Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 20.3s

Alternatives: 15

Speedup: 0.8×

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, 0.8× speedup

Math

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

FPCore

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

Julia

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

Derivation

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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate--l+ 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -50.0)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= (- a 0.5) -0.5)
     (- (+ (log z) (+ (log (+ x y)) (* -0.5 (log t)))) t)
     (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -50.0) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if ((a - 0.5) <= -0.5) {
		tmp = (log(z) + (log((x + y)) + (-0.5 * log(t)))) - t;
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -50.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (Float64(a - 0.5) <= -0.5)
		tmp = Float64(Float64(log(z) + Float64(log(Float64(x + y)) + Float64(-0.5 * log(t)))) - t);
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -50.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a 1/2) < -50

   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. Step-by-step derivation

      1. cancel-sign-sub 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. associate--l+ 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

   if -50 < (-.f64 a 1/2) < -0.5

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 99.0%

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

   if -0.5 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. fma-neg 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -50.0)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= (- a 0.5) -0.5)
     (- (+ (log z) (+ (log y) (* -0.5 (log t)))) t)
     (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -50.0) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if ((a - 0.5) <= -0.5) {
		tmp = (log(z) + (log(y) + (-0.5 * log(t)))) - t;
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -50.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (Float64(a - 0.5) <= -0.5)
		tmp = Float64(Float64(log(z) + Float64(log(y) + Float64(-0.5 * log(t)))) - t);
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -50.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a 1/2) < -50

   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. Step-by-step derivation

      1. cancel-sign-sub 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. associate--l+ 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

   if -50 < (-.f64 a 1/2) < -0.5

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 63.6%

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

   5. Taylor expanded in a around 0 63.6%

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

   if -0.5 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. fma-neg 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 4: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -50.0)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= (- a 0.5) -0.5)
     (- (+ (log y) (log (* z (pow t -0.5)))) t)
     (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -50.0) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if ((a - 0.5) <= -0.5) {
		tmp = (log(y) + log((z * pow(t, -0.5)))) - t;
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -50.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (Float64(a - 0.5) <= -0.5)
		tmp = Float64(Float64(log(y) + log(Float64(z * (t ^ -0.5)))) - t);
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -50.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a 1/2) < -50

   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. Step-by-step derivation

      1. cancel-sign-sub 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. associate--l+ 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

   if -50 < (-.f64 a 1/2) < -0.5

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 63.6%

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

      1. +-commutative 63.6%

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

      2. associate--l+ 63.6%

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

      3. remove-double-neg 63.6%

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

      4. log-rec 63.6%

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

      5. mul-1-neg 63.6%

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

      6. associate--l+ 63.6%

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

      7. +-commutative 63.6%

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

   6. Simplified 59.9%

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

   7. Taylor expanded in a around 0 59.9%

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

      1. *-commutative 59.9%

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

      2. unpow1/2 59.9%

         \[\leadsto \left(\log \left(z \cdot \color{blue}{{\left(\frac{1}{t}\right)}^{0.5}}\right) + \log y\right) - t \]

      3. exp-to-pow 59.8%

         \[\leadsto \left(\log \left(z \cdot \color{blue}{e^{\log \left(\frac{1}{t}\right) \cdot 0.5}}\right) + \log y\right) - t \]

      4. log-rec 59.8%

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

      5. distribute-lft-neg-out 59.8%

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

      6. distribute-rgt-neg-in 59.8%

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

      7. metadata-eval 59.8%

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

      8. exp-to-pow 59.9%

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

   9. Simplified 59.9%

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

   if -0.5 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. fma-neg 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. cancel-sign-sub 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. associate--l+ 99.5%

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

   4. remove-double-neg 99.5%

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

   5. sub-neg 99.5%

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

   6. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.7:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 5.35 \cdot 10^{-17}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -0.7)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= a 5.35e-17)
     (- (+ (log z) (log (* y (pow t -0.5)))) t)
     (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -0.7) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if (a <= 5.35e-17) {
		tmp = (log(z) + log((y * pow(t, -0.5)))) - t;
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -0.7)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (a <= 5.35e-17)
		tmp = Float64(Float64(log(z) + log(Float64(y * (t ^ -0.5)))) - t);
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -0.7], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 5.35e-17], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.69999999999999996

   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. Step-by-step derivation

      1. cancel-sign-sub 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. associate--l+ 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

   if -0.69999999999999996 < a < 5.35000000000000004e-17

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 63.6%

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

   5. Taylor expanded in a around 0 63.6%

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

      1. add-log-exp 63.6%

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

      2. sum-log 53.7%

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

      3. *-commutative 53.7%

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

      4. exp-to-pow 53.7%

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

   7. Applied egg-rr 53.7%

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

   if 5.35000000000000004e-17 < a

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. fma-neg 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.7:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 5.35 \cdot 10^{-17}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 7: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate--l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 70.6%

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

5. Final simplification 70.6%

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

Alternative 8: 71.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (- a 0.5) -50.0)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= (- a 0.5) -0.5)
     (- (log (* z (* y (pow t -0.5)))) t)
     (fma (log t) a (- t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a - 0.5) <= -50.0) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if ((a - 0.5) <= -0.5) {
		tmp = log((z * (y * pow(t, -0.5)))) - t;
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(a - 0.5) <= -50.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (Float64(a - 0.5) <= -0.5)
		tmp = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t);
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(a - 0.5), $MachinePrecision], -50.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(a - 0.5), $MachinePrecision], -0.5], N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 a 1/2) < -50

   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. Step-by-step derivation

      1. cancel-sign-sub 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. associate--l+ 99.6%

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

      4. remove-double-neg 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 98.5%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.5%

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

   6. Simplified 98.5%

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

   if -50 < (-.f64 a 1/2) < -0.5

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 63.6%

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

   5. Taylor expanded in a around 0 63.6%

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

      1. add-log-exp 53.4%

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

      2. sum-log 45.2%

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

      3. exp-sum 45.3%

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

      4. add-exp-log 45.8%

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

      5. *-commutative 45.8%

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

      6. exp-to-pow 45.8%

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

   7. Applied egg-rr 45.8%

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

   if -0.5 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 97.7%

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

      1. *-commutative 97.7%

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

   7. Simplified 97.7%

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

      1. fma-neg 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -50:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a - 0.5 \leq -0.5:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 9: 87.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.5e-22)
   (+ (* (+ a -0.5) (log t)) (log (* z (+ x y))))
   (fma (log t) a (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.5e-22) {
		tmp = ((a + -0.5) * log(t)) + log((z * (x + y)));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.5e-22)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + log(Float64(z * Float64(x + y))));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.5e-22], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.49999999999999977e-22

   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. Step-by-step derivation

      1. cancel-sign-sub 99.2%

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

      2. cancel-sign-sub-inv 99.2%

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

      3. associate--l+ 99.2%

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

      4. remove-double-neg 99.2%

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

      5. sub-neg 99.2%

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

      6. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. log-prod 73.5%

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

      3. +-commutative 73.5%

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

   6. Simplified 73.5%

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

   if 2.49999999999999977e-22 < 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. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. fma-def 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 75.7%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. fma-neg 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.5 \cdot 10^{-22}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 10: 74.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.7e-21)
   (+ (* (log t) (- a 0.5)) (log (* y z)))
   (fma (log t) a (- t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.7e-21) {
		tmp = (log(t) * (a - 0.5)) + log((y * z));
	} else {
		tmp = fma(log(t), a, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.7e-21)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(y * z)));
	else
		tmp = fma(log(t), a, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.7e-21], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] * a + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.7e-21

   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. Step-by-step derivation

      1. +-commutative 99.2%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. associate--l+ 99.2%

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

   3. Simplified 99.2%

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

      1. fma-udef 99.2%

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

      2. +-commutative 99.2%

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

      3. add-sqr-sqrt 61.0%

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

      4. pow2 61.0%

         \[\leadsto \color{blue}{{\left(\sqrt{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t}\right)}^{2}} \]

      5. +-commutative 61.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\left(a + -0.5\right) \cdot \log t + \left(\log \left(x + y\right) + \left(\log z - t\right)\right)}}\right)}^{2} \]

      6. fma-udef 61.0%

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right) + \left(\log z - t\right)\right)}}\right)}^{2} \]

      7. associate-+r- 61.0%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\left(\log \left(x + y\right) + \log z\right) - t}\right)}\right)}^{2} \]

      8. sum-log 44.2%

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \color{blue}{\log \left(\left(x + y\right) \cdot z\right)} - t\right)}\right)}^{2} \]

   5. Applied egg-rr 44.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(a + -0.5, \log t, \log \left(\left(x + y\right) \cdot z\right) - t\right)}\right)}^{2}} \]

   6. Taylor expanded in x around 0 25.5%

      \[\leadsto {\color{blue}{\left(\sqrt{\left(\log \left(y \cdot z\right) + \left(a - 0.5\right) \cdot \log t\right) - t}\right)}}^{2} \]

   7. Taylor expanded in t around 0 47.3%

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

   if 1.7e-21 < 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. Step-by-step derivation

      1. +-commutative 99.8%

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

      2. fma-def 99.8%

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

      3. sub-neg 99.8%

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

      4. metadata-eval 99.8%

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

      5. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 75.7%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

      1. fma-neg 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.7 \cdot 10^{-21}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log t, a, -t\right)\\ \end{array} \]

Alternative 11: 77.6% accurate, 1.5× 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate--l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in t around inf 80.1%

   \[\leadsto \mathsf{fma}\left(a + -0.5, \log t, \color{blue}{-1 \cdot t}\right) \]
5. Step-by-step derivation

   1. neg-mul-1 80.1%

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

6. Simplified 80.1%

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

7. Final simplification 80.1%

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

Alternative 12: 61.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -19000000000000 \lor \neg \left(a \leq 5.5 \cdot 10^{+113}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -19000000000000.0) (not (<= a 5.5e+113))) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -19000000000000.0) || !(a <= 5.5e+113)) {
		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 ((a <= (-19000000000000.0d0)) .or. (.not. (a <= 5.5d+113))) 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 ((a <= -19000000000000.0) || !(a <= 5.5e+113)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -19000000000000.0) or not (a <= 5.5e+113):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -19000000000000.0) || !(a <= 5.5e+113))
		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 ((a <= -19000000000000.0) || ~((a <= 5.5e+113)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -19000000000000.0], N[Not[LessEqual[a, 5.5e+113]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -1.9e13 or 5.5000000000000001e113 < a

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around inf 84.8%

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

      1. *-commutative 84.8%

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

   6. Simplified 84.8%

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

   if -1.9e13 < a < 5.5000000000000001e113

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.5%

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

      3. sub-neg 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate--l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 65.7%

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

   5. Taylor expanded in a around inf 59.6%

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

      1. *-commutative 59.6%

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

   7. Simplified 59.6%

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

   8. Taylor expanded in t around inf 53.9%

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

      1. neg-mul-1 53.9%

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

   10. Simplified 53.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -19000000000000 \lor \neg \left(a \leq 5.5 \cdot 10^{+113}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 77.6% 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 99.5%

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

   1. cancel-sign-sub 99.5%

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

   2. cancel-sign-sub-inv 99.5%

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

   3. associate--l+ 99.5%

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

   4. remove-double-neg 99.5%

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

   5. sub-neg 99.5%

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

   6. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in t around inf 80.1%

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. neg-mul-1 80.1%

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

6. Simplified 80.1%

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

7. Final simplification 80.1%

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

Alternative 14: 75.0% accurate, 3.0× 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate--l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 70.6%

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

5. Taylor expanded in a around inf 77.8%

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

   1. *-commutative 77.8%

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

7. Simplified 77.8%

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

8. Final simplification 77.8%

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

Alternative 15: 37.6% accurate, 156.5× 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 99.5%

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

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

   3. sub-neg 99.5%

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

   4. metadata-eval 99.5%

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

   5. associate--l+ 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 70.6%

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

5. Taylor expanded in a around inf 77.8%

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

   1. *-commutative 77.8%

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

7. Simplified 77.8%

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

8. Taylor expanded in t around inf 37.3%

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

   1. neg-mul-1 37.3%

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

10. Simplified 37.3%

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

11. Final simplification 37.3%

    \[\leadsto -t \]

Developer target: 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 2023268 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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