, If[LessEqual[t$95$1, 2000000000.0], N[(N[(N[(y + x), $MachinePrecision] + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.9999999999999999e-67 or 2e9 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 84.1

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

   4. Applied rewrites 84.1%

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

   if -4.9999999999999999e-67 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2e9

   1. Initial program 99.8%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 96.8

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

   4. Applied rewrites 96.8%

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

Alternative 7: 85.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot z\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+156}:\\ \;\;\;\;\left(x + z\right) - t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (log t) z)))
   (if (<= z -4.2e+156)
     (- (+ x z) t_1)
     (if (<= z 6e+158) (+ (fma (- a 0.5) b y) x) (- (+ y z) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log(t) * z;
	double tmp;
	if (z <= -4.2e+156) {
		tmp = (x + z) - t_1;
	} else if (z <= 6e+158) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = (y + z) - t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(log(t) * z)
	tmp = 0.0
	if (z <= -4.2e+156)
		tmp = Float64(Float64(x + z) - t_1);
	elseif (z <= 6e+158)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(Float64(y + z) - t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.2e+156], N[(N[(x + z), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[z, 6e+158], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(y + z), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.19999999999999963e156

   1. Initial program 99.6%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 74.9

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 66.9%

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

   if -4.19999999999999963e156 < z < 6e158

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.6

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

   4. Applied rewrites 90.6%

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

   if 6e158 < z

   1. Initial program 99.2%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 79.7

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

   4. Applied rewrites 79.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 70.6%

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

Alternative 8: 85.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + z\right) - \log t \cdot z\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.9 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ x z) (* (log t) z))))
   (if (<= z -4.2e+156)
     t_1
     (if (<= z 7.9e+164) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + z) - (log(t) * z);
	double tmp;
	if (z <= -4.2e+156) {
		tmp = t_1;
	} else if (z <= 7.9e+164) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + z) - Float64(log(t) * z))
	tmp = 0.0
	if (z <= -4.2e+156)
		tmp = t_1;
	elseif (z <= 7.9e+164)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.2e+156], t$95$1, If[LessEqual[z, 7.9e+164], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999963e156 or 7.90000000000000037e164 < z

   1. Initial program 99.3%

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

   2. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. lower--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-log.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 68.7%

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

   if -4.19999999999999963e156 < z < 7.90000000000000037e164

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.6

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

   4. 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. Add Preprocessing

Alternative 9: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+164}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -6.5e+156) t_1 (if (<= z 9e+164) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -6.5e+156) {
		tmp = t_1;
	} else if (z <= 9e+164) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -6.5e+156)
		tmp = t_1;
	elseif (z <= 9e+164)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -6.5e+156], t$95$1, If[LessEqual[z, 9e+164], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.50000000000000027e156 or 8.9999999999999995e164 < z

   1. Initial program 99.3%

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

   2. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lift-log.f64 59.6

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

   4. Applied rewrites 59.6%

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

   if -6.50000000000000027e156 < z < 8.9999999999999995e164

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift--.f64 90.6

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

   4. 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. Add Preprocessing

Alternative 10: 78.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lift--.f64 78.0

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

4. Applied rewrites 78.0%

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

Alternative 11: 57.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -5 \cdot 10^{-100}:\\ \;\;\;\;x + t\_1\\ \mathbf{else}:\\ \;\;\;\;y + t\_1\\ \end{array} \end{array} \]

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-100) (+ 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-100) {
		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-100)) 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-100) {
		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-100:
		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-100)
		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-100)
		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-100], 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))) < -5.0000000000000001e-100

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 55.8%

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

   if -5.0000000000000001e-100 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. Taylor expanded in y around inf

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

   4. Applied rewrites 59.1%

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

Alternative 12: 57.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq 0.2:\\ \;\;\;\;x + \left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) 0.2)
   (+ x (* (- a 0.5) b))
   (+ y (* a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((((x + y) + z) - (z * log(t))) <= 0.2) {
		tmp = x + ((a - 0.5) * b);
	} else {
		tmp = y + (a * b);
	}
	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))) <= 0.2d0) then
        tmp = x + ((a - 0.5d0) * b)
    else
        tmp = y + (a * b)
    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))) <= 0.2) {
		tmp = x + ((a - 0.5) * b);
	} else {
		tmp = y + (a * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (((x + y) + z) - (z * math.log(t))) <= 0.2:
		tmp = x + ((a - 0.5) * b)
	else:
		tmp = y + (a * b)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(Float64(x + y) + z) - Float64(z * log(t))) <= 0.2)
		tmp = Float64(x + Float64(Float64(a - 0.5) * b));
	else
		tmp = Float64(y + Float64(a * b));
	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))) <= 0.2)
		tmp = x + ((a - 0.5) * b);
	else
		tmp = y + (a * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(N[(x + y), $MachinePrecision] + z), $MachinePrecision] - N[(z * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(x + N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < 0.20000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 59.6%

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

   if 0.20000000000000001 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   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. Taylor expanded in x around inf

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 46.3%

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

   8. Taylor expanded in y around inf

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

      1. associate--l+ 44.9

         \[\leadsto y + a \cdot b \]

      2. +-commutative 44.9

         \[\leadsto y + a \cdot b \]

      3. *-commutative 44.9

         \[\leadsto y + a \cdot b \]

      4. associate--l+ 44.9

         \[\leadsto y + a \cdot b \]

   10. Applied rewrites 44.9%

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

Alternative 13: 56.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 0.2:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + a \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -5e-45)
   (fma a b x)
   (if (<= (+ x y) 0.2) (* (- a 0.5) b) (+ y (* a b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -5e-45) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= 0.2) {
		tmp = (a - 0.5) * b;
	} else {
		tmp = y + (a * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -5e-45)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= 0.2)
		tmp = Float64(Float64(a - 0.5) * b);
	else
		tmp = Float64(y + Float64(a * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-45], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.2], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(y + N[(a * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -4.99999999999999976e-45

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 46.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 46.7

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

      5. associate--l+ 46.7

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

      6. +-commutative 46.7

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

      7. *-commutative 46.7

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

      8. associate--l+ 46.7

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

   9. Applied rewrites 46.7%

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

   if -4.99999999999999976e-45 < (+.f64 x y) < 0.20000000000000001

   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. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 55.9

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

   4. Applied rewrites 55.9%

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

   if 0.20000000000000001 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 49.4%

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

   8. Taylor expanded in y around inf

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

      1. associate--l+ 47.8

         \[\leadsto y + a \cdot b \]

      2. +-commutative 47.8

         \[\leadsto y + a \cdot b \]

      3. *-commutative 47.8

         \[\leadsto y + a \cdot b \]

      4. associate--l+ 47.8

         \[\leadsto y + a \cdot b \]

   10. Applied rewrites 47.8%

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

Alternative 14: 53.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 10^{+42}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + -0.5 \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -5e-45)
   (fma a b x)
   (if (<= (+ x y) 1e+42) (* (- a 0.5) b) (+ y (* -0.5 b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -5e-45) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= 1e+42) {
		tmp = (a - 0.5) * b;
	} else {
		tmp = y + (-0.5 * b);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -5e-45)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= 1e+42)
		tmp = Float64(Float64(a - 0.5) * b);
	else
		tmp = Float64(y + Float64(-0.5 * b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-45], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+42], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(y + N[(-0.5 * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -4.99999999999999976e-45

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 46.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 46.7

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

      5. associate--l+ 46.7

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

      6. +-commutative 46.7

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

      7. *-commutative 46.7

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

      8. associate--l+ 46.7

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

   9. Applied rewrites 46.7%

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

   if -4.99999999999999976e-45 < (+.f64 x y) < 1.00000000000000004e42

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 55.2

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

   4. Applied rewrites 55.2%

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

   if 1.00000000000000004e42 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 58.4%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 37.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 35.8%

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

Alternative 15: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq -5 \cdot 10^{-45}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 10^{+42}:\\ \;\;\;\;\left(a - 0.5\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) -5e-45)
   (fma a b x)
   (if (<= (+ x y) 1e+42) (* (- a 0.5) b) (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= -5e-45) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= 1e+42) {
		tmp = (a - 0.5) * b;
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= -5e-45)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= 1e+42)
		tmp = Float64(Float64(a - 0.5) * b);
	else
		tmp = Float64(y + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(x + y), $MachinePrecision], -5e-45], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 1e+42], N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < -4.99999999999999976e-45

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 56.9%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 46.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 46.7

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

      5. associate--l+ 46.7

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

      6. +-commutative 46.7

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

      7. *-commutative 46.7

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

      8. associate--l+ 46.7

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

   9. Applied rewrites 46.7%

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

   if -4.99999999999999976e-45 < (+.f64 x y) < 1.00000000000000004e42

   1. Initial program 99.8%

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

   2. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 55.2

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

   4. Applied rewrites 55.2%

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

   if 1.00000000000000004e42 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 54.7%

      \[\leadsto y + x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + y \leq 2 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(a, b, x\right)\\ \mathbf{elif}\;x + y \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ x y) 2e-110)
   (fma a b x)
   (if (<= (+ x y) 0.2) (fma -0.5 b x) (+ y x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((x + y) <= 2e-110) {
		tmp = fma(a, b, x);
	} else if ((x + y) <= 0.2) {
		tmp = fma(-0.5, b, x);
	} else {
		tmp = y + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(x + y) <= 2e-110)
		tmp = fma(a, b, x);
	elseif (Float64(x + y) <= 0.2)
		tmp = fma(-0.5, 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], 2e-110], N[(a * b + x), $MachinePrecision], If[LessEqual[N[(x + y), $MachinePrecision], 0.2], N[(-0.5 * b + x), $MachinePrecision], N[(y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x y) < 2.0000000000000001e-110

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 57.6%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 44.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 44.5

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

      5. associate--l+ 44.5

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

      6. +-commutative 44.5

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

      7. *-commutative 44.5

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

      8. associate--l+ 44.5

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

   9. Applied rewrites 44.5%

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

   if 2.0000000000000001e-110 < (+.f64 x y) < 0.20000000000000001

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 57.9%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 27.1%

      \[\leadsto x + \color{blue}{-0.5} \cdot b \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b} + x \]

      4. lower-fma.f64 27.1

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

   9. Applied rewrites 27.1%

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

   if 0.20000000000000001 < (+.f64 x y)

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 51.7%

      \[\leadsto y + x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 45.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+269}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, b, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -2e+269)
     (* b a)
     (if (<= t_1 -2e+128)
       (fma -0.5 b x)
       (if (<= t_1 2e+192) (+ y x) (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -2e+269) {
		tmp = b * a;
	} else if (t_1 <= -2e+128) {
		tmp = fma(-0.5, b, x);
	} else if (t_1 <= 2e+192) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -2e+269)
		tmp = Float64(b * a);
	elseif (t_1 <= -2e+128)
		tmp = fma(-0.5, b, x);
	elseif (t_1 <= 2e+192)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+269], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, -2e+128], N[(-0.5 * b + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+192], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000001e269 or 2.00000000000000008e192 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   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. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.2

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

   4. Applied rewrites 69.2%

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

   if -2.0000000000000001e269 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -2.0000000000000002e128

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 42.2%

      \[\leadsto x + \color{blue}{-0.5} \cdot b \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{-1}{2} \cdot b} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot b} + x \]

      4. lower-fma.f64 42.2

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

   9. Applied rewrites 42.2%

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

   if -2.0000000000000002e128 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.00000000000000008e192

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 56.4%

      \[\leadsto y + x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 45.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+127}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -4e+127) (* b a) (if (<= t_1 2e+192) (+ y x) (* b a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -4e+127) {
		tmp = b * a;
	} else if (t_1 <= 2e+192) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a - 0.5d0) * b
    if (t_1 <= (-4d+127)) then
        tmp = b * a
    else if (t_1 <= 2d+192) then
        tmp = y + x
    else
        tmp = b * a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - 0.5) * b;
	double tmp;
	if (t_1 <= -4e+127) {
		tmp = b * a;
	} else if (t_1 <= 2e+192) {
		tmp = y + x;
	} else {
		tmp = b * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - 0.5) * b
	tmp = 0
	if t_1 <= -4e+127:
		tmp = b * a
	elif t_1 <= 2e+192:
		tmp = y + x
	else:
		tmp = b * a
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - 0.5) * b)
	tmp = 0.0
	if (t_1 <= -4e+127)
		tmp = Float64(b * a);
	elseif (t_1 <= 2e+192)
		tmp = Float64(y + x);
	else
		tmp = Float64(b * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - 0.5) * b;
	tmp = 0.0;
	if (t_1 <= -4e+127)
		tmp = b * a;
	elseif (t_1 <= 2e+192)
		tmp = y + x;
	else
		tmp = b * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+127], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 2e+192], N[(y + x), $MachinePrecision], N[(b * a), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -3.99999999999999982e127 or 2.00000000000000008e192 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b)

   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. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.6

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

   4. Applied rewrites 56.6%

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

   if -3.99999999999999982e127 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 2.00000000000000008e192

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. *-commutative N/A

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

      11. lift--.f64 N/A

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

      12. lift-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 56.4%

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

Alternative 19: 41.5% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. *-commutative N/A

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

   11. lift--.f64 N/A

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

   12. lift-*.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around inf

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

7. Applied rewrites 41.5%

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

Alternative 20: 21.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b \leq -5 \cdot 10^{-100}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)) -5e-100) 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-100) {
		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-100)) 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-100) {
		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-100:
		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-100)
		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-100)
		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-100], 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)) < -5.0000000000000001e-100

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 21.5%

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

   if -5.0000000000000001e-100 < (+.f64 (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) (*.f64 (-.f64 a #s(literal 1/2 binary64)) b))

   1. Initial program 99.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 21.3%

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

Alternative 21: 21.4% accurate, 26.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 21.9%

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

Reproduce

herbie shell --seed 2025106 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (+ (+ x y) z) (* z (log t))) (* (- a 0.5) b)))