Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 7.5s

Alternatives: 21

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

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z around inf

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

   1. lower--.f64 N/A

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

4. Applied rewrites 99.6%

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

Alternative 2: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\left(\log z + \log \left(y + x\right)\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (fma (log t) a (log y)) (log z)) t)))
   (if (<= t_1 -1000000.0)
     t_2
     (if (<= t_1 2000.0) (+ (+ (log z) (log (+ y x))) (* -0.5 (log t))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = (fma(log(t), a, log(y)) + log(z)) - t;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = (log(z) + log((y + x))) + (-0.5 * log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = Float64(Float64(log(z) + log(Float64(y + x))) + Float64(-0.5 * log(t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 2000.0], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

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))) < -1e6 or 2e3 < (+.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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 74.5%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 74.0%

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

   if -1e6 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.1%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 72.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. log-prod N/A

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

      5. lower-+.f64 N/A

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

      6. lift-log.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 93.7

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

   9. Applied rewrites 93.7%

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

Alternative 3: 67.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\left(\log y + \log z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (fma (log t) a (log y)) (log z)) t)))
   (if (<= t_1 -1000000.0)
     t_2
     (if (<= t_1 2000.0) (+ (+ (log y) (log z)) (* -0.5 (log t))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = (fma(log(t), a, log(y)) + log(z)) - t;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2000.0) {
		tmp = (log(y) + log(z)) + (-0.5 * log(t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = t_2;
	elseif (t_1 <= 2000.0)
		tmp = Float64(Float64(log(y) + log(z)) + Float64(-0.5 * log(t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], t$95$2, If[LessEqual[t$95$1, 2000.0], N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

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))) < -1e6 or 2e3 < (+.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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 74.5%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 74.0%

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

   if -1e6 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.1%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 37.9%

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

       1. lift-log.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. sum-log N/A

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

       5. lift-log.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lift-log.f64 49.3

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

   12. Applied rewrites 49.3%

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

Alternative 4: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 2000:\\ \;\;\;\;\left(\log y + \log z\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -1000000.0)
     (- (+ (* a (log t)) (log z)) t)
     (if (<= t_1 2000.0)
       (+ (+ (log y) (log z)) (* -0.5 (log t)))
       (- (+ (* (log t) a) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = ((a * log(t)) + log(z)) - t;
	} else if (t_1 <= 2000.0) {
		tmp = (log(y) + log(z)) + (-0.5 * log(t));
	} else {
		tmp = ((log(t) * a) + log((y + x))) - 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) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-1000000.0d0)) then
        tmp = ((a * log(t)) + log(z)) - t
    else if (t_1 <= 2000.0d0) then
        tmp = (log(y) + log(z)) + ((-0.5d0) * log(t))
    else
        tmp = ((log(t) * a) + log((y + x))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = ((a * Math.log(t)) + Math.log(z)) - t;
	} else if (t_1 <= 2000.0) {
		tmp = (Math.log(y) + Math.log(z)) + (-0.5 * Math.log(t));
	} else {
		tmp = ((Math.log(t) * a) + Math.log((y + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = ((a * math.log(t)) + math.log(z)) - t
	elif t_1 <= 2000.0:
		tmp = (math.log(y) + math.log(z)) + (-0.5 * math.log(t))
	else:
		tmp = ((math.log(t) * a) + math.log((y + x))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	elseif (t_1 <= 2000.0)
		tmp = Float64(Float64(log(y) + log(z)) + Float64(-0.5 * log(t)));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(Float64(y + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = ((a * log(t)) + log(z)) - t;
	elseif (t_1 <= 2000.0)
		tmp = (log(y) + log(z)) + (-0.5 * log(t));
	else
		tmp = ((log(t) * a) + log((y + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 2000.0], N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $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 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))) < -1e6

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -1e6 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 2e3

   1. Initial program 99.1%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 37.9%

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

       1. lift-log.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. sum-log N/A

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

       5. lift-log.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lift-log.f64 49.3

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

   12. Applied rewrites 49.3%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 98.2

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

   7. Applied rewrites 98.2%

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

Alternative 5: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000000:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\log \left(z \cdot \left(y + x\right)\right)}{t}\right) - 1\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -200000000.0)
     (- (+ (* a (log t)) (log z)) t)
     (if (<= t_1 1000.0)
       (*
        (-
         (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 (/ (log (* z (+ y x))) t))
         1.0)
        t)
       (- (+ (* (log t) a) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -200000000.0) {
		tmp = ((a * log(t)) + log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = (fma((-log(t) * ((a - 0.5) / t)), -1.0, (log((z * (y + x))) / t)) - 1.0) * t;
	} else {
		tmp = ((log(t) * a) + log((y + x))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -200000000.0)
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, Float64(log(Float64(z * Float64(y + x))) / t)) - 1.0) * 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_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000000.0], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * t), $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 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))) < -2e8

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if -2e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.1%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 6: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -200000000:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\left(\mathsf{fma}\left(\left(-\log t\right) \cdot \frac{a - 0.5}{t}, -1, \frac{\log \left(z \cdot y\right)}{t}\right) - 1\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -200000000.0)
     (- (+ (* a (log t)) (log z)) t)
     (if (<= t_1 1000.0)
       (*
        (- (fma (* (- (log t)) (/ (- a 0.5) t)) -1.0 (/ (log (* z y)) t)) 1.0)
        t)
       (- (+ (* (log t) a) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -200000000.0) {
		tmp = ((a * log(t)) + log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = (fma((-log(t) * ((a - 0.5) / t)), -1.0, (log((z * y)) / t)) - 1.0) * t;
	} else {
		tmp = ((log(t) * a) + log((y + x))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -200000000.0)
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(fma(Float64(Float64(-log(t)) * Float64(Float64(a - 0.5) / t)), -1.0, Float64(log(Float64(z * y)) / t)) - 1.0) * 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_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -200000000.0], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[(N[(N[((-N[Log[t], $MachinePrecision]) * N[(N[(a - 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * t), $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 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))) < -2e8

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if -2e8 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.1%

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

   2. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 44.9%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 7: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -600:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\left(\log \left(y \cdot z\right) - t\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -600.0)
     (- (+ (* a (log t)) (log z)) t)
     (if (<= t_1 1000.0)
       (+ (- (log (* y z)) t) (* -0.5 (log t)))
       (- (+ (* (log t) a) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -600.0) {
		tmp = ((a * log(t)) + log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = (log((y * z)) - t) + (-0.5 * log(t));
	} else {
		tmp = ((log(t) * a) + log((y + x))) - 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) :: t_1
    real(8) :: tmp
    t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
    if (t_1 <= (-600.0d0)) then
        tmp = ((a * log(t)) + log(z)) - t
    else if (t_1 <= 1000.0d0) then
        tmp = (log((y * z)) - t) + ((-0.5d0) * log(t))
    else
        tmp = ((log(t) * a) + log((y + x))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
	double tmp;
	if (t_1 <= -600.0) {
		tmp = ((a * Math.log(t)) + Math.log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = (Math.log((y * z)) - t) + (-0.5 * Math.log(t));
	} else {
		tmp = ((Math.log(t) * a) + Math.log((y + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))
	tmp = 0
	if t_1 <= -600.0:
		tmp = ((a * math.log(t)) + math.log(z)) - t
	elif t_1 <= 1000.0:
		tmp = (math.log((y * z)) - t) + (-0.5 * math.log(t))
	else:
		tmp = ((math.log(t) * a) + math.log((y + x))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -600.0)
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = Float64(Float64(log(Float64(y * z)) - t) + Float64(-0.5 * log(t)));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(Float64(y + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	tmp = 0.0;
	if (t_1 <= -600.0)
		tmp = ((a * log(t)) + log(z)) - t;
	elseif (t_1 <= 1000.0)
		tmp = (log((y * z)) - t) + (-0.5 * log(t));
	else
		tmp = ((log(t) * a) + log((y + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -600.0], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $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 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 99.9%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 97.4

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

   9. Applied rewrites 97.4%

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

   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))) < 1e3

   1. Initial program 99.0%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 88.7%

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

   8. Taylor expanded in x around 0

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

      1. sum-log N/A

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

      2. lower--.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-*.f64 46.0

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

   10. Applied rewrites 46.0%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 8: 83.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + t\_1\\ \mathbf{if}\;t\_2 \leq -450:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \mathbf{elif}\;t\_2 \leq 1000:\\ \;\;\;\;\log \left(z \cdot y\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- a 0.5) (log t)))
        (t_2 (+ (- (+ (log (+ x y)) (log z)) t) t_1)))
   (if (<= t_2 -450.0)
     (- (+ (* a (log t)) (log z)) t)
     (if (<= t_2 1000.0)
       (+ (log (* z y)) t_1)
       (- (+ (* (log t) a) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * log(t);
	double t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -450.0) {
		tmp = ((a * log(t)) + log(z)) - t;
	} else if (t_2 <= 1000.0) {
		tmp = log((z * y)) + t_1;
	} else {
		tmp = ((log(t) * a) + log((y + x))) - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a - 0.5d0) * log(t)
    t_2 = ((log((x + y)) + log(z)) - t) + t_1
    if (t_2 <= (-450.0d0)) then
        tmp = ((a * log(t)) + log(z)) - t
    else if (t_2 <= 1000.0d0) then
        tmp = log((z * y)) + t_1
    else
        tmp = ((log(t) * a) + log((y + x))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (a - 0.5) * Math.log(t);
	double t_2 = ((Math.log((x + y)) + Math.log(z)) - t) + t_1;
	double tmp;
	if (t_2 <= -450.0) {
		tmp = ((a * Math.log(t)) + Math.log(z)) - t;
	} else if (t_2 <= 1000.0) {
		tmp = Math.log((z * y)) + t_1;
	} else {
		tmp = ((Math.log(t) * a) + Math.log((y + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (a - 0.5) * math.log(t)
	t_2 = ((math.log((x + y)) + math.log(z)) - t) + t_1
	tmp = 0
	if t_2 <= -450.0:
		tmp = ((a * math.log(t)) + math.log(z)) - t
	elif t_2 <= 1000.0:
		tmp = math.log((z * y)) + t_1
	else:
		tmp = ((math.log(t) * a) + math.log((y + x))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(a - 0.5) * log(t))
	t_2 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + t_1)
	tmp = 0.0
	if (t_2 <= -450.0)
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	elseif (t_2 <= 1000.0)
		tmp = Float64(log(Float64(z * y)) + t_1);
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(Float64(y + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (a - 0.5) * log(t);
	t_2 = ((log((x + y)) + log(z)) - t) + t_1;
	tmp = 0.0;
	if (t_2 <= -450.0)
		tmp = ((a * log(t)) + log(z)) - t;
	elseif (t_2 <= 1000.0)
		tmp = log((z * y)) + t_1;
	else
		tmp = ((log(t) * a) + log((y + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -450.0], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$2, 1000.0], N[(N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision] + t$95$1), $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 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))) < -450

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 96.5

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

   9. Applied rewrites 96.5%

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

   if -450 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.0%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.8

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

   7. Applied rewrites 46.8%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 9: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ \mathbf{if}\;t\_1 \leq -450:\\ \;\;\;\;\left(a \cdot \log t + \log z\right) - t\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot a + \log \left(y + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))))
   (if (<= t_1 -450.0)
     (- (+ (* a (log t)) (log z)) t)
     (if (<= t_1 1000.0)
       (fma (log t) -0.5 (log (* y z)))
       (- (+ (* (log t) a) (log (+ y x))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double tmp;
	if (t_1 <= -450.0) {
		tmp = ((a * log(t)) + log(z)) - t;
	} else if (t_1 <= 1000.0) {
		tmp = fma(log(t), -0.5, log((y * z)));
	} else {
		tmp = ((log(t) * a) + log((y + x))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	tmp = 0.0
	if (t_1 <= -450.0)
		tmp = Float64(Float64(Float64(a * log(t)) + log(z)) - t);
	elseif (t_1 <= 1000.0)
		tmp = fma(log(t), -0.5, log(Float64(y * z)));
	else
		tmp = Float64(Float64(Float64(log(t) * a) + log(Float64(y + x))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -450.0], N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t$95$1, 1000.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(y * z), $MachinePrecision]], $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 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))) < -450

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 96.5

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

   9. Applied rewrites 96.5%

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

   if -450 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.0%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lift-log.f64 45.9

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 45.9

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

   12. Applied rewrites 45.9%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-log.f64 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 10: 83.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\\ t_2 := \left(a \cdot \log t + \log z\right) - t\\ \mathbf{if}\;t\_1 \leq -450:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1000:\\ \;\;\;\;\mathsf{fma}\left(\log t, -0.5, \log \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))
        (t_2 (- (+ (* a (log t)) (log z)) t)))
   (if (<= t_1 -450.0)
     t_2
     (if (<= t_1 1000.0) (fma (log t) -0.5 (log (* y z))) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
	double t_2 = ((a * log(t)) + log(z)) - t;
	double tmp;
	if (t_1 <= -450.0) {
		tmp = t_2;
	} else if (t_1 <= 1000.0) {
		tmp = fma(log(t), -0.5, log((y * z)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
	t_2 = Float64(Float64(Float64(a * log(t)) + log(z)) - t)
	tmp = 0.0
	if (t_1 <= -450.0)
		tmp = t_2;
	elseif (t_1 <= 1000.0)
		tmp = fma(log(t), -0.5, log(Float64(y * z)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t$95$1, -450.0], t$95$2, If[LessEqual[t$95$1, 1000.0], N[(N[Log[t], $MachinePrecision] * -0.5 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

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))) < -450 or 1e3 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lift-log.f64 93.0

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

   9. Applied rewrites 93.0%

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

   if -450 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < 1e3

   1. Initial program 99.0%

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

   2. Taylor expanded in t around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 91.6

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

   4. Applied rewrites 91.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 89.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.9%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lift-log.f64 45.9

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 45.9

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

   12. Applied rewrites 45.9%

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

Alternative 11: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -33500000:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{elif}\;a \leq 0.35:\\ \;\;\;\;\left(\mathsf{fma}\left(-0.5, \log t, \log z\right) + \log \left(y + x\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log y + \log z\right) - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -33500000.0)
   (- (+ (fma (log t) a (log y)) (log z)) t)
   (if (<= a 0.35)
     (- (+ (fma -0.5 (log t) (log z)) (log (+ y x))) t)
     (+ (- (+ (log y) (log z)) t) (* a (log t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -33500000.0) {
		tmp = (fma(log(t), a, log(y)) + log(z)) - t;
	} else if (a <= 0.35) {
		tmp = (fma(-0.5, log(t), log(z)) + log((y + x))) - t;
	} else {
		tmp = ((log(y) + log(z)) - t) + (a * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -33500000.0)
		tmp = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t);
	elseif (a <= 0.35)
		tmp = Float64(Float64(fma(-0.5, log(t), log(z)) + log(Float64(y + x))) - t);
	else
		tmp = Float64(Float64(Float64(log(y) + log(z)) - t) + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -33500000.0], N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.35], 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[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.35e7

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 76.4%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 76.2%

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

   if -3.35e7 < a < 0.34999999999999998

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in a around 0

      \[\leadsto \left(\left(\log z + \frac{-1}{2} \cdot \log t\right) + \log \left(y + x\right)\right) - t \]
   6. 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 97.9

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

   7. Applied rewrites 97.9%

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

   if 0.34999999999999998 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 72.6%

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

Alternative 12: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log y + \log z\right) - t\\ \mathbf{if}\;a \leq -33500000:\\ \;\;\;\;\left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{elif}\;a \leq 0.35:\\ \;\;\;\;t\_1 + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_1 + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (log y) (log z)) t)))
   (if (<= a -33500000.0)
     (- (+ (fma (log t) a (log y)) (log z)) t)
     (if (<= a 0.35) (+ t_1 (* -0.5 (log t))) (+ t_1 (* a (log t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(y) + log(z)) - t;
	double tmp;
	if (a <= -33500000.0) {
		tmp = (fma(log(t), a, log(y)) + log(z)) - t;
	} else if (a <= 0.35) {
		tmp = t_1 + (-0.5 * log(t));
	} else {
		tmp = t_1 + (a * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(y) + log(z)) - t)
	tmp = 0.0
	if (a <= -33500000.0)
		tmp = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t);
	elseif (a <= 0.35)
		tmp = Float64(t_1 + Float64(-0.5 * log(t)));
	else
		tmp = Float64(t_1 + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -33500000.0], N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 0.35], N[(t$95$1 + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.35e7

   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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 76.4%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 76.2%

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

   if -3.35e7 < a < 0.34999999999999998

   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. Taylor expanded in x around 0

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 61.7%

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

   if 0.34999999999999998 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 72.6%

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

Alternative 13: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\mathsf{fma}\left(\log t, a, \log y\right) + \log z\right) - t\\ \mathbf{if}\;a \leq -33500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.35:\\ \;\;\;\;\left(\left(\log y + \log z\right) - t\right) + -0.5 \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ (fma (log t) a (log y)) (log z)) t)))
   (if (<= a -33500000.0)
     t_1
     (if (<= a 0.35) (+ (- (+ (log y) (log z)) t) (* -0.5 (log t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (fma(log(t), a, log(y)) + log(z)) - t;
	double tmp;
	if (a <= -33500000.0) {
		tmp = t_1;
	} else if (a <= 0.35) {
		tmp = ((log(y) + log(z)) - t) + (-0.5 * log(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(fma(log(t), a, log(y)) + log(z)) - t)
	tmp = 0.0
	if (a <= -33500000.0)
		tmp = t_1;
	elseif (a <= 0.35)
		tmp = Float64(Float64(Float64(log(y) + log(z)) - t) + Float64(-0.5 * log(t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -33500000.0], t$95$1, If[LessEqual[a, 0.35], N[(N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.35e7 or 0.34999999999999998 < 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. Taylor expanded in z around inf

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

      1. lower--.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. +-commutative N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. neg-log N/A

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

      14. +-commutative N/A

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

      15. associate-+r+ N/A

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 74.8%

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

   10. Taylor expanded in a around inf

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

   12. Applied rewrites 74.4%

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

   if -3.35e7 < a < 0.34999999999999998

   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. Taylor expanded in x around 0

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

   4. Applied rewrites 62.6%

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

   5. Taylor expanded in a around 0

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

   7. Applied rewrites 61.7%

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

Alternative 14: 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]

Derivation

1. Initial program 99.6%

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

Alternative 15: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\left(\log z + \left(\log \left(x + y\right) + \left(\frac{-1}{2} \cdot \log t + a \cdot \log t\right)\right)\right) - t} \]
3. 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 99.6

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

4. Applied rewrites 99.6%

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

Alternative 16: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 68.6%

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

Alternative 17: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z around inf

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

   1. lower--.f64 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift-log.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. lift-log.f64 N/A

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

   10. +-commutative N/A

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

   11. +-commutative N/A

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

   12. mul-1-neg N/A

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

   13. neg-log N/A

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

   14. +-commutative N/A

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

   15. associate-+r+ N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 68.6%

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

Alternative 18: 76.8% 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 99.6%

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

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a - \frac{1}{2}\right) \cdot \log t \]
3. 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 76.8

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

4. Applied rewrites 76.8%

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

Alternative 19: 62.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.65 \cdot 10^{+18}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.65e+18) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.65e+18) {
		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 <= 2.65d+18) 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 <= 2.65e+18) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.65e+18:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.65e+18)
		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 <= 2.65e+18)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.65e+18], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.65e18

   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. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \log t} \]
   3. 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 50.2

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

   4. Applied rewrites 50.2%

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

   if 2.65e18 < 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. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.8

         \[\leadsto -t \]

   4. Applied rewrites 75.8%

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

Alternative 20: 74.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in z around inf

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

   1. lower--.f64 N/A

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in a around inf

   \[\leadsto a \cdot \log t - t \]
6. 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 74.2

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

7. Applied rewrites 74.2%

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

Alternative 21: 37.7% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.7

      \[\leadsto -t \]

4. Applied rewrites 37.7%

   \[\leadsto \color{blue}{-t} \]
5. 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 2025105 
(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))))