Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.6% → 99.8%

Time: 16.1s

Alternatives: 15

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Initial Program: 88.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x - 1.0d0) * log(y)) + ((z - 1.0d0) * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x - 1.0) * Math.log(y)) + ((z - 1.0) * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x - 1.0) * log(y)) + Float64(Float64(z - 1.0) * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * log((1.0 - y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(z - 1.0), $MachinePrecision] * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \log y \cdot \left(x + -1\right) - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (+ z -1.0) (log1p (- y)) (- (* (log y) (+ x -1.0)) t)))

C

double code(double x, double y, double z, double t) {
	return fma((z + -1.0), log1p(-y), ((log(y) * (x + -1.0)) - t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(z + -1.0), log1p(Float64(-y)), Float64(Float64(log(y) * Float64(x + -1.0)) - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z + -1.0), $MachinePrecision] * N[Log[1 + (-y)], $MachinePrecision] + N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
2. Step-by-step derivation

   1. +-commutative 86.9%

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

   2. associate--l+ 86.9%

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

   3. fma-def 86.9%

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

   4. sub-neg 86.9%

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

   5. log1p-def 99.8%

      \[\leadsto \mathsf{fma}\left(z - 1, \color{blue}{\mathsf{log1p}\left(-y\right)}, \left(x - 1\right) \cdot \log y - t\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \left(x - 1\right) \cdot \log y - t\right)} \]

4. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(z + -1, \mathsf{log1p}\left(-y\right), \log y \cdot \left(x + -1\right) - t\right) \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\log y \cdot \left(x + -1\right) + z \cdot \mathsf{log1p}\left(-y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (log y) (+ x -1.0)) (* z (log1p (- y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (x + -1.0)) + (z * log1p(-y))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (x + -1.0)) + (z * Math.log1p(-y))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) + (z * math.log1p(-y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(z * log1p(Float64(-y)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

2. Taylor expanded in z around inf 86.5%

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

   1. *-commutative 86.5%

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

   2. sub-neg 86.5%

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

   3. mul-1-neg 86.5%

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

   4. log1p-def 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

   5. mul-1-neg 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

4. Simplified 99.5%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]

5. Final simplification 99.5%

   \[\leadsto \left(\log y \cdot \left(x + -1\right) + z \cdot \mathsf{log1p}\left(-y\right)\right) - t \]

Alternative 3: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (log y) (+ x -1.0)) (* (+ z -1.0) (- (* y (* y -0.5)) y))) t))

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (x + -1.0)) + ((z + -1.0) * ((y * (y * -0.5)) - y))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * (x + (-1.0d0))) + ((z + (-1.0d0)) * ((y * (y * (-0.5d0))) - y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (x + -1.0)) + ((z + -1.0) * ((y * (y * -0.5)) - y))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) + ((z + -1.0) * ((y * (y * -0.5)) - y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(Float64(z + -1.0) * Float64(Float64(y * Float64(y * -0.5)) - y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (x + -1.0)) + ((z + -1.0) * ((y * (y * -0.5)) - y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * N[(N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

2. Taylor expanded in y around 0 99.4%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)}\right) - t \]
3. Step-by-step derivation

   1. mul-1-neg 99.4%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right)\right) - t \]

   2. unsub-neg 99.4%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)}\right) - t \]

   3. unpow2 99.4%

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

   4. associate-*r* 99.4%

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

4. Simplified 99.4%

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

5. Final simplification 99.4%

   \[\leadsto \left(\log y \cdot \left(x + -1\right) + \left(z + -1\right) \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\right) - t \]

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(\log y \cdot \left(x + -1\right) + z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (log y) (+ x -1.0)) (* z (- (* y (* y -0.5)) y))) t))

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (x + -1.0)) + (z * ((y * (y * -0.5)) - y))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * (x + (-1.0d0))) + (z * ((y * (y * (-0.5d0))) - y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (x + -1.0)) + (z * ((y * (y * -0.5)) - y))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) + (z * ((y * (y * -0.5)) - y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(z * Float64(Float64(y * Float64(y * -0.5)) - y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (x + -1.0)) + (z * ((y * (y * -0.5)) - y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

2. Taylor expanded in z around inf 86.5%

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

   1. *-commutative 86.5%

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

   2. sub-neg 86.5%

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

   3. mul-1-neg 86.5%

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

   4. log1p-def 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

   5. mul-1-neg 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

4. Simplified 99.5%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]

5. Taylor expanded in y around 0 99.2%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + -1 \cdot \left(y \cdot z\right)\right)}\right) - t \]
6. Step-by-step derivation

   1. +-commutative 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)}\right) - t \]

   2. associate-*r* 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-1 \cdot y\right) \cdot z} + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)\right) - t \]

   3. associate-*r* 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\left(-1 \cdot y\right) \cdot z + \color{blue}{\left(-0.5 \cdot {y}^{2}\right) \cdot z}\right)\right) - t \]

   4. distribute-rgt-out 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{z \cdot \left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

   5. neg-mul-1 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + z \cdot \left(\color{blue}{\left(-y\right)} + -0.5 \cdot {y}^{2}\right)\right) - t \]

   6. unpow2 99.2%

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

   7. associate-*r* 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

   \[\leadsto \left(\log y \cdot \left(x + -1\right) + z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\right) - t \]

Alternative 5: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\log y \cdot \left(x + -1\right) - \left(z + -1\right) \cdot y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (- (* (log y) (+ x -1.0)) (* (+ z -1.0) y)) t))

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (x + -1.0)) - ((z + -1.0) * y)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * (x + (-1.0d0))) - ((z + (-1.0d0)) * y)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (x + -1.0)) - ((z + -1.0) * y)) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) - ((z + -1.0) * y)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(y) * Float64(x + -1.0)) - Float64(Float64(z + -1.0) * y)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (x + -1.0)) - ((z + -1.0) * y)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(z + -1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

2. Taylor expanded in y around 0 99.1%

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

   1. mul-1-neg 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 6: 55.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -2.75 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.08 \cdot 10^{-287}:\\ \;\;\;\;\left(z + -1\right) \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= t -2.75e+48)
     (- t)
     (if (<= t -1.05e-195)
       t_1
       (if (<= t -1.08e-287)
         (* (+ z -1.0) (- (* -0.5 (* y y)) y))
         (if (<= t 1.25e+69) t_1 (- t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (t <= -2.75e+48) {
		tmp = -t;
	} else if (t <= -1.05e-195) {
		tmp = t_1;
	} else if (t <= -1.08e-287) {
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y);
	} else if (t <= 1.25e+69) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (t <= (-2.75d+48)) then
        tmp = -t
    else if (t <= (-1.05d-195)) then
        tmp = t_1
    else if (t <= (-1.08d-287)) then
        tmp = (z + (-1.0d0)) * (((-0.5d0) * (y * y)) - y)
    else if (t <= 1.25d+69) then
        tmp = t_1
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (t <= -2.75e+48) {
		tmp = -t;
	} else if (t <= -1.05e-195) {
		tmp = t_1;
	} else if (t <= -1.08e-287) {
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y);
	} else if (t <= 1.25e+69) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if t <= -2.75e+48:
		tmp = -t
	elif t <= -1.05e-195:
		tmp = t_1
	elif t <= -1.08e-287:
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y)
	elif t <= 1.25e+69:
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (t <= -2.75e+48)
		tmp = Float64(-t);
	elseif (t <= -1.05e-195)
		tmp = t_1;
	elseif (t <= -1.08e-287)
		tmp = Float64(Float64(z + -1.0) * Float64(Float64(-0.5 * Float64(y * y)) - y));
	elseif (t <= 1.25e+69)
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (t <= -2.75e+48)
		tmp = -t;
	elseif (t <= -1.05e-195)
		tmp = t_1;
	elseif (t <= -1.08e-287)
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y);
	elseif (t <= 1.25e+69)
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.75e+48], (-t), If[LessEqual[t, -1.05e-195], t$95$1, If[LessEqual[t, -1.08e-287], N[(N[(z + -1.0), $MachinePrecision] * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.25e+69], t$95$1, (-t)]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7500000000000001e48 or 1.25000000000000009e69 < t

   1. Initial program 97.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in t around inf 80.0%

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

      1. neg-mul-1 80.0%

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

   4. Simplified 80.0%

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

   if -2.7500000000000001e48 < t < -1.05e-195 or -1.08e-287 < t < 1.25000000000000009e69

   1. Initial program 83.7%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in z around inf 83.0%

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

      1. *-commutative 83.0%

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

      2. sub-neg 83.0%

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

      3. mul-1-neg 83.0%

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

      4. log1p-def 99.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

      5. mul-1-neg 99.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

   4. Simplified 99.0%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]

   5. Taylor expanded in x around inf 46.5%

      \[\leadsto \color{blue}{\log y \cdot x} \]

   if -1.05e-195 < t < -1.08e-287

   1. Initial program 62.1%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 100.0%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right)\right) - t \]

      2. unsub-neg 100.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)}\right) - t \]

      3. unpow2 100.0%

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

      4. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 43.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 43.8%

         \[\leadsto \color{blue}{-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)} \]

      2. *-commutative 43.8%

         \[\leadsto \color{blue}{\left(\left(z - 1\right) \cdot {y}^{2}\right) \cdot -0.5} + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      3. sub-neg 43.8%

         \[\leadsto \left(\color{blue}{\left(z + \left(-1\right)\right)} \cdot {y}^{2}\right) \cdot -0.5 + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      4. metadata-eval 43.8%

         \[\leadsto \left(\left(z + \color{blue}{-1}\right) \cdot {y}^{2}\right) \cdot -0.5 + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      5. associate-*l* 43.8%

         \[\leadsto \color{blue}{\left(z + -1\right) \cdot \left({y}^{2} \cdot -0.5\right)} + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      6. unpow2 43.8%

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

      7. associate-*r* 43.8%

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

      8. mul-1-neg 43.8%

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

      9. sub-neg 43.8%

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

      10. metadata-eval 43.8%

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

      11. distribute-rgt-neg-in 43.8%

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

      12. distribute-lft-in 43.9%

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

      13. associate-*r* 43.9%

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

      14. unpow2 43.9%

          \[\leadsto \left(z + -1\right) \cdot \left(\color{blue}{{y}^{2}} \cdot -0.5 + \left(-y\right)\right) \]

      15. *-commutative 43.9%

          \[\leadsto \left(z + -1\right) \cdot \left(\color{blue}{-0.5 \cdot {y}^{2}} + \left(-y\right)\right) \]

      16. sub-neg 43.9%

          \[\leadsto \left(z + -1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)} \]

      17. unpow2 43.9%

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

   7. Simplified 43.9%

      \[\leadsto \color{blue}{\left(z + -1\right) \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.75 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-195}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;t \leq -1.08 \cdot 10^{-287}:\\ \;\;\;\;\left(z + -1\right) \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+69}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 7: 87.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+133}:\\ \;\;\;\;\left(y + \log y \cdot \left(x + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.25e+133)
   (- (+ y (* (log y) (+ x -1.0))) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.25e+133) {
		tmp = (y + (log(y) * (x + -1.0))) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.25e+133) {
		tmp = (y + (Math.log(y) * (x + -1.0))) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.25e+133:
		tmp = (y + (math.log(y) * (x + -1.0))) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.25e+133)
		tmp = Float64(Float64(y + Float64(log(y) * Float64(x + -1.0))) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.25e+133], N[(N[(y + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2499999999999999e133

   1. Initial program 93.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in z around 0 93.4%

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

      1. sub-neg 93.4%

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

      2. metadata-eval 93.4%

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

      3. mul-1-neg 93.4%

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

      4. unsub-neg 93.4%

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

      5. *-commutative 93.4%

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

      6. +-commutative 93.4%

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

      7. sub-neg 93.4%

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

      8. mul-1-neg 93.4%

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

      9. log1p-def 93.4%

         \[\leadsto \left(\log y \cdot \left(-1 + x\right) - \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)}\right) - t \]

      10. mul-1-neg 93.4%

          \[\leadsto \left(\log y \cdot \left(-1 + x\right) - \mathsf{log1p}\left(\color{blue}{-y}\right)\right) - t \]

   4. Simplified 93.4%

      \[\leadsto \color{blue}{\left(\log y \cdot \left(-1 + x\right) - \mathsf{log1p}\left(-y\right)\right) - t} \]

   5. Taylor expanded in y around 0 93.1%

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

   if 1.2499999999999999e133 < z

   1. Initial program 46.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in z around inf 46.6%

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

      1. *-commutative 46.6%

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

      2. sub-neg 46.6%

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

      3. mul-1-neg 46.6%

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

      4. log1p-def 100.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

      5. mul-1-neg 100.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

   4. Simplified 100.0%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\log y \cdot \left(x - 1\right)} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      2. flip-- 89.7%

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      3. associate-*r/ 89.7%

         \[\leadsto \left(\color{blue}{\frac{\log y \cdot \left(x \cdot x - 1 \cdot 1\right)}{x + 1}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      4. metadata-eval 89.7%

         \[\leadsto \left(\frac{\log y \cdot \left(x \cdot x - \color{blue}{1}\right)}{x + 1} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      5. fma-neg 89.7%

         \[\leadsto \left(\frac{\log y \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)}}{x + 1} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      6. metadata-eval 89.7%

         \[\leadsto \left(\frac{\log y \cdot \mathsf{fma}\left(x, x, \color{blue}{-1}\right)}{x + 1} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      7. +-commutative 89.7%

         \[\leadsto \left(\frac{\log y \cdot \mathsf{fma}\left(x, x, -1\right)}{\color{blue}{1 + x}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   6. Applied egg-rr 89.7%

      \[\leadsto \left(\color{blue}{\frac{\log y \cdot \mathsf{fma}\left(x, x, -1\right)}{1 + x}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]
   7. Step-by-step derivation

      1. associate-/l* 89.7%

         \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{1 + x}{\mathsf{fma}\left(x, x, -1\right)}}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   8. Simplified 89.7%

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{1 + x}{\mathsf{fma}\left(x, x, -1\right)}}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   9. Taylor expanded in x around inf 89.9%

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{x}}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   10. Taylor expanded in x around 0 31.2%

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

       1. sub-neg 31.2%

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

       2. log1p-def 80.6%

          \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   12. Simplified 80.6%

       \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+133}:\\ \;\;\;\;\left(y + \log y \cdot \left(x + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 8: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\log y \cdot \left(x + -1\right) - z \cdot y\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (- (* (log y) (+ x -1.0)) (* z y)) t))

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (x + -1.0)) - (z * y)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * (x + (-1.0d0))) - (z * y)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log(y) * (x + -1.0)) - (z * y)) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) - (z * y)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(y) * Float64(x + -1.0)) - Float64(z * y)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((log(y) * (x + -1.0)) - (z * y)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

2. Taylor expanded in z around inf 86.5%

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

   1. *-commutative 86.5%

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

   2. sub-neg 86.5%

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

   3. mul-1-neg 86.5%

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

   4. log1p-def 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

   5. mul-1-neg 99.5%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

4. Simplified 99.5%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]

5. Taylor expanded in y around 0 99.2%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + -1 \cdot \left(y \cdot z\right)\right)}\right) - t \]
6. Step-by-step derivation

   1. +-commutative 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)}\right) - t \]

   2. associate-*r* 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\color{blue}{\left(-1 \cdot y\right) \cdot z} + -0.5 \cdot \left({y}^{2} \cdot z\right)\right)\right) - t \]

   3. associate-*r* 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(\left(-1 \cdot y\right) \cdot z + \color{blue}{\left(-0.5 \cdot {y}^{2}\right) \cdot z}\right)\right) - t \]

   4. distribute-rgt-out 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{z \cdot \left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

   5. neg-mul-1 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + z \cdot \left(\color{blue}{\left(-y\right)} + -0.5 \cdot {y}^{2}\right)\right) - t \]

   6. unpow2 99.2%

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

   7. associate-*r* 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. mul-1-neg 98.9%

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

   5. unsub-neg 98.9%

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

   6. *-commutative 98.9%

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

   7. +-commutative 98.9%

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

10. Simplified 98.9%

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

11. Final simplification 98.9%

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

Alternative 9: 86.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -125000000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -125000000000.0) (not (<= x 1.0)))
   (- (* x (log y)) t)
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -125000000000.0) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-125000000000.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -125000000000.0) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -125000000000.0) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -125000000000.0) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -125000000000.0) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -125000000000.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25e11 or 1 < x

   1. Initial program 90.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 89.1%

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

   3. Taylor expanded in x around inf 87.6%

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

   if -1.25e11 < x < 1

   1. Initial program 83.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 82.4%

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

   3. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

   5. Simplified 81.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -125000000000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 10: 75.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+66}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.1e+48) (- t) (if (<= t 3.5e+66) (* (log y) (+ x -1.0)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.1e+48) {
		tmp = -t;
	} else if (t <= 3.5e+66) {
		tmp = log(y) * (x + -1.0);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.1d+48)) then
        tmp = -t
    else if (t <= 3.5d+66) then
        tmp = log(y) * (x + (-1.0d0))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.1e+48) {
		tmp = -t;
	} else if (t <= 3.5e+66) {
		tmp = Math.log(y) * (x + -1.0);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.1e+48:
		tmp = -t
	elif t <= 3.5e+66:
		tmp = math.log(y) * (x + -1.0)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.1e+48)
		tmp = Float64(-t);
	elseif (t <= 3.5e+66)
		tmp = Float64(log(y) * Float64(x + -1.0));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -4.1e+48)
		tmp = -t;
	elseif (t <= 3.5e+66)
		tmp = log(y) * (x + -1.0);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.1e+48], (-t), If[LessEqual[t, 3.5e+66], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -4.1000000000000003e48 or 3.4999999999999997e66 < t

   1. Initial program 97.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in t around inf 80.0%

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

      1. neg-mul-1 80.0%

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

   4. Simplified 80.0%

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

   if -4.1000000000000003e48 < t < 3.4999999999999997e66

   1. Initial program 80.4%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 78.5%

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

   3. Taylor expanded in t around 0 74.1%

      \[\leadsto \color{blue}{\left(x - 1\right) \cdot \log y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+66}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 87.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+133}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.25e+133) (- (* (log y) (+ x -1.0)) t) (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.25e+133) {
		tmp = (log(y) * (x + -1.0)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.25e+133) {
		tmp = (Math.log(y) * (x + -1.0)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.25e+133:
		tmp = (math.log(y) * (x + -1.0)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.25e+133)
		tmp = Float64(Float64(log(y) * Float64(x + -1.0)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.25e+133], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2499999999999999e133

   1. Initial program 93.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 92.9%

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

   if 1.2499999999999999e133 < z

   1. Initial program 46.6%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in z around inf 46.6%

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

      1. *-commutative 46.6%

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

      2. sub-neg 46.6%

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

      3. mul-1-neg 46.6%

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

      4. log1p-def 100.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

      5. mul-1-neg 100.0%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

   4. Simplified 100.0%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(\color{blue}{\log y \cdot \left(x - 1\right)} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      2. flip-- 89.7%

         \[\leadsto \left(\log y \cdot \color{blue}{\frac{x \cdot x - 1 \cdot 1}{x + 1}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      3. associate-*r/ 89.7%

         \[\leadsto \left(\color{blue}{\frac{\log y \cdot \left(x \cdot x - 1 \cdot 1\right)}{x + 1}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      4. metadata-eval 89.7%

         \[\leadsto \left(\frac{\log y \cdot \left(x \cdot x - \color{blue}{1}\right)}{x + 1} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      5. fma-neg 89.7%

         \[\leadsto \left(\frac{\log y \cdot \color{blue}{\mathsf{fma}\left(x, x, -1\right)}}{x + 1} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      6. metadata-eval 89.7%

         \[\leadsto \left(\frac{\log y \cdot \mathsf{fma}\left(x, x, \color{blue}{-1}\right)}{x + 1} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

      7. +-commutative 89.7%

         \[\leadsto \left(\frac{\log y \cdot \mathsf{fma}\left(x, x, -1\right)}{\color{blue}{1 + x}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   6. Applied egg-rr 89.7%

      \[\leadsto \left(\color{blue}{\frac{\log y \cdot \mathsf{fma}\left(x, x, -1\right)}{1 + x}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]
   7. Step-by-step derivation

      1. associate-/l* 89.7%

         \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{1 + x}{\mathsf{fma}\left(x, x, -1\right)}}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   8. Simplified 89.7%

      \[\leadsto \left(\color{blue}{\frac{\log y}{\frac{1 + x}{\mathsf{fma}\left(x, x, -1\right)}}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   9. Taylor expanded in x around inf 89.9%

      \[\leadsto \left(\frac{\log y}{\color{blue}{\frac{1}{x}}} + \mathsf{log1p}\left(-y\right) \cdot z\right) - t \]

   10. Taylor expanded in x around 0 31.2%

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

       1. sub-neg 31.2%

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

       2. log1p-def 80.6%

          \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]

   12. Simplified 80.6%

       \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.25 \cdot 10^{+133}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 12: 74.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1350000000000 \lor \neg \left(x \leq 1.9 \cdot 10^{+97}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1350000000000.0) (not (<= x 1.9e+97)))
   (* x (log y))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1350000000000.0) || !(x <= 1.9e+97)) {
		tmp = x * log(y);
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1350000000000.0d0)) .or. (.not. (x <= 1.9d+97))) then
        tmp = x * log(y)
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1350000000000.0) || !(x <= 1.9e+97)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1350000000000.0) or not (x <= 1.9e+97):
		tmp = x * math.log(y)
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1350000000000.0) || !(x <= 1.9e+97))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1350000000000.0) || ~((x <= 1.9e+97)))
		tmp = x * log(y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1350000000000.0], N[Not[LessEqual[x, 1.9e+97]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35e12 or 1.90000000000000018e97 < x

   1. Initial program 91.9%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in z around inf 91.9%

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

      1. *-commutative 91.9%

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

      2. sub-neg 91.9%

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

      3. mul-1-neg 91.9%

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

      4. log1p-def 99.7%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z\right) - t \]

      5. mul-1-neg 99.7%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \mathsf{log1p}\left(\color{blue}{-y}\right) \cdot z\right) - t \]

   4. Simplified 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\mathsf{log1p}\left(-y\right) \cdot z}\right) - t \]

   5. Taylor expanded in x around inf 67.4%

      \[\leadsto \color{blue}{\log y \cdot x} \]

   if -1.35e12 < x < 1.90000000000000018e97

   1. Initial program 83.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 81.9%

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

   3. Taylor expanded in x around 0 76.0%

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

      1. mul-1-neg 76.0%

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

   5. Simplified 76.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1350000000000 \lor \neg \left(x \leq 1.9 \cdot 10^{+97}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 13: 41.4% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;\left(z + -1\right) \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.55e+48)
   (- t)
   (if (<= t 1.4e+61) (* (+ z -1.0) (- (* -0.5 (* y y)) y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.55e+48) {
		tmp = -t;
	} else if (t <= 1.4e+61) {
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.55d+48)) then
        tmp = -t
    else if (t <= 1.4d+61) then
        tmp = (z + (-1.0d0)) * (((-0.5d0) * (y * y)) - y)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.55e+48) {
		tmp = -t;
	} else if (t <= 1.4e+61) {
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.55e+48:
		tmp = -t
	elif t <= 1.4e+61:
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.55e+48)
		tmp = Float64(-t);
	elseif (t <= 1.4e+61)
		tmp = Float64(Float64(z + -1.0) * Float64(Float64(-0.5 * Float64(y * y)) - y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.55e+48)
		tmp = -t;
	elseif (t <= 1.4e+61)
		tmp = (z + -1.0) * ((-0.5 * (y * y)) - y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.55e+48], (-t), If[LessEqual[t, 1.4e+61], N[(N[(z + -1.0), $MachinePrecision] * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55000000000000003e48 or 1.4000000000000001e61 < t

   1. Initial program 97.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in t around inf 77.8%

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

      1. neg-mul-1 77.8%

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

   4. Simplified 77.8%

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

   if -1.55000000000000003e48 < t < 1.4000000000000001e61

   1. Initial program 80.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 99.1%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right)\right) - t \]

      2. unsub-neg 99.1%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)}\right) - t \]

      3. unpow2 99.1%

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

      4. associate-*r* 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in y around inf 23.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\left(z - 1\right) \cdot y\right) + -0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative 23.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left(\left(z - 1\right) \cdot {y}^{2}\right) + -1 \cdot \left(\left(z - 1\right) \cdot y\right)} \]

      2. *-commutative 23.0%

         \[\leadsto \color{blue}{\left(\left(z - 1\right) \cdot {y}^{2}\right) \cdot -0.5} + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      3. sub-neg 23.0%

         \[\leadsto \left(\color{blue}{\left(z + \left(-1\right)\right)} \cdot {y}^{2}\right) \cdot -0.5 + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      4. metadata-eval 23.0%

         \[\leadsto \left(\left(z + \color{blue}{-1}\right) \cdot {y}^{2}\right) \cdot -0.5 + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      5. associate-*l* 23.0%

         \[\leadsto \color{blue}{\left(z + -1\right) \cdot \left({y}^{2} \cdot -0.5\right)} + -1 \cdot \left(\left(z - 1\right) \cdot y\right) \]

      6. unpow2 23.0%

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

      7. associate-*r* 23.0%

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

      8. mul-1-neg 23.0%

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

      9. sub-neg 23.0%

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

      10. metadata-eval 23.0%

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

      11. distribute-rgt-neg-in 23.0%

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

      12. distribute-lft-in 23.0%

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

      13. associate-*r* 23.0%

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

      14. unpow2 23.0%

          \[\leadsto \left(z + -1\right) \cdot \left(\color{blue}{{y}^{2}} \cdot -0.5 + \left(-y\right)\right) \]

      15. *-commutative 23.0%

          \[\leadsto \left(z + -1\right) \cdot \left(\color{blue}{-0.5 \cdot {y}^{2}} + \left(-y\right)\right) \]

      16. sub-neg 23.0%

          \[\leadsto \left(z + -1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)} \]

      17. unpow2 23.0%

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

   7. Simplified 23.0%

      \[\leadsto \color{blue}{\left(z + -1\right) \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;\left(z + -1\right) \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 41.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.55e+48)
   (- t)
   (if (<= t 1.4e+61) (* z (- (* -0.5 (* y y)) y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.55e+48) {
		tmp = -t;
	} else if (t <= 1.4e+61) {
		tmp = z * ((-0.5 * (y * y)) - y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.55d+48)) then
        tmp = -t
    else if (t <= 1.4d+61) then
        tmp = z * (((-0.5d0) * (y * y)) - y)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.55e+48) {
		tmp = -t;
	} else if (t <= 1.4e+61) {
		tmp = z * ((-0.5 * (y * y)) - y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.55e+48:
		tmp = -t
	elif t <= 1.4e+61:
		tmp = z * ((-0.5 * (y * y)) - y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.55e+48)
		tmp = Float64(-t);
	elseif (t <= 1.4e+61)
		tmp = Float64(z * Float64(Float64(-0.5 * Float64(y * y)) - y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.55e+48)
		tmp = -t;
	elseif (t <= 1.4e+61)
		tmp = z * ((-0.5 * (y * y)) - y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.55e+48], (-t), If[LessEqual[t, 1.4e+61], N[(z * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.55000000000000003e48 or 1.4000000000000001e61 < t

   1. Initial program 97.3%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in t around inf 77.8%

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

      1. neg-mul-1 77.8%

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

   4. Simplified 77.8%

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

   if -1.55000000000000003e48 < t < 1.4000000000000001e61

   1. Initial program 80.0%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. Taylor expanded in y around 0 99.1%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)}\right) - t \]
   3. Step-by-step derivation

      1. mul-1-neg 99.1%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right)\right) - t \]

      2. unsub-neg 99.1%

         \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)}\right) - t \]

      3. unpow2 99.1%

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

      4. associate-*r* 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in z around inf 22.4%

      \[\leadsto \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right) \cdot z} \]
   6. Step-by-step derivation

      1. *-commutative 22.4%

         \[\leadsto \color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} - y\right)} \]

      2. unpow2 22.4%

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

   7. Simplified 22.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{+48}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+61}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 15: 35.2% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

2. Taylor expanded in t around inf 33.8%

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

   1. neg-mul-1 33.8%

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

4. Simplified 33.8%

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

5. Final simplification 33.8%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))