Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 16.1s

Alternatives: 13

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: 92.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 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 96.5

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

   7. Simplified 96.5%

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

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

   1. Initial program 99.2%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sum-log N/A

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

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

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

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

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

      11. +-lowering-+.f64 93.8

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.7

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

   7. Simplified 93.7%

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

4. Final simplification 95.9%

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

Alternative 3: 83.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 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 96.5

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

   7. Simplified 96.5%

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

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

   1. Initial program 99.2%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sum-log N/A

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

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

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

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

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

      11. +-lowering-+.f64 93.8

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.7

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

   7. Simplified 93.7%

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

   8. Taylor expanded in x around 0

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

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

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

      2. *-lowering-*.f64 52.0

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

   10. Simplified 52.0%

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

4. Final simplification 87.5%

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

Alternative 4: 92.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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 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 96.5

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

   7. Simplified 96.5%

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

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

   1. Initial program 99.2%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing
   3. 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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sum-log N/A

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

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

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

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

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

      11. +-lowering-+.f64 93.8

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 93.7

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

   7. Simplified 93.7%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 90.8

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

   10. Simplified 90.8%

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

4. Final simplification 95.4%

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

Alternative 5: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -10000000000000:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log z + \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<=
      (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
      -10000000000000.0)
   (+ (- (log z) t) (* a (log t)))
   (fma (+ a -0.5) (log t) (+ (log z) (log y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t))) <= -10000000000000.0) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = fma((a + -0.5), log(t), (log(z) + log(y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t))) <= -10000000000000.0)
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = fma(Float64(a + -0.5), log(t), Float64(log(z) + log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[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], -10000000000000.0], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e13

   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 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 99.7

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

   7. Simplified 99.7%

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

   if -1e13 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.4%

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

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

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

      3. sub-neg N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sum-log N/A

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

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

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

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

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

      11. +-lowering-+.f64 82.7

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 82.6

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

   7. Simplified 82.6%

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

   8. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. log-rec N/A

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

      4. remove-double-neg N/A

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

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

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

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

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

      7. log-lowering-log.f64 61.5

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

   10. Simplified 61.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -10000000000000:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log z + \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -10000000000000:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<=
      (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t)))
      -10000000000000.0)
   (+ (- (log z) t) (* a (log t)))
   (+ (log y) (fma (log t) (+ a -0.5) (log z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t))) <= -10000000000000.0) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = log(y) + fma(log(t), (a + -0.5), log(z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t))) <= -10000000000000.0)
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(log(y) + fma(log(t), Float64(a + -0.5), log(z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[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], -10000000000000.0], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t))) < -1e13

   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 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 99.7

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

   7. Simplified 99.7%

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

   if -1e13 < (+.f64 (-.f64 (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) t) (*.f64 (-.f64 a #s(literal 1/2 binary64)) (log.f64 t)))

   1. Initial program 99.4%

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

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

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

      3. sub-neg N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sum-log N/A

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

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

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

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

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

      11. +-lowering-+.f64 82.7

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 82.6

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

   7. Simplified 82.6%

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

   8. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. associate-+r+ N/A

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

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

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

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

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

      9. +-commutative N/A

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

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

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

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

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

      15. log-lowering-log.f64 61.5

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

   10. Simplified 61.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \leq -10000000000000:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log y + \mathsf{fma}\left(\log t, a + -0.5, \log z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (+ (log (+ x y)) (log z)) 720.0)
   (fma (+ a -0.5) (log t) (- (log (* (+ x y) z)) t))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((log((x + y)) + log(z)) <= 720.0) {
		tmp = fma((a + -0.5), log(t), (log(((x + y) * z)) - t));
	} else {
		tmp = (log(z) - t) + (a * log(t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(log(Float64(x + y)) + log(z)) <= 720.0)
		tmp = fma(Float64(a + -0.5), log(t), Float64(log(Float64(Float64(x + y) * z)) - t));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision], 720.0], N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (log.f64 (+.f64 x y)) (log.f64 z)) < 720

   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. accelerator-lowering-fma.f64 N/A

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

      3. sub-neg N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

      8. sum-log N/A

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

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

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

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

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

      11. +-lowering-+.f64 97.5

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

   4. Applied egg-rr 97.5%

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

   if 720 < (+.f64 (log.f64 (+.f64 x y)) (log.f64 z))

   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 82.3

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

   7. Simplified 82.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(x + y\right) + \log z \leq 720:\\ \;\;\;\;\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 99.6

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

4. Applied egg-rr 99.6%

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

Alternative 9: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $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. accelerator-lowering-fma.f64 N/A

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

   3. sub-neg N/A

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

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

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

   5. metadata-eval N/A

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

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

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

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

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

   8. sum-log N/A

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

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

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

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

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

   11. +-lowering-+.f64 81.8

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

4. Applied egg-rr 81.8%

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

5. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. log-rec N/A

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

   4. remove-double-neg N/A

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

   5. associate--l+ N/A

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

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

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

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

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

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

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

   9. log-lowering-log.f64 68.5

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

7. Simplified 68.5%

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

8. Final simplification 68.5%

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

Alternative 10: 68.5% 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 99.6

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

4. Applied egg-rr 99.6%

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

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 68.4

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

7. Simplified 68.4%

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

8. Final simplification 68.4%

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

Alternative 11: 61.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 16000000000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 16000000000000.0) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 16000000000000.0) {
		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 <= 16000000000000.0d0) 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 <= 16000000000000.0) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 16000000000000.0:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 16000000000000.0)
		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 <= 16000000000000.0)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 16000000000000.0], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.6e13

   1. Initial program 99.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 56.8

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

   5. Simplified 56.8%

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

   if 1.6e13 < 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 76.8

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

   5. Simplified 76.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 16000000000000:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

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

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

   3. sub-neg N/A

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

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

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

   5. metadata-eval N/A

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

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

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

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

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

   8. sum-log N/A

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

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

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

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

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

   11. +-lowering-+.f64 81.8

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

4. Applied egg-rr 81.8%

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

5. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 79.7

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

7. Simplified 79.7%

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

Alternative 13: 37.1% 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 37.8

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

5. Simplified 37.8%

   \[\leadsto \color{blue}{-t} \]
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 2024204 
(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))))