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

Percentage Accurate: 99.8% → 99.9%

Time: 5.3s

Alternatives: 13

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: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -400000:\\ \;\;\;\;x + \left(\mathsf{fma}\left(a - 0.5, b, z\right) - \log t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(b, a - 0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -400000.0)
   (+ x (- (fma (- a 0.5) b z) (* (log t) z)))
   (fma (- 1.0 (log t)) z (fma b (- a 0.5) 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))) <= -400000.0) {
		tmp = x + (fma((a - 0.5), b, z) - (log(t) * z));
	} else {
		tmp = fma((1.0 - log(t)), z, fma(b, (a - 0.5), y));
	}
	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))) <= -400000.0)
		tmp = Float64(x + Float64(fma(Float64(a - 0.5), b, z) - Float64(log(t) * z)));
	else
		tmp = fma(Float64(1.0 - log(t)), z, fma(b, Float64(a - 0.5), y));
	end
	return 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], -400000.0], N[(x + N[(N[(N[(a - 0.5), $MachinePrecision] * b + z), $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -4e5

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 79.3

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

   4. Applied rewrites 79.3%

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

   if -4e5 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in 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 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

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

      3. *-commutative N/A

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

      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. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 79.5

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

   7. Applied rewrites 79.5%

      \[\leadsto \mathsf{fma}\left(1 - \log t, \color{blue}{z}, \mathsf{fma}\left(b, a - 0.5, y\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, \mathsf{fma}\left(b, a - 0.5, y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.75e+75)
   (+ (fma (- a 0.5) b y) x)
   (fma (- 1.0 (log t)) z (fma b (- a 0.5) y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.75e+75) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = fma((1.0 - log(t)), z, fma(b, (a - 0.5), y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.75e+75)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = fma(Float64(1.0 - log(t)), z, fma(b, Float64(a - 0.5), y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.75e+75], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.75e75

   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 78.4

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

   4. Applied rewrites 78.4%

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

   if -2.75e75 < x

   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 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

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

      3. *-commutative N/A

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

      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. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 79.5

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

   7. Applied rewrites 79.5%

      \[\leadsto \mathsf{fma}\left(1 - \log t, \color{blue}{z}, \mathsf{fma}\left(b, a - 0.5, y\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{if}\;b \leq -3.1 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(1 - \log t, z, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (fma (- a 0.5) b y) x)))
   (if (<= b -3.1e+94)
     t_1
     (if (<= b 2.3e+27) (+ (fma (- 1.0 (log t)) z y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a - 0.5), b, y) + x;
	double tmp;
	if (b <= -3.1e+94) {
		tmp = t_1;
	} else if (b <= 2.3e+27) {
		tmp = fma((1.0 - log(t)), z, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(Float64(a - 0.5), b, y) + x)
	tmp = 0.0
	if (b <= -3.1e+94)
		tmp = t_1;
	elseif (b <= 2.3e+27)
		tmp = Float64(fma(Float64(1.0 - log(t)), z, 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[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[b, -3.1e+94], t$95$1, If[LessEqual[b, 2.3e+27], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.09999999999999991e94 or 2.3000000000000001e27 < 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 78.4

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

   4. Applied rewrites 78.4%

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

   if -3.09999999999999991e94 < b < 2.3000000000000001e27

   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 b around 0

      \[\leadsto \left(y + z \cdot \left(1 - \log t\right)\right) + x \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 62.3

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

   7. Applied rewrites 62.3%

      \[\leadsto \mathsf{fma}\left(1 - \log t, z, y\right) + x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+127}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z + x\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;y + \left(z - \log t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.75e+127)
   (+ (* (- 1.0 (log t)) z) x)
   (if (<= z 1.45e+160) (+ (fma (- a 0.5) b y) x) (+ y (- z (* (log t) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.75e+127) {
		tmp = ((1.0 - log(t)) * z) + x;
	} else if (z <= 1.45e+160) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = y + (z - (log(t) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.75e+127)
		tmp = Float64(Float64(Float64(1.0 - log(t)) * z) + x);
	elseif (z <= 1.45e+160)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(y + Float64(z - Float64(log(t) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.75e+127], N[(N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.45e+160], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.75000000000000021e127

   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 b around 0

      \[\leadsto \left(y + z \cdot \left(1 - \log t\right)\right) + x \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift--.f64 62.3

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

   7. Applied rewrites 62.3%

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

   8. Taylor expanded in y around 0

      \[\leadsto z \cdot \left(1 - \log t\right) + x \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift--.f64 42.3

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

   10. Applied rewrites 42.3%

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

   if -2.75000000000000021e127 < z < 1.45e160

   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 78.4

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

   4. Applied rewrites 78.4%

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

   if 1.45e160 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

Alternative 6: 83.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+174}:\\ \;\;\;\;\left(1 - \log t\right) \cdot z\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;y + \left(z - \log t \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.5e+174)
   (* (- 1.0 (log t)) z)
   (if (<= z 1.45e+160) (+ (fma (- a 0.5) b y) x) (+ y (- z (* (log t) z))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.5e+174) {
		tmp = (1.0 - log(t)) * z;
	} else if (z <= 1.45e+160) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = y + (z - (log(t) * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.5e+174)
		tmp = Float64(Float64(1.0 - log(t)) * z);
	elseif (z <= 1.45e+160)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = Float64(y + Float64(z - Float64(log(t) * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.5e+174], N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[z, 1.45e+160], N[(N[(N[(a - 0.5), $MachinePrecision] * b + y), $MachinePrecision] + x), $MachinePrecision], N[(y + N[(z - N[(N[Log[t], $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5e174

   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 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 22.5

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

   4. Applied rewrites 22.5%

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

   if -1.5e174 < z < 1.45e160

   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 78.4

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

   4. Applied rewrites 78.4%

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

   if 1.45e160 < z

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-log.f64 79.4

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

   4. Applied rewrites 79.4%

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

   5. Taylor expanded in b around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

Alternative 7: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - \log t\right) \cdot z\\ \mathbf{if}\;z \leq -1.5 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+160}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, b, y\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- 1.0 (log t)) z)))
   (if (<= z -1.5e+174)
     t_1
     (if (<= z 1.45e+160) (+ (fma (- a 0.5) b y) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (1.0 - log(t)) * z;
	double tmp;
	if (z <= -1.5e+174) {
		tmp = t_1;
	} else if (z <= 1.45e+160) {
		tmp = fma((a - 0.5), b, y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(1.0 - log(t)) * z)
	tmp = 0.0
	if (z <= -1.5e+174)
		tmp = t_1;
	elseif (z <= 1.45e+160)
		tmp = Float64(fma(Float64(a - 0.5), b, y) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(1.0 - N[Log[t], $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.5e+174], t$95$1, If[LessEqual[z, 1.45e+160], 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 < -1.5e174 or 1.45e160 < z

   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 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 22.5

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

   4. Applied rewrites 22.5%

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

   if -1.5e174 < z < 1.45e160

   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 78.4

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

   4. Applied rewrites 78.4%

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

Alternative 8: 78.4% 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 78.4

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

4. Applied rewrites 78.4%

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

Alternative 9: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x + y\right) + z\right) - z \cdot \log t \leq -400000:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, a - 0.5, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= (- (+ (+ x y) z) (* z (log t))) -400000.0)
   (fma b (- a 0.5) x)
   (fma b (- a 0.5) 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))) <= -400000.0) {
		tmp = fma(b, (a - 0.5), x);
	} else {
		tmp = fma(b, (a - 0.5), y);
	}
	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))) <= -400000.0)
		tmp = fma(b, Float64(a - 0.5), x);
	else
		tmp = fma(b, Float64(a - 0.5), y);
	end
	return 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], -400000.0], N[(b * N[(a - 0.5), $MachinePrecision] + x), $MachinePrecision], N[(b * N[(a - 0.5), $MachinePrecision] + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t))) < -4e5

   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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift--.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if -4e5 < (-.f64 (+.f64 (+.f64 x y) z) (*.f64 z (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in 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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 58.5

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

   7. Applied rewrites 58.5%

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

Alternative 10: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ t_2 := \mathsf{fma}\left(b, a - 0.5, x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+207}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+20}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)) (t_2 (fma b (- a 0.5) x)))
   (if (<= t_1 -5e+207) t_2 (if (<= t_1 4e+20) (+ y x) 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(b, (a - 0.5), x);
	double tmp;
	if (t_1 <= -5e+207) {
		tmp = t_2;
	} else if (t_1 <= 4e+20) {
		tmp = y + x;
	} 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 = fma(b, Float64(a - 0.5), x)
	tmp = 0.0
	if (t_1 <= -5e+207)
		tmp = t_2;
	elseif (t_1 <= 4e+20)
		tmp = Float64(y + x);
	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[(b * N[(a - 0.5), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+207], t$95$2, If[LessEqual[t$95$1, 4e+20], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < -4.9999999999999999e207 or 4e20 < (*.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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift--.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if -4.9999999999999999e207 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 4e20

   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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto y + x \]

      2. lower-+.f64 41.3

         \[\leadsto y + x \]

   7. Applied rewrites 41.3%

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

Alternative 11: 57.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+208}:\\ \;\;\;\;b \cdot a\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+264}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;b \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) b)))
   (if (<= t_1 -4e+208) (* b a) (if (<= t_1 5e+264) (+ 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+208) {
		tmp = b * a;
	} else if (t_1 <= 5e+264) {
		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+208)) then
        tmp = b * a
    else if (t_1 <= 5d+264) 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+208) {
		tmp = b * a;
	} else if (t_1 <= 5e+264) {
		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+208:
		tmp = b * a
	elif t_1 <= 5e+264:
		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+208)
		tmp = Float64(b * a);
	elseif (t_1 <= 5e+264)
		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+208)
		tmp = b * a;
	elseif (t_1 <= 5e+264)
		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+208], N[(b * a), $MachinePrecision], If[LessEqual[t$95$1, 5e+264], 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) < -3.9999999999999999e208 or 5.00000000000000033e264 < (*.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 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 26.9

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

   4. Applied rewrites 26.9%

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

   if -3.9999999999999999e208 < (*.f64 (-.f64 a #s(literal 1/2 binary64)) b) < 5.00000000000000033e264

   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 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in b around 0

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

      1. +-commutative N/A

         \[\leadsto y + x \]

      2. lower-+.f64 41.3

         \[\leadsto y + x \]

   7. Applied rewrites 41.3%

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

Alternative 12: 41.3% 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

   \[\leadsto \color{blue}{x + \left(y + b \cdot \left(a - \frac{1}{2}\right)\right)} \]
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 78.4

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in b around 0

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

   1. +-commutative N/A

      \[\leadsto y + x \]

   2. lower-+.f64 41.3

      \[\leadsto y + x \]

7. Applied rewrites 41.3%

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

Alternative 13: 21.7% accurate, 26.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 78.4

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in b around 0

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

   1. +-commutative N/A

      \[\leadsto y + x \]

   2. lower-+.f64 41.3

      \[\leadsto y + x \]

7. Applied rewrites 41.3%

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

8. Taylor expanded in x around 0

   \[\leadsto y \]
9. Step-by-step derivation

10. Applied rewrites 21.7%

    \[\leadsto y \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))