Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 8.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift--.f64 N/A

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

   4. sub-negate-rev N/A

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

   5. lift-+.f64 N/A

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

   6. associate--r+ N/A

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

   7. sub-negate N/A

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

   8. associate-+r- N/A

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

   9. lower--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower--.f64 99.6%

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 0.00115:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= t 0.00115)
  (+ (log z) (+ (log (+ x y)) (* (log t) (- a 0.5))))
  (fma (- a 0.5) (log t) (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00115) {
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00115)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00115], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.00115

   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. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower--.f64 61.4%

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

   4. Applied rewrites 61.4%

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

   if 0.00115 < t

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2.2:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;a \leq 85:\\ \;\;\;\;\left(\left(\log \left(\mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= a -2.2)
  (- (- t) (* (- 0.5 a) (log t)))
  (if (<= a 85.0)
    (+ (- (+ (log (fmax x y)) (log z)) t) (* -0.5 (log t)))
    (fma (- a 0.5) (log t) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.2) {
		tmp = -t - ((0.5 - a) * log(t));
	} else if (a <= 85.0) {
		tmp = ((log(fmax(x, y)) + log(z)) - t) + (-0.5 * log(t));
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.2)
		tmp = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)));
	elseif (a <= 85.0)
		tmp = Float64(Float64(Float64(log(fmax(x, y)) + log(z)) - t) + Float64(-0.5 * log(t)));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.2], N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 85.0], N[(N[(N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if 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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - \frac{1}{2}\right)\right)\right)\right) \cdot \log t \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \mathsf{Rewrite<=}\left(sub-negate-rev, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\frac{1}{2} - a\right) \cdot \log t\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

   6. Applied rewrites 77.4%

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

   if -2.2000000000000002 < a < 85

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 41.8%

      \[\leadsto -1 \cdot t + \color{blue}{-0.5} \cdot \log t \]

   8. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-log.f64 42.4%

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

   10. Applied rewrites 42.4%

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

   if 85 < a

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 4: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;a \leq -2.2:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;a \leq 85:\\ \;\;\;\;\left(\log z + \left(\log \left(\mathsf{max}\left(x, y\right)\right) - \log \left(\sqrt{t}\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= a -2.2)
  (- (- t) (* (- 0.5 a) (log t)))
  (if (<= a 85.0)
    (- (+ (log z) (- (log (fmax x y)) (log (sqrt t)))) t)
    (fma (- a 0.5) (log t) (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.2) {
		tmp = -t - ((0.5 - a) * log(t));
	} else if (a <= 85.0) {
		tmp = (log(z) + (log(fmax(x, y)) - log(sqrt(t)))) - t;
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.2)
		tmp = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)));
	elseif (a <= 85.0)
		tmp = Float64(Float64(log(z) + Float64(log(fmax(x, y)) - log(sqrt(t)))) - t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.2], N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 85.0], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[Max[x, y], $MachinePrecision]], $MachinePrecision] - N[Log[N[Sqrt[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if 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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - \frac{1}{2}\right)\right)\right)\right) \cdot \log t \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \mathsf{Rewrite<=}\left(sub-negate-rev, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\frac{1}{2} - a\right) \cdot \log t\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

   6. Applied rewrites 77.4%

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

   if -2.2000000000000002 < a < 85

   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. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 42.4%

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

   7. Applied rewrites 42.4%

      \[\leadsto \left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lift-log.f64 N/A

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

      7. log-pow-rev N/A

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

      8. pow1/2 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lower-log.f64 42.4%

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

   9. Applied rewrites 42.4%

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

   if 85 < a

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 5: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \log \left(x + y\right)\\ t_2 := \left(\left(t\_1 + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_2 \leq -200000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;t\_2 \leq 1400:\\ \;\;\;\;\left(t\_1 - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (log (+ x y)))
       (t_2 (+ (- (+ t_1 (log z)) t) (* (- a 0.5) (log t)))))
  (if (<= t_2 -200000.0)
    (- (- t) (* (- 0.5 a) (log t)))
    (if (<= t_2 1400.0)
      (- (- t_1 (log (/ (sqrt t) z))) t)
      (fma (- a 0.5) (log t) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y));
	double t_2 = ((t_1 + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_2 <= -200000.0) {
		tmp = -t - ((0.5 - a) * log(t));
	} else if (t_2 <= 1400.0) {
		tmp = (t_1 - log((sqrt(t) / z))) - t;
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = log(Float64(x + y))
	t_2 = Float64(Float64(Float64(t_1 + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_2 <= -200000.0)
		tmp = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)));
	elseif (t_2 <= 1400.0)
		tmp = Float64(Float64(t_1 - log(Float64(sqrt(t) / z))) - t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -200000.0], N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1400.0], N[(N[(t$95$1 - N[Log[N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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))) < -2e5

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - \frac{1}{2}\right)\right)\right)\right) \cdot \log t \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \mathsf{Rewrite<=}\left(sub-negate-rev, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\frac{1}{2} - a\right) \cdot \log t\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

   6. Applied rewrites 77.4%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1400

   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. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. add-flip N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-+l- N/A

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

      9. lower--.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-log.f64 N/A

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

      15. log-pow-rev N/A

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

      16. neg-log N/A

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

      17. lift-log.f64 N/A

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

      18. diff-log N/A

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

      19. lower-log.f64 N/A

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

      20. lower-/.f64 N/A

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

   6. Applied rewrites 55.9%

      \[\leadsto \left(\log \left(x + y\right) - \log \left(\frac{\sqrt{t}}{z}\right)\right) - t \]

   if 1400 < (+.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 99.6%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 6: 93.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;\left(\log \left(\frac{\mathsf{max}\left(x, y\right)}{\sqrt{t}}\right) - t\right) + \log z\\ \mathbf{elif}\;t\_1 \leq 710:\\ \;\;\;\;\log \left(z \cdot \mathsf{max}\left(x, y\right)\right) - \mathsf{fma}\left(0.5 - a, \log t, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (+ (log (+ (fmin x y) (fmax x y))) (log z))))
  (if (<= t_1 -750.0)
    (+ (- (log (/ (fmax x y) (sqrt t))) t) (log z))
    (if (<= t_1 710.0)
      (- (log (* z (fmax x y))) (fma (- 0.5 a) (log t) t))
      (fma (- a 0.5) (log t) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((fmin(x, y) + fmax(x, y))) + log(z);
	double tmp;
	if (t_1 <= -750.0) {
		tmp = (log((fmax(x, y) / sqrt(t))) - t) + log(z);
	} else if (t_1 <= 710.0) {
		tmp = log((z * fmax(x, y))) - fma((0.5 - a), log(t), t);
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z))
	tmp = 0.0
	if (t_1 <= -750.0)
		tmp = Float64(Float64(log(Float64(fmax(x, y) / sqrt(t))) - t) + log(z));
	elseif (t_1 <= 710.0)
		tmp = Float64(log(Float64(z * fmax(x, y))) - fma(Float64(0.5 - a), log(t), t));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -750.0], N[(N[(N[Log[N[(N[Max[x, y], $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 710.0], N[(N[Log[N[(z * N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -750

   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. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 42.4%

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

   7. Applied rewrites 42.4%

      \[\leadsto \left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   9. Applied rewrites 37.7%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 710

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\left(1 - \frac{\left(0.5 - a\right) \cdot \log t}{-t}\right) \cdot \left(-t\right)} \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

   9. Applied rewrites 69.3%

      \[\leadsto \color{blue}{\left(1 - \frac{\log t \cdot \left(0.5 - a\right)}{\left(\log y + \log z\right) - t}\right) \cdot \left(\left(\log y + \log z\right) - t\right)} \]
   10. Step-by-step derivation

       1. sub-to-mult-rev 69.3%

          \[\leadsto \color{blue}{\left(1 - \frac{\log t \cdot \left(0.5 - a\right)}{\left(\log y + \log z\right) - t}\right)} \cdot \left(\left(\log y + \log z\right) - t\right) \]

       2. sub-negate-rev 69.3%

          \[\leadsto \left(1 - \frac{\color{blue}{\log t \cdot \left(0.5 - a\right)}}{\left(\log y + \log z\right) - t}\right) \cdot \left(\left(\log y + \log z\right) - t\right) \]

       3. fp-cancel-sign-sub-inv 69.3%

          \[\leadsto \color{blue}{\left(1 - \frac{\log t \cdot \left(0.5 - a\right)}{\left(\log y + \log z\right) - t}\right)} \cdot \left(\left(\log y + \log z\right) - t\right) \]

       4. lift-*.f64 N/A

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

       5. lift--.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. sub-to-mult-rev N/A

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

       8. lift--.f64 N/A

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

       9. lift-log.f64 N/A

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

   11. Applied rewrites 52.7%

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

   if 710 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 7: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;t\_1 \leq 1300:\\ \;\;\;\;\left(\log z + \left(-\log \left(\frac{\sqrt{t}}{\mathsf{max}\left(x, y\right)}\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1
        (+
         (- (+ (log (+ (fmin x y) (fmax x y))) (log z)) t)
         (* (- a 0.5) (log t)))))
  (if (<= t_1 -200000.0)
    (- (- t) (* (- 0.5 a) (log t)))
    (if (<= t_1 1300.0)
      (- (+ (log z) (- (log (/ (sqrt t) (fmax x y))))) t)
      (fma (- a 0.5) (log t) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((fmin(x, y) + fmax(x, y))) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = -t - ((0.5 - a) * log(t));
	} else if (t_1 <= 1300.0) {
		tmp = (log(z) + -log((sqrt(t) / fmax(x, y)))) - t;
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)));
	elseif (t_1 <= 1300.0)
		tmp = Float64(Float64(log(z) + Float64(-log(Float64(sqrt(t) / fmax(x, y))))) - t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000.0], N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1300.0], N[(N[(N[Log[z], $MachinePrecision] + (-N[Log[N[(N[Sqrt[t], $MachinePrecision] / N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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))) < -2e5

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - \frac{1}{2}\right)\right)\right)\right) \cdot \log t \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \mathsf{Rewrite<=}\left(sub-negate-rev, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\frac{1}{2} - a\right) \cdot \log t\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

   6. Applied rewrites 77.4%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1300

   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. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 42.4%

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

   7. Applied rewrites 42.4%

      \[\leadsto \left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. sub-negate-rev N/A

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

      5. lower-neg.f64 N/A

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

      6. metadata-eval N/A

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

      7. lift-log.f64 N/A

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

      8. log-pow-rev N/A

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

      9. pow1/2 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. diff-log N/A

          \[\leadsto \left(\log z + \left(-\log \left(\frac{\sqrt{t}}{y}\right)\right)\right) - t \]

      13. lower-log.f64 N/A

          \[\leadsto \left(\log z + \left(-\log \left(\frac{\sqrt{t}}{y}\right)\right)\right) - t \]

      14. lower-/.f64 37.6%

          \[\leadsto \left(\log z + \left(-\log \left(\frac{\sqrt{t}}{y}\right)\right)\right) - t \]

   9. Applied rewrites 37.6%

      \[\leadsto \left(\log z + \left(-\log \left(\frac{\sqrt{t}}{y}\right)\right)\right) - t \]

   if 1300 < (+.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 99.6%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 8: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\log \left(\mathsf{min}\left(x, y\right) + \mathsf{max}\left(x, y\right)\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;t\_1 \leq 1200:\\ \;\;\;\;\left(\log \left(\frac{\mathsf{max}\left(x, y\right)}{\sqrt{t}}\right) - t\right) + \log z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1
        (+
         (- (+ (log (+ (fmin x y) (fmax x y))) (log z)) t)
         (* (- a 0.5) (log t)))))
  (if (<= t_1 -200000.0)
    (- (- t) (* (- 0.5 a) (log t)))
    (if (<= t_1 1200.0)
      (+ (- (log (/ (fmax x y) (sqrt t))) t) (log z))
      (fma (- a 0.5) (log t) (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((fmin(x, y) + fmax(x, y))) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -200000.0) {
		tmp = -t - ((0.5 - a) * log(t));
	} else if (t_1 <= 1200.0) {
		tmp = (log((fmax(x, y) / sqrt(t))) - t) + log(z);
	} else {
		tmp = fma((a - 0.5), log(t), -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(fmin(x, y) + fmax(x, y))) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)));
	elseif (t_1 <= 1200.0)
		tmp = Float64(Float64(log(Float64(fmax(x, y) / sqrt(t))) - t) + log(z));
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(N[Min[x, y], $MachinePrecision] + N[Max[x, y], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000.0], N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1200.0], N[(N[(N[Log[N[(N[Max[x, y], $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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))) < -2e5

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - \frac{1}{2}\right)\right)\right)\right) \cdot \log t \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \mathsf{Rewrite<=}\left(sub-negate-rev, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\frac{1}{2} - a\right) \cdot \log t\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

   6. Applied rewrites 77.4%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1200

   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. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 42.4%

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

   7. Applied rewrites 42.4%

      \[\leadsto \left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   9. Applied rewrites 37.7%

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

   if 1200 < (+.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 99.6%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 9: 91.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000:\\ \;\;\;\;\left(-t\right) - \left(0.5 - a\right) \cdot \log t\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1
        (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
  (if (<= t_1 -200000.0)
    (- (- t) (* (- 0.5 a) (log t)))
    (if (<= t_1 700.0)
      (- (log (/ (* (+ x y) z) (sqrt t))) t)
      (fma (- a 0.5) (log t) (- 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 tmp;
	if (t_1 <= -200000.0) {
		tmp = -t - ((0.5 - a) * log(t));
	} else if (t_1 <= 700.0) {
		tmp = log((((x + y) * z) / sqrt(t))) - t;
	} else {
		tmp = fma((a - 0.5), log(t), -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)))
	tmp = 0.0
	if (t_1 <= -200000.0)
		tmp = Float64(Float64(-t) - Float64(Float64(0.5 - a) * log(t)));
	elseif (t_1 <= 700.0)
		tmp = Float64(log(Float64(Float64(Float64(x + y) * z) / sqrt(t))) - t);
	else
		tmp = fma(Float64(a - 0.5), log(t), Float64(-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]}, If[LessEqual[t$95$1, -200000.0], N[((-t) - N[(N[(0.5 - a), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 700.0], N[(N[Log[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $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))) < -2e5

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift--.f64, \left(a - \frac{1}{2}\right)\right)\right)\right) \cdot \log t \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(-t\right) - \mathsf{Rewrite<=}\left(sub-negate-rev, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

      10. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower-*.f64, \left(\left(\frac{1}{2} - a\right) \cdot \log t\right)\right) \]

      11. lower-neg.f64 N/A

          \[\leadsto \left(-t\right) - \mathsf{Rewrite=>}\left(lower--.f64, \left(\frac{1}{2} - a\right)\right) \cdot \log t \]

   6. Applied rewrites 77.4%

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

   if -2e5 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 700

   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. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 62.9%

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

   4. Applied rewrites 62.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. add-flip N/A

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-+.f64 N/A

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

      7. associate-+r- N/A

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

      8. lift-log.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. sum-log N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-log.f64 N/A

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

      17. log-pow-rev N/A

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

      18. neg-log N/A

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

      19. diff-log N/A

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

      20. lower-log.f64 N/A

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

   6. Applied rewrites 44.7%

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

   if 700 < (+.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 99.6%

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

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 77.4%

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 77.4%

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

   6. Applied rewrites 77.4%

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

Alternative 10: 77.4% accurate, 2.2× speedup

Math

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

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. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
3. Step-by-step derivation

   1. lower-*.f64 77.4%

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

4. Applied rewrites 77.4%

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 77.4%

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

   5. lift-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-neg.f64 77.4%

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

6. Applied rewrites 77.4%

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

Alternative 11: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -2 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (* a (log t))))
  (if (<= a -2e+59)
    t_1
    (if (<= a 2e+14) (fma (log t) -0.5 (- t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (a <= -2e+59) {
		tmp = t_1;
	} else if (a <= 2e+14) {
		tmp = fma(log(t), -0.5, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -2e+59)
		tmp = t_1;
	elseif (a <= 2e+14)
		tmp = fma(log(t), -0.5, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2e+59], t$95$1, If[LessEqual[a, 2e+14], N[(N[Log[t], $MachinePrecision] * -0.5 + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9999999999999999e59 or 2e14 < a

   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. 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. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 37.9%

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

   6. Applied rewrites 37.9%

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

   if -1.9999999999999999e59 < a < 2e14

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 41.8%

      \[\leadsto -1 \cdot t + \color{blue}{-0.5} \cdot \log t \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{-1 \cdot t + \frac{-1}{2} \cdot \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log t + -1 \cdot t} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \log t} + -1 \cdot t \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\log t \cdot \frac{-1}{2}} + -1 \cdot t \]

      5. lift-*.f64 N/A

         \[\leadsto \log t \cdot \frac{-1}{2} + -1 \cdot \color{blue}{t} \]

      6. mul-1-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 41.8%

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

   9. Applied rewrites 41.8%

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

Alternative 12: 62.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(-t\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= t 2.2e+38) (* a (log t)) (* 1.0 (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.2e+38) {
		tmp = a * log(t);
	} else {
		tmp = 1.0 * -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 (t <= 2.2d+38) then
        tmp = a * log(t)
    else
        tmp = 1.0d0 * -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 (t <= 2.2e+38) {
		tmp = a * Math.log(t);
	} else {
		tmp = 1.0 * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.2e+38:
		tmp = a * math.log(t)
	else:
		tmp = 1.0 * -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.2e+38)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(1.0 * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.2e+38)
		tmp = a * log(t);
	else
		tmp = 1.0 * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.2e+38], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(1.0 * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.2000000000000001e38

   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. 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. associate-+l- N/A

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

      4. lift-+.f64 N/A

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

      5. add-flip N/A

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

      6. associate--l- N/A

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

      7. sub-to-mult N/A

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

      8. lower-unsound-*.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 37.9%

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

   6. Applied rewrites 37.9%

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

   if 2.2000000000000001e38 < t

   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. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   3. Step-by-step derivation

      1. lower-*.f64 77.4%

         \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

   4. Applied rewrites 77.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. sub-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\left(1 - \frac{\left(0.5 - a\right) \cdot \log t}{-t}\right) \cdot \left(-t\right)} \]

   7. Taylor expanded in t around inf

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

   9. Applied rewrites 39.0%

      \[\leadsto \color{blue}{1} \cdot \left(-t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 39.0% accurate, 7.0× speedup

Math

\[1 \cdot \left(-t\right) \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return 1.0 * -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 = 1.0d0 * -t
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 1.0 * -t;
}

Python

def code(x, y, z, t, a):
	return 1.0 * -t

Julia

function code(x, y, z, t, a)
	return Float64(1.0 * Float64(-t))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 1.0 * -t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(1.0 * (-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. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
3. Step-by-step derivation

   1. lower-*.f64 77.4%

      \[\leadsto -1 \cdot \color{blue}{t} + \left(a - 0.5\right) \cdot \log t \]

4. Applied rewrites 77.4%

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - 0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. sub-to-mult N/A

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

   5. lower-unsound-*.f64 N/A

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

6. Applied rewrites 66.1%

   \[\leadsto \color{blue}{\left(1 - \frac{\left(0.5 - a\right) \cdot \log t}{-t}\right) \cdot \left(-t\right)} \]

7. Taylor expanded in t around inf

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

9. Applied rewrites 39.0%

   \[\leadsto \color{blue}{1} \cdot \left(-t\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))