Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 11.8s

Alternatives: 13

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.2551581507720492e-222
  2. y = 2.273827020251864e-29
  3. z = 2.1819749800179985e+280
  4. t = 1.9759195960006264e-219
  5. a = -7.075902191942023e+239

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

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

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

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

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

real(8) function code(x, y, z, t, a)
    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
3. Add Preprocessing

Alternative 2: 74.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower--.f64 N/A

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-log.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 51.3

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

   9. Applied rewrites 51.3%

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

   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 99.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 96.8%

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

   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 99.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 98.2

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

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\log t \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -800.0)
     (- t)
     (if (<= t_1 700.0)
       (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
       (* (log t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= -800.0) {
		tmp = -t;
	} else if (t_1 <= 700.0) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = Float64(-t);
	elseif (t_1 <= 700.0)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = Float64(log(t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], (-t), If[LessEqual[t$95$1, 700.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.4

         \[\leadsto \color{blue}{-t} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{-t} \]

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.6

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 39.4

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

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\log t \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 58.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -800:\\ \;\;\;\;-t\\ \mathbf{elif}\;t\_1 \leq 700:\\ \;\;\;\;\mathsf{fma}\left(a - 0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -800.0)
     (- t)
     (if (<= t_1 700.0)
       (- (fma (- a 0.5) (log t) (log (* z y))) t)
       (* (log t) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + log(z);
	double tmp;
	if (t_1 <= -800.0) {
		tmp = -t;
	} else if (t_1 <= 700.0) {
		tmp = fma((a - 0.5), log(t), log((z * y))) - t;
	} else {
		tmp = log(t) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + log(z))
	tmp = 0.0
	if (t_1 <= -800.0)
		tmp = Float64(-t);
	elseif (t_1 <= 700.0)
		tmp = Float64(fma(Float64(a - 0.5), log(t), log(Float64(z * y))) - t);
	else
		tmp = Float64(log(t) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -800.0], (-t), If[LessEqual[t$95$1, 700.0], N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < -800

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 44.4

         \[\leadsto \color{blue}{-t} \]

   5. Applied rewrites 44.4%

      \[\leadsto \color{blue}{-t} \]

   if -800 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 700

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower--.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

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

      1. lower-log.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.5

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

   9. Applied rewrites 56.5%

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

   if 700 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 39.4

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

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\log t \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log y\right) + \left(\log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.5e-6)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (if (<= t 1.6e+227)
     (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
     (+ (fma -0.5 (log t) (log y)) (- (log z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.5e-6) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else if (t <= 1.6e+227) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = fma(-0.5, log(t), log(y)) + (log(z) - t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.5e-6)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	elseif (t <= 1.6e+227)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = Float64(fma(-0.5, log(t), log(y)) + Float64(log(z) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.5e-6], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+227], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 7.50000000000000019e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.4%

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

   if 7.50000000000000019e-6 < t < 1.59999999999999994e227

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 79.7

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

   4. Applied rewrites 79.7%

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

   if 1.59999999999999994e227 < t

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 62.9

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 55.8%

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

Alternative 6: 72.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log z\right) + \log y\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a - 0.5, \log \left(z \cdot \left(y + x\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.5e-6)
   (+ (fma (log t) (- a 0.5) (log z)) (log y))
   (if (<= t 1.6e+227)
     (- (fma (log t) (- a 0.5) (log (* z (+ y x)))) t)
     (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.5e-6) {
		tmp = fma(log(t), (a - 0.5), log(z)) + log(y);
	} else if (t <= 1.6e+227) {
		tmp = fma(log(t), (a - 0.5), log((z * (y + x)))) - t;
	} else {
		tmp = -t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7.5e-6)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(z)) + log(y));
	elseif (t <= 1.6e+227)
		tmp = Float64(fma(log(t), Float64(a - 0.5), log(Float64(z * Float64(y + x)))) - t);
	else
		tmp = Float64(-t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.5e-6], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.6e+227], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision] + N[Log[N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < 7.50000000000000019e-6

   1. Initial program 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-out-- N/A

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

      5. metadata-eval N/A

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

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

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

      7. +-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-log.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-log.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.4%

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

   if 7.50000000000000019e-6 < t < 1.59999999999999994e227

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. sum-log N/A

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

      13. lower-log.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 79.7

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 79.7

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

   4. Applied rewrites 79.7%

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

   if 1.59999999999999994e227 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.9

         \[\leadsto \color{blue}{-t} \]

   5. Applied rewrites 92.9%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 8: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[Log[z], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. associate-+r- N/A

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

   4. lower--.f64 N/A

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

6. Applied rewrites 74.0%

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

7. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

   5. lower-+.f64 N/A

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

   6. lower-log.f64 N/A

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

   7. lower-log.f64 62.7

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

9. Applied rewrites 62.7%

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

Alternative 9: 68.2% accurate, 1.0× speedup

Math

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

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(fma(log(t), Float64(a - 0.5), log(y)) + Float64(log(z) - t))
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. associate--l+ N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 N/A

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

   14. lower--.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-log.f64 62.7

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

7. Applied rewrites 62.7%

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

Alternative 10: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. lower-+.f64 N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-out-- N/A

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

   6. metadata-eval N/A

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

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

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

   8. distribute-rgt-out N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-log.f64 62.7

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

5. Applied rewrites 62.7%

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

Alternative 11: 61.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+21} \lor \neg \left(a - 0.5 \leq 40000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -1e+21) (not (<= (- a 0.5) 40000000.0)))
   (* (log t) a)
   (- (fma -0.5 (log t) (log (* z y))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -1e+21) || !((a - 0.5) <= 40000000.0)) {
		tmp = log(t) * a;
	} else {
		tmp = fma(-0.5, log(t), log((z * y))) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -1e+21) || !(Float64(a - 0.5) <= 40000000.0))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(fma(-0.5, log(t), log(Float64(z * y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+21], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 40000000.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a #s(literal 1/2 binary64)) < -1e21 or 4e7 < (-.f64 a #s(literal 1/2 binary64))

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -1e21 < (-.f64 a #s(literal 1/2 binary64)) < 4e7

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. associate--l+ N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. associate-+r- N/A

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

      4. lower--.f64 N/A

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

   6. Applied rewrites 73.8%

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

   7. Taylor expanded in x around 0

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

      1. lower-log.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.5

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

   9. Applied rewrites 41.5%

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

   10. Taylor expanded in a around 0

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

   12. Applied rewrites 40.2%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+21} \lor \neg \left(a - 0.5 \leq 40000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \log t, \log \left(z \cdot y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+65} \lor \neg \left(a \leq 110000000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7e+65) (not (<= a 110000000000.0))) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7e+65) || !(a <= 110000000000.0)) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-7d+65)) .or. (.not. (a <= 110000000000.0d0))) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7e+65) || !(a <= 110000000000.0)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7e+65) or not (a <= 110000000000.0):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7e+65) || !(a <= 110000000000.0))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -7e+65) || ~((a <= 110000000000.0)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7e+65], N[Not[LessEqual[a, 110000000000.0]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -7.0000000000000002e65 or 1.1e11 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -7.0000000000000002e65 < a < 1.1e11

   1. Initial program 99.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 48.9

         \[\leadsto \color{blue}{-t} \]

   5. Applied rewrites 48.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{+65} \lor \neg \left(a \leq 110000000000\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.4% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 35.4

      \[\leadsto \color{blue}{-t} \]

5. Applied rewrites 35.4%

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

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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 2024329 
(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))))