Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.0s

Alternatives: 17

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

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 99.7%

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

Alternative 2: 64.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{elif}\;t\_1 \leq 675:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(y \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -750.0)
     (fma (log t) a (- (log z) t))
     (if (<= t_1 675.0)
       (fma (+ a -0.5) (log t) (- (log (* y z)) t))
       (- (fma (log t) (+ a -0.5) (log y)) t)))))

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 <= -750.0) {
		tmp = fma(log(t), a, (log(z) - t));
	} else if (t_1 <= 675.0) {
		tmp = fma((a + -0.5), log(t), (log((y * z)) - t));
	} else {
		tmp = fma(log(t), (a + -0.5), log(y)) - t;
	}
	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 <= -750.0)
		tmp = fma(log(t), a, Float64(log(z) - t));
	elseif (t_1 <= 675.0)
		tmp = fma(Float64(a + -0.5), log(t), Float64(log(Float64(y * z)) - t));
	else
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(y)) - t);
	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, -750.0], N[(N[Log[t], $MachinePrecision] * a + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 675.0], N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 79.5

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

   7. Simplified 79.5%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 79.7

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

   9. Applied egg-rr 79.7%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 675

   1. Initial program 99.7%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   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. accelerator-lowering-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. log-lowering-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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 N/A

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

      7. log-lowering-log.f64 71.1

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

   7. Simplified 71.1%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. +-lowering-+.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. associate-+r- N/A

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

      7. --lowering--.f64 N/A

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

      8. sum-log N/A

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

      9. log-lowering-log.f64 N/A

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

      10. *-lowering-*.f64 70.3

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

   9. Applied egg-rr 70.3%

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

   if 675 < (+.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   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. accelerator-lowering-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. log-lowering-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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 N/A

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

      7. log-lowering-log.f64 73.6

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

   7. Simplified 73.6%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 65.0

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

   10. Simplified 65.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 675:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(y \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log z\\ \mathbf{if}\;t\_1 \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{elif}\;t\_1 \leq 675:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (log z))))
   (if (<= t_1 -750.0)
     (fma (log t) a (- (log z) t))
     (if (<= t_1 675.0)
       (- (fma (log t) (+ a -0.5) (log (* y z))) t)
       (- (fma (log t) (+ a -0.5) (log y)) t)))))

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 <= -750.0) {
		tmp = fma(log(t), a, (log(z) - t));
	} else if (t_1 <= 675.0) {
		tmp = fma(log(t), (a + -0.5), log((y * z))) - t;
	} else {
		tmp = fma(log(t), (a + -0.5), log(y)) - t;
	}
	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 <= -750.0)
		tmp = fma(log(t), a, Float64(log(z) - t));
	elseif (t_1 <= 675.0)
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(Float64(y * z))) - t);
	else
		tmp = Float64(fma(log(t), Float64(a + -0.5), log(y)) - t);
	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, -750.0], N[(N[Log[t], $MachinePrecision] * a + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 675.0], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.6%

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 79.5

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

   7. Simplified 79.5%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 79.7

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

   9. Applied egg-rr 79.7%

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

   if -750 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 675

   1. Initial program 99.7%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 70.2

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

   7. Simplified 70.2%

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

   if 675 < (+.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. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   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. accelerator-lowering-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. log-lowering-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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 N/A

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

      7. log-lowering-log.f64 73.6

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

   7. Simplified 73.6%

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

   8. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 65.0

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

   10. Simplified 65.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq -750:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \mathbf{elif}\;\log \left(x + y\right) + \log z \leq 675:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a + -0.5, \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.45:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \mathsf{fma}\left(\log t, a, \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.45)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (+ (- (log z) t) (fma (log t) a (log y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.45) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else {
		tmp = (log(z) - t) + fma(log(t), a, log(y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.45)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	else
		tmp = Float64(Float64(log(z) - t) + fma(log(t), a, log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.45], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.450000000000000011

   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 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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 N/A

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

      9. log-lowering-log.f64 64.2

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

   8. Simplified 64.2%

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

   if 0.450000000000000011 < 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   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. accelerator-lowering-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. log-lowering-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. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-lowering-+.f64 N/A

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

      7. log-lowering-log.f64 77.8

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

   7. Simplified 77.8%

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

   8. Taylor expanded in a around inf

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

   10. Simplified 77.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.45:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + \mathsf{fma}\left(\log t, a, \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 26:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, \log z - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 26.0)
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))
   (fma (log t) a (- (log z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 26.0) {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	} else {
		tmp = fma(log(t), a, (log(z) - t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 26.0)
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	else
		tmp = fma(log(t), a, Float64(log(z) - t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 26.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 26

   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 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. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-lowering-+.f64 N/A

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

      9. log-lowering-log.f64 64.2

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

   8. Simplified 64.2%

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

   if 26 < 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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-+r+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. +-lowering-+.f64 N/A

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

      8. metadata-eval N/A

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

      9. log-lowering-log.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. log-lowering-log.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 99.1

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

   7. Simplified 99.1%

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

      1. accelerator-lowering-fma.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. log-lowering-log.f64 99.1

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

   9. Applied egg-rr 99.1%

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

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

Alternative 7: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

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. accelerator-lowering-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. log-lowering-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. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-lowering-+.f64 N/A

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

   7. log-lowering-log.f64 71.4

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

7. Simplified 71.4%

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

8. Final simplification 71.4%

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

Alternative 8: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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. associate--l+ N/A

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

   2. +-lowering-+.f64 N/A

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

   3. log-lowering-log.f64 N/A

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

   4. +-commutative N/A

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

   5. associate--l+ N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. log-lowering-log.f64 N/A

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

   10. +-lowering-+.f64 N/A

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

   11. --lowering--.f64 N/A

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

   12. log-lowering-log.f64 71.4

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

5. Simplified 71.4%

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

Alternative 9: 77.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 78.5

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

7. Simplified 78.5%

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

8. Final simplification 78.5%

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

Alternative 10: 57.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

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. accelerator-lowering-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. log-lowering-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. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-lowering-+.f64 N/A

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

   7. log-lowering-log.f64 71.4

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

7. Simplified 71.4%

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

8. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 60.4

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

10. Simplified 60.4%

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

11. Final simplification 60.4%

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

Alternative 11: 76.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. log-lowering-log.f64 77.9

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

7. Simplified 77.9%

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

   1. accelerator-lowering-fma.f64 N/A

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

   2. log-lowering-log.f64 N/A

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

   3. --lowering--.f64 N/A

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

   4. log-lowering-log.f64 77.9

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

9. Applied egg-rr 77.9%

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

Alternative 12: 76.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. log-lowering-log.f64 77.9

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

7. Simplified 77.9%

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

   1. associate-+r- N/A

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

   2. --lowering--.f64 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. log-lowering-log.f64 N/A

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

   5. log-lowering-log.f64 77.9

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

9. Applied egg-rr 77.9%

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

Alternative 13: 61.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.35e+26) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.35e+26) {
		tmp = a * log(t);
	} 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 (t <= 1.35d+26) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.35e+26) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.35e+26:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.35e+26)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.35e+26)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.35e+26], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.35e26

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 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. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 50.4

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

   5. Simplified 50.4%

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

   if 1.35e26 < 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. neg-lowering-neg.f64 80.0

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

   5. Simplified 80.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.35 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 77.6

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

5. Simplified 77.6%

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

6. Final simplification 77.6%

   \[\leadsto \left(a - 0.5\right) \cdot \log t - t \]
7. Add Preprocessing

Alternative 15: 74.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. log-lowering-log.f64 77.9

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

7. Simplified 77.9%

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

8. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \log t \cdot a + \color{blue}{\left(\mathsf{neg}\left(t\right)\right)} \]

   2. neg-lowering-neg.f64 75.3

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

10. Simplified 75.3%

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

11. Final simplification 75.3%

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

Alternative 16: 40.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return log(z) - 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(z) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-+r+ N/A

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

   4. +-lowering-+.f64 N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. +-lowering-+.f64 N/A

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

   8. metadata-eval N/A

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

   9. log-lowering-log.f64 N/A

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

   10. log-lowering-log.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. --lowering--.f64 N/A

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

   13. log-lowering-log.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

   3. log-lowering-log.f64 77.9

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

7. Simplified 77.9%

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

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\log z - t} \]
9. Step-by-step derivation

   1. --lowering--.f64 N/A

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

   2. log-lowering-log.f64 43.9

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

10. Simplified 43.9%

    \[\leadsto \color{blue}{\log z - t} \]
11. Add Preprocessing

Alternative 17: 37.3% accurate, 107.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 99.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 40.9

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

5. Simplified 40.9%

   \[\leadsto \color{blue}{-t} \]
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 2024198 
(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))))