Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.7s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 2: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -600 \lor \neg \left(t\_1 \leq 1050\right):\\ \;\;\;\;\left(\log t \cdot a + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (or (<= t_1 -600.0) (not (<= t_1 1050.0)))
     (- (+ (* (log t) a) (log z)) t)
     (fma (log t) (- a 0.5) (log (* z y))))))

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 <= -600.0) || !(t_1 <= 1050.0)) {
		tmp = ((log(t) * a) + log(z)) - t;
	} else {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	}
	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 <= -600.0) || !(t_1 <= 1050.0))
		tmp = Float64(Float64(Float64(log(t) * a) + log(z)) - t);
	else
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	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[Or[LessEqual[t$95$1, -600.0], N[Not[LessEqual[t$95$1, 1050.0]], $MachinePrecision]], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 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))) < -600 or 1050 < (+.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 99.8%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 96.5

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

   8. Applied rewrites 96.5%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1050

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 43.5

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. log-prod N/A

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

      5. log-pow-rev N/A

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

      6. *-commutative N/A

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

      7. sum-log N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sum-log N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 48.9

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

   8. Applied rewrites 48.9%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -600 \lor \neg \left(\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq 1050\right):\\ \;\;\;\;\left(\log t \cdot a + \log z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\left(t\_2 + \log \left(y + x\right)\right) - t\\ \mathbf{elif}\;t\_1 \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -5e+15)
     (- (+ t_2 (log (+ y x))) t)
     (if (<= t_1 1050.0)
       (fma -0.5 (log t) (- (log (* z y)) t))
       (- (+ t_2 (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -5e+15) {
		tmp = (t_2 + log((y + x))) - t;
	} else if (t_1 <= 1050.0) {
		tmp = fma(-0.5, log(t), (log((z * y)) - t));
	} else {
		tmp = (t_2 + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -5e+15)
		tmp = Float64(Float64(t_2 + log(Float64(y + x))) - t);
	elseif (t_1 <= 1050.0)
		tmp = fma(-0.5, log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(t_2 + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+15], N[(N[(t$95$2 + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1050.0], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -5e15

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -5e15 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1050

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 89.7

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

   4. Applied rewrites 89.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 48.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 47.8%

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

   if 1050 < (+.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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 91.5

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

   8. Applied rewrites 91.5%

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

Alternative 4: 83.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;\left(t\_2 + \log \left(y + x\right)\right) - t\\ \mathbf{elif}\;t\_1 \leq 1050:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 + \log z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (* (log t) a)))
   (if (<= t_1 -600.0)
     (- (+ t_2 (log (+ y x))) t)
     (if (<= t_1 1050.0)
       (fma (log t) (- a 0.5) (log (* z y)))
       (- (+ t_2 (log z)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = log(t) * a;
	double tmp;
	if (t_1 <= -600.0) {
		tmp = (t_2 + log((y + x))) - t;
	} else if (t_1 <= 1050.0) {
		tmp = fma(log(t), (a - 0.5), log((z * y)));
	} else {
		tmp = (t_2 + log(z)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = Float64(Float64(t_2 + log(Float64(y + x))) - t);
	elseif (t_1 <= 1050.0)
		tmp = fma(log(t), Float64(a - 0.5), log(Float64(z * y)));
	else
		tmp = Float64(Float64(t_2 + log(z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], N[(N[(t$95$2 + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1050.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -600

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 98.0

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

   8. Applied rewrites 98.0%

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

   if -600 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1050

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. associate-+r+ N/A

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

      3. sum-log N/A

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

      4. *-commutative N/A

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

      5. log-pow-rev N/A

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

      6. sum-log N/A

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

      7. lower-log.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift--.f64 43.5

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in t around 0

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

      1. pow-to-exp N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. log-prod N/A

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

      5. log-pow-rev N/A

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

      6. *-commutative N/A

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

      7. sum-log N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift--.f64 N/A

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

      11. sum-log N/A

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

      12. lower-log.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 48.9

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

   8. Applied rewrites 48.9%

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

   if 1050 < (+.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. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 91.5

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

   8. Applied rewrites 91.5%

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

Alternative 5: 65.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 705:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 705.0)
   (fma (- a 0.5) (log t) (- (log (* z y)) t))
   (- (+ (* (log t) a) (log (+ y x))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 705.0) {
		tmp = fma((a - 0.5), log(t), (log((z * y)) - t));
	} else {
		tmp = ((log(t) * a) + log((y + x))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 705.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(log(Float64(z * y)) - t));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(Float64(y + x))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 705.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 705

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 66.9%

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

   if 705 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lift-log.f64 N/A

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

      3. lift-*.f64 89.7

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

   8. Applied rewrites 89.7%

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

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -110000:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 0.00033:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log \left(y + x\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -110000.0)
   (fma (- a 0.5) (log t) (- t))
   (if (<= a 0.00033)
     (- (+ (fma -0.5 (log t) (log z)) (log (+ y x))) t)
     (- (* (log t) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -110000.0) {
		tmp = fma((a - 0.5), log(t), -t);
	} else if (a <= 0.00033) {
		tmp = (fma(-0.5, log(t), log(z)) + log((y + x))) - t;
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -110000.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	elseif (a <= 0.00033)
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(Float64(y + x))) - t);
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -110000.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[a, 0.00033], N[(N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1e5

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 72.8

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -1.1e5 < a < 3.3e-4

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. lift-log.f64 99.5

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

   8. Applied rewrites 99.5%

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

   if 3.3e-4 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 99.7

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

   8. Applied rewrites 99.7%

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

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -110000:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, -t\right)\\ \mathbf{elif}\;a \leq 0.00033:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \left(\log y + \log z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -110000.0)
   (fma (- a 0.5) (log t) (- t))
   (if (<= a 0.00033)
     (fma -0.5 (log t) (- (+ (log y) (log z)) t))
     (- (* (log t) a) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -110000.0) {
		tmp = fma((a - 0.5), log(t), -t);
	} else if (a <= 0.00033) {
		tmp = fma(-0.5, log(t), ((log(y) + log(z)) - t));
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -110000.0)
		tmp = fma(Float64(a - 0.5), log(t), Float64(-t));
	elseif (a <= 0.00033)
		tmp = fma(-0.5, log(t), Float64(Float64(log(y) + log(z)) - t));
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -110000.0], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[a, 0.00033], N[(-0.5 * N[Log[t], $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1e5

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 72.8

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

   4. Applied rewrites 72.8%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lift-neg.f64 98.8

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

   7. Applied rewrites 98.8%

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

   if -1.1e5 < a < 3.3e-4

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-log.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. sum-log N/A

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

      17. lower-log.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 75.6

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 51.1%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. sum-log N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 67.1

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

      9. +-commutative 67.1

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

   9. Applied rewrites 67.1%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 66.8%

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

   if 3.3e-4 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-out N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lift-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 99.7

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

   8. Applied rewrites 99.7%

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

Alternative 8: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. sum-log N/A

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

   17. lower-log.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 76.3

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

4. Applied rewrites 76.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 55.7%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. sum-log N/A

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

   6. lift-log.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-+.f64 72.1

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

   9. +-commutative 72.1

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

9. Applied rewrites 72.1%

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

Alternative 9: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. associate-+r+ N/A

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

   3. sum-log N/A

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

   4. *-commutative N/A

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

   5. log-pow-rev N/A

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

   6. sum-log N/A

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

   7. lower-log.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift--.f64 23.5

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

5. Applied rewrites 23.5%

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

   1. lift-log.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*r* N/A

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

   7. pow-to-exp N/A

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

   8. sum-log N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 N/A

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

7. Applied rewrites 72.1%

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

Alternative 10: 77.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-log.f64 79.5

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

8. Applied rewrites 79.5%

   \[\leadsto \left(\log t \cdot a + \log z\right) - t \]
9. Add Preprocessing

Alternative 11: 77.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return -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 = -t + ((a - 0.5d0) * log(t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 79.4

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

5. Applied rewrites 79.4%

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

Alternative 12: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.6 \cdot 10^{+28}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.6e+28) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.6e+28) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.6d+28) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.6e+28) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.6e+28:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.6e+28)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.6e+28)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.6e+28], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.6e28

   1. Initial program 99.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 54.1

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

   5. Applied rewrites 54.1%

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

   if 1.6e28 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 79.0

         \[\leadsto -t \]

   5. Applied rewrites 79.0%

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

Alternative 13: 77.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-log.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift--.f64 N/A

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

   13. lift-log.f64 N/A

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

   14. lower--.f64 N/A

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

   15. +-commutative N/A

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

   16. sum-log N/A

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

   17. lower-log.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 76.3

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

4. Applied rewrites 76.3%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lift-neg.f64 79.4

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

7. Applied rewrites 79.4%

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

Alternative 14: 74.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in a around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lift-log.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-log.f64 76.9

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

8. Applied rewrites 76.9%

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

Alternative 15: 37.9% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 39.7

      \[\leadsto -t \]

5. Applied rewrites 39.7%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return Float64(log(Float64(x + y)) + Float64(Float64(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[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025064 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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