Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 6.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lift-log.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y + x\right) + \left(-t\right)\right) + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -5e+19)
     (- t_2 t)
     (if (<= t_1 1000.0)
       (fma -0.5 (log t) (- (log (* z y)) t))
       (+ (+ (log (+ y x)) (- t)) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = t_2 - t;
	} else if (t_1 <= 1000.0) {
		tmp = fma(-0.5, log(t), (log((z * y)) - t));
	} else {
		tmp = (log((y + x)) + -t) + t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -5e+19)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 1000.0)
		tmp = fma(-0.5, log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-t)) + t_2);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+19], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + (-t)), $MachinePrecision] + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e19

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \log t \cdot a - t \]

      2. lift-log.f64 N/A

         \[\leadsto \log t \cdot a - t \]

      3. lift-*.f64 99.8

         \[\leadsto \log t \cdot a - t \]

   8. Applied rewrites 99.8%

      \[\leadsto \log t \cdot a - t \]

   if -5e19 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 46.2

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

   7. Applied rewrites 46.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 46.2%

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

   if 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 20.0

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

   7. Applied rewrites 20.0%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 6.1

         \[\leadsto \left(\log \left(y + x\right) + \left(-t\right)\right) + -0.5 \cdot \log t \]

   10. Applied rewrites 6.1%

       \[\leadsto \left(\log \left(y + x\right) + \color{blue}{\left(-t\right)}\right) + -0.5 \cdot \log t \]

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 83.2

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

   13. Applied rewrites 83.2%

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

Alternative 3: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_2 - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -5e+19)
     (- t_2 t)
     (if (<= t_1 1000.0)
       (fma -0.5 (log t) (- (log (* z y)) t))
       (- (+ t_2 (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -5e+19) {
		tmp = t_2 - t;
	} else if (t_1 <= 1000.0) {
		tmp = fma(-0.5, log(t), (log((z * y)) - t));
	} else {
		tmp = (t_2 + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -5e+19)
		tmp = Float64(t_2 - t);
	elseif (t_1 <= 1000.0)
		tmp = fma(-0.5, log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(t_2 + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+19], N[(t$95$2 - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e19

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \log t \cdot a - t \]

      2. lift-log.f64 N/A

         \[\leadsto \log t \cdot a - t \]

      3. lift-*.f64 99.8

         \[\leadsto \log t \cdot a - t \]

   8. Applied rewrites 99.8%

      \[\leadsto \log t \cdot a - t \]

   if -5e19 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 46.2

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

   5. Applied rewrites 46.2%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 46.2

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

   7. Applied rewrites 46.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 46.2%

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

   if 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 83.0

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

   8. Applied rewrites 83.0%

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

Alternative 4: 91.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 705.36:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot \left(y + x\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y + x\right) + \left(-t\right)\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 705.36)
   (fma (- a 0.5) (log t) (- (log (* z (+ y x))) t))
   (+ (+ (log (+ y x)) (- t)) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 705.36) {
		tmp = fma((a - 0.5), log(t), (log((z * (y + x))) - t));
	} else {
		tmp = (log((y + x)) + -t) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 705.36)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * Float64(y + x))) - t));
	else
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-t)) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 705.36], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + (-t)), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705.360000000000014

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 96.8

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

   4. Applied rewrites 96.8%

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

   if 705.360000000000014 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 5: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 705.36:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y + x\right) + \left(-t\right)\right) + \left(a - 0.5\right) \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 705.36)
   (fma (- a 0.5) (log t) (- (log (* z y)) t))
   (+ (+ (log (+ y x)) (- t)) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 705.36) {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	} else {
		tmp = (log((y + x)) + -t) + ((a - 0.5) * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 705.36)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-t)) + Float64(Float64(a - 0.5) * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 705.36], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + (-t)), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705.360000000000014

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if 705.360000000000014 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 81.8

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

   7. Applied rewrites 81.8%

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

Alternative 6: 66.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 705.36:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y + x\right) + \left(-t\right)\right) + \log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 705.36)
   (fma (- a 0.5) (log t) (- (log (* z y)) t))
   (+ (+ (log (+ y x)) (- t)) (* (log t) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 705.36) {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	} else {
		tmp = (log((y + x)) + -t) + (log(t) * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 705.36)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(log(Float64(y + x)) + Float64(-t)) + Float64(log(t) * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 705.36], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] + (-t)), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705.360000000000014

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if 705.360000000000014 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 70.6

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

   7. Applied rewrites 70.6%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 52.6

         \[\leadsto \left(\log \left(y + x\right) + \left(-t\right)\right) + -0.5 \cdot \log t \]

   10. Applied rewrites 52.6%

       \[\leadsto \left(\log \left(y + x\right) + \color{blue}{\left(-t\right)}\right) + -0.5 \cdot \log t \]

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-log.f64 N/A

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

       3. lift-*.f64 81.6

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

   13. Applied rewrites 81.6%

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

Alternative 7: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.92:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 2.2:\\ \;\;\;\;\left(\log y + \left(\log z - t\right)\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.92)
   (fma (- a 0.5) (log t) (- t))
   (if (<= a 2.2)
     (+ (+ (log y) (- (log z) t)) (* -0.5 (log t)))
     (- (* (log t) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.92) {
		tmp = fma((a - 0.5), log(t), -t);
	} else if (a <= 2.2) {
		tmp = (log(y) + (log(z) - t)) + (-0.5 * log(t));
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.92)
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	elseif (a <= 2.2)
		tmp = Float64(Float64(log(y) + Float64(log(z) - t)) + Float64(-0.5 * log(t)));
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.92], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[a, 2.2], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.92000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \color{blue}{-1 \cdot t}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.2

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

   10. Applied rewrites 98.2%

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

   if -0.92000000000000004 < a < 2.2000000000000002

   1. Initial program 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lift-log.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 65.4%

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

   if 2.2000000000000002 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \log t \cdot a - t \]

      2. lift-log.f64 N/A

         \[\leadsto \log t \cdot a - t \]

      3. lift-*.f64 99.5

         \[\leadsto \log t \cdot a - t \]

   8. Applied rewrites 99.5%

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

Alternative 8: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.92:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 2.2:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, -0.5, \log y\right) + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.92)
   (fma (- a 0.5) (log t) (- t))
   (if (<= a 2.2)
     (- (+ (fma (log t) -0.5 (log y)) (log z)) t)
     (- (* (log t) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.92) {
		tmp = fma((a - 0.5), log(t), -t);
	} else if (a <= 2.2) {
		tmp = (fma(log(t), -0.5, log(y)) + log(z)) - t;
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.92)
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	elseif (a <= 2.2)
		tmp = Float64(Float64(fma(log(t), -0.5, log(y)) + log(z)) - t);
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.92], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[a, 2.2], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.92000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sum-log N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 60.2

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

   5. Applied rewrites 60.2%

      \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-log.f64 60.3

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

   7. Applied rewrites 60.3%

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

   8. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \color{blue}{-1 \cdot t}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+r- N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 98.2

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

   10. Applied rewrites 98.2%

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

   if -0.92000000000000004 < a < 2.2000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in a around 0

      \[\leadsto \left(\mathsf{fma}\left(\log t, \frac{-1}{2}, \log y\right) + \log z\right) - t \]
   10. Step-by-step derivation

   11. Applied rewrites 65.4%

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

   if 2.2000000000000002 < a

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \log t \cdot a - t \]

      2. lift-log.f64 N/A

         \[\leadsto \log t \cdot a - t \]

      3. lift-*.f64 99.5

         \[\leadsto \log t \cdot a - t \]

   8. Applied rewrites 99.5%

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

Alternative 9: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 10: 69.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 69.9%

   \[\leadsto \left(\mathsf{fma}\left(\log t, -0.5 + a, \log y\right) + \log z\right) - t \]
9. Add Preprocessing

Alternative 11: 62.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+38} \lor \neg \left(a - 0.5 \leq 10^{+16}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -5e+38) (not (<= (- a 0.5) 1e+16)))
   (* (log t) a)
   (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -5e+38) || !((a - 0.5) <= 1e+16)) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	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)
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) :: tmp
    if (((a - 0.5d0) <= (-5d+38)) .or. (.not. ((a - 0.5d0) <= 1d+16))) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -5e+38) || !((a - 0.5) <= 1e+16)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -5e+38) or not ((a - 0.5) <= 1e+16):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -5e+38) || !(Float64(a - 0.5) <= 1e+16))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -5e+38) || ~(((a - 0.5) <= 1e+16)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -5e+38], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 1e+16]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -4.9999999999999997e38 or 1e16 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

         \[\leadsto \log t \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \log t \cdot \color{blue}{a} \]

      3. lift-log.f64 85.8

         \[\leadsto \log t \cdot a \]

   5. Applied rewrites 85.8%

      \[\leadsto \color{blue}{\log t \cdot a} \]

   if -4.9999999999999997e38 < (-.f64 a #s(literal 1/2 binary64)) < 1e16

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 56.4

         \[\leadsto -t \]

   5. Applied rewrites 56.4%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -5 \cdot 10^{+38} \lor \neg \left(a - 0.5 \leq 10^{+16}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a - 0.5, \log t, -t\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. sum-log N/A

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

   3. lower-log.f64 N/A

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

   4. lower-*.f64 53.7

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

5. Applied rewrites 53.7%

   \[\leadsto \left(\color{blue}{\log \left(z \cdot y\right)} - t\right) + \left(a - 0.5\right) \cdot \log t \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-log.f64 53.7

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

7. Applied rewrites 53.7%

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

8. Taylor expanded in t around inf

   \[\leadsto \mathsf{fma}\left(a - \frac{1}{2}, \log t, \color{blue}{-1 \cdot t}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+r- N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 78.6

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

10. Applied rewrites 78.6%

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{-t}\right) \]
11. Add Preprocessing

Alternative 13: 75.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \log t \cdot a - t \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
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
    code = (log(t) * a) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in a around inf

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

   1. *-commutative N/A

      \[\leadsto \log t \cdot a - t \]

   2. lift-log.f64 N/A

      \[\leadsto \log t \cdot a - t \]

   3. lift-*.f64 76.1

      \[\leadsto \log t \cdot a - t \]

8. Applied rewrites 76.1%

   \[\leadsto \log t \cdot a - t \]
9. Add Preprocessing

Alternative 14: 38.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.2

      \[\leadsto -t \]

5. Applied rewrites 39.2%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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