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

Percentage Accurate: 89.3% → 99.8%

Time: 20.2s

Alternatives: 14

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: 89.3% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate--l+ 91.1%

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

   2. fma-def 91.2%

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

   3. sub-neg 91.2%

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

   4. metadata-eval 91.2%

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

   5. fma-neg 91.2%

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

   6. sub-neg 91.2%

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

   7. metadata-eval 91.2%

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

   8. sub-neg 91.2%

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

   9. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. +-commutative 91.1%

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

   2. fma-def 91.1%

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

   3. sub-neg 91.1%

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

   4. metadata-eval 91.1%

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

   5. sub-neg 91.1%

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

   6. log1p-def 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in z around inf 90.9%

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

   1. *-commutative 90.9%

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

   2. sub-neg 90.9%

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

   3. mul-1-neg 90.9%

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

   4. log1p-def 99.5%

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

   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 x around 0 90.9%

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

   1. +-commutative 90.9%

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

   2. *-commutative 90.9%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in z around inf 90.9%

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

   1. *-commutative 90.9%

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

   2. sub-neg 90.9%

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

   3. mul-1-neg 90.9%

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

   4. log1p-def 99.5%

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

   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%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x + -1.0) * log(y)) + (((y * y) * (0.5 + (z * -0.5))) - (y * (-1.0 + z)))) - 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)) + (((y * y) * (0.5d0 + (z * (-0.5d0)))) - (y * ((-1.0d0) + z)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

   2. unsub-neg 98.9%

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

   3. associate-*r* 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

   6. sub-neg 98.9%

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

   7. metadata-eval 98.9%

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

   8. +-commutative 98.9%

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

   9. distribute-lft-in 98.9%

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

   10. metadata-eval 98.9%

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

   11. *-commutative 98.9%

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

   12. sub-neg 98.9%

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

   13. metadata-eval 98.9%

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

   14. +-commutative 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 6: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x + -1.0) * log(y)) + (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 = (((x + (-1.0d0)) * log(y)) + (z * (((y * y) * (-0.5d0)) - 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 * (((y * y) * -0.5) - y))) - t;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in z around inf 90.9%

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

   1. *-commutative 90.9%

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

   2. sub-neg 90.9%

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

   3. mul-1-neg 90.9%

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

   4. log1p-def 99.5%

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

   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 98.8%

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

   1. +-commutative 98.8%

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

   2. associate-*r* 98.8%

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

   3. associate-*r* 98.8%

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

   4. distribute-rgt-out 98.8%

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

   5. neg-mul-1 98.8%

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

   6. unpow2 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 7: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((x + -1.0) * log(y)) + (y * (1.0 - z))) - 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)) + (y * (1.0d0 - z))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in y around 0 98.9%

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

   1. mul-1-neg 98.9%

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

   2. unsub-neg 98.9%

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

   3. associate-*r* 98.9%

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

   4. *-commutative 98.9%

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

   5. unpow2 98.9%

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

   6. sub-neg 98.9%

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

   7. metadata-eval 98.9%

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

   8. +-commutative 98.9%

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

   9. distribute-lft-in 98.9%

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

   10. metadata-eval 98.9%

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

   11. *-commutative 98.9%

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

   12. sub-neg 98.9%

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

   13. metadata-eval 98.9%

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

   14. +-commutative 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in y around 0 98.4%

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

6. Final simplification 98.4%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in z around inf 90.9%

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

   1. *-commutative 90.9%

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

   2. sub-neg 90.9%

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

   3. mul-1-neg 90.9%

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

   4. log1p-def 99.5%

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

   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 98.3%

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

   1. +-commutative 98.3%

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

   2. sub-neg 98.3%

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

   3. metadata-eval 98.3%

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

   4. mul-1-neg 98.3%

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

   5. unsub-neg 98.3%

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

   6. *-commutative 98.3%

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

   7. +-commutative 98.3%

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

7. Simplified 98.3%

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

8. Final simplification 98.3%

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

Alternative 9: 75.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+18} \lor \neg \left(x \leq 3.7 \cdot 10^{+87}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.9e+18) (not (<= x 3.7e+87)))
   (* x (log y))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.9e+18) || !(x <= 3.7e+87)) {
		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 <= (-2.9d+18)) .or. (.not. (x <= 3.7d+87))) 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 <= -2.9e+18) || !(x <= 3.7e+87)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.9e+18) or not (x <= 3.7e+87):
		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 <= -2.9e+18) || !(x <= 3.7e+87))
		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 <= -2.9e+18) || ~((x <= 3.7e+87)))
		tmp = x * log(y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.9e+18], N[Not[LessEqual[x, 3.7e+87]], $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 < -2.9e18 or 3.70000000000000003e87 < x

   1. Initial program 96.5%

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

   2. Taylor expanded in z around inf 96.5%

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

      1. *-commutative 96.5%

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

      2. sub-neg 96.5%

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

      3. mul-1-neg 96.5%

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

      4. log1p-def 99.6%

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

      5. mul-1-neg 99.6%

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

   4. Simplified 99.6%

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

   5. Taylor expanded in x around inf 75.8%

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

   if -2.9e18 < x < 3.70000000000000003e87

   1. Initial program 87.3%

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

   2. Taylor expanded in y around 0 84.6%

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

      1. fma-neg 84.6%

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

      2. sub-neg 84.6%

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

      3. metadata-eval 84.6%

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

      4. +-commutative 84.6%

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

   4. Simplified 84.6%

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

   5. Taylor expanded in x around 0 77.7%

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

      1. mul-1-neg 77.7%

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

   7. Simplified 77.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+18} \lor \neg \left(x \leq 3.7 \cdot 10^{+87}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 10: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+31}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2e+31) (- t) (if (<= t 1.22e+23) (* x (log y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2e+31) {
		tmp = -t;
	} else if (t <= 1.22e+23) {
		tmp = x * log(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 <= (-2d+31)) then
        tmp = -t
    else if (t <= 1.22d+23) then
        tmp = x * log(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 <= -2e+31) {
		tmp = -t;
	} else if (t <= 1.22e+23) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2e+31:
		tmp = -t
	elif t <= 1.22e+23:
		tmp = x * math.log(y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2e+31)
		tmp = Float64(-t);
	elseif (t <= 1.22e+23)
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2e+31)
		tmp = -t;
	elseif (t <= 1.22e+23)
		tmp = x * log(y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2e+31], (-t), If[LessEqual[t, 1.22e+23], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.9999999999999999e31 or 1.22e23 < t

   1. Initial program 97.4%

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

   2. Taylor expanded in t around inf 80.4%

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

      1. mul-1-neg 80.4%

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

   4. Simplified 80.4%

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

   if -1.9999999999999999e31 < t < 1.22e23

   1. Initial program 85.9%

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

   2. Taylor expanded in z around inf 85.4%

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

      1. *-commutative 85.4%

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

      2. sub-neg 85.4%

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

      3. mul-1-neg 85.4%

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

      4. log1p-def 99.2%

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

      5. mul-1-neg 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in x around inf 53.1%

      \[\leadsto \color{blue}{\log y \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+31}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 88.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

2. Taylor expanded in y around 0 89.2%

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

3. Final simplification 89.2%

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

Alternative 12: 42.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -44:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 370000:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -44.0)
   (- t)
   (if (<= t 370000.0) (* z (- (* y (* y -0.5)) y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -44.0) {
		tmp = -t;
	} else if (t <= 370000.0) {
		tmp = z * ((y * (y * -0.5)) - 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 <= (-44.0d0)) then
        tmp = -t
    else if (t <= 370000.0d0) then
        tmp = z * ((y * (y * (-0.5d0))) - 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 <= -44.0) {
		tmp = -t;
	} else if (t <= 370000.0) {
		tmp = z * ((y * (y * -0.5)) - y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -44.0:
		tmp = -t
	elif t <= 370000.0:
		tmp = z * ((y * (y * -0.5)) - y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -44.0)
		tmp = Float64(-t);
	elseif (t <= 370000.0)
		tmp = Float64(z * Float64(Float64(y * Float64(y * -0.5)) - y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -44.0)
		tmp = -t;
	elseif (t <= 370000.0)
		tmp = z * ((y * (y * -0.5)) - y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -44.0], (-t), If[LessEqual[t, 370000.0], N[(z * N[(N[(y * N[(y * -0.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -44 or 3.7e5 < t

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf 74.7%

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

      1. mul-1-neg 74.7%

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

   4. Simplified 74.7%

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

   if -44 < t < 3.7e5

   1. Initial program 85.2%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. associate-*r* 99.2%

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

      4. *-commutative 99.2%

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

      5. unpow2 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in z around inf 17.2%

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

      1. *-commutative 17.2%

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

      2. *-commutative 17.2%

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

      3. unpow2 17.2%

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

      4. associate-*l* 17.2%

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

   7. Simplified 17.2%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -44:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 370000:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 13: 42.2% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -80:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 320000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -80.0) (- t) (if (<= t 320000.0) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -80.0) {
		tmp = -t;
	} else if (t <= 320000.0) {
		tmp = y * -z;
	} 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 <= (-80.0d0)) then
        tmp = -t
    else if (t <= 320000.0d0) then
        tmp = y * -z
    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 <= -80.0) {
		tmp = -t;
	} else if (t <= 320000.0) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -80.0:
		tmp = -t
	elif t <= 320000.0:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -80.0)
		tmp = Float64(-t);
	elseif (t <= 320000.0)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -80.0)
		tmp = -t;
	elseif (t <= 320000.0)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -80.0], (-t), If[LessEqual[t, 320000.0], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -80 or 3.2e5 < t

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf 74.7%

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

      1. mul-1-neg 74.7%

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

   4. Simplified 74.7%

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

   if -80 < t < 3.2e5

   1. Initial program 85.2%

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

   2. Taylor expanded in y around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. unsub-neg 99.2%

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

      3. associate-*r* 99.2%

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

      4. *-commutative 99.2%

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

      5. unpow2 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

      8. +-commutative 99.2%

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

      9. distribute-lft-in 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. sub-neg 99.2%

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

      13. metadata-eval 99.2%

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

      14. +-commutative 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in z around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. associate-*l* 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in z around inf 16.5%

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

      1. associate-*r* 16.5%

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

      2. mul-1-neg 16.5%

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

   10. Simplified 16.5%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -80:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 320000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 35.1% 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 91.1%

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

2. Taylor expanded in t around inf 39.3%

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

   1. mul-1-neg 39.3%

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

4. Simplified 39.3%

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

5. Final simplification 39.3%

   \[\leadsto -t \]

Reproduce

herbie shell --seed 2023279 
(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))