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

Percentage Accurate: 88.8% → 99.8%

Time: 19.3s

Alternatives: 18

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: 88.8% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. sub-neg 90.2%

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

   2. +-commutative 90.2%

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

   3. associate-+l+ 90.2%

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

   4. fma-define 90.2%

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

   5. sub-neg 90.2%

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

   6. metadata-eval 90.2%

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

   7. sub-neg 90.2%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

   1. +-commutative 90.2%

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

   2. fma-define 90.2%

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

   3. sub-neg 90.2%

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

   4. metadata-eval 90.2%

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

   5. sub-neg 90.2%

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

   6. log1p-define 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. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) + \log y \cdot \left(-1 + x\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (*
    (+ z -1.0)
    (* y (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
   (* (log y) (+ -1.0 x)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) + (log(y) * (-1.0 + x))) - 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 = (((z + (-1.0d0)) * (y * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) + (log(y) * ((-1.0d0) + x))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) + (math.log(y) * (-1.0 + x))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(z + -1.0) * Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) + Float64(log(y) * Float64(-1.0 + x))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) + (log(y) * (-1.0 + x))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}\right) - t \]

4. Final simplification 99.8%

   \[\leadsto \left(\left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) + \log y \cdot \left(-1 + x\right)\right) - t \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (-
   (* (+ z -1.0) (* y (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5)))))
   (* (+ -1.0 x) (log (/ 1.0 y))))
  t))

C

double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - ((-1.0 + x) * 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 = (((z + (-1.0d0)) * (y * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - (((-1.0d0) + x) * log((1.0d0 / y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - ((-1.0 + x) * Math.log((1.0 / y)))) - t;
}

Python

def code(x, y, z, t):
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - ((-1.0 + x) * math.log((1.0 / y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(z + -1.0) * Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5))))) - Float64(Float64(-1.0 + x) * log(Float64(1.0 / y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - ((-1.0 + x) * log((1.0 / y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 + x), $MachinePrecision] * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)}\right) - t \]

4. Taylor expanded in y around inf 99.8%

   \[\leadsto \left(\color{blue}{-1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot \left(x - 1\right)\right)} + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)\right) - t \]

5. Final simplification 99.8%

   \[\leadsto \left(\left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - \left(-1 + x\right) \cdot \log \left(\frac{1}{y}\right)\right) - t \]
6. Add Preprocessing

Alternative 5: 95.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right) - t\\ \mathbf{if}\;-1 + x \leq -1.00000002:\\ \;\;\;\;y + t\_1\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) (+ -1.0 x)) t)))
   (if (<= (+ -1.0 x) -1.00000002)
     (+ y t_1)
     (if (<= (+ -1.0 x) -1.0) (- (* y (- 1.0 z)) (+ (log y) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * (-1.0 + x)) - t;
	double tmp;
	if ((-1.0 + x) <= -1.00000002) {
		tmp = y + t_1;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = (y * (1.0 - z)) - (log(y) + t);
	} else {
		tmp = t_1;
	}
	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 = (log(y) * ((-1.0d0) + x)) - t
    if (((-1.0d0) + x) <= (-1.00000002d0)) then
        tmp = y + t_1
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = (y * (1.0d0 - z)) - (log(y) + t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * (-1.0 + x)) - t;
	double tmp;
	if ((-1.0 + x) <= -1.00000002) {
		tmp = y + t_1;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = (y * (1.0 - z)) - (Math.log(y) + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * (-1.0 + x)) - t
	tmp = 0
	if (-1.0 + x) <= -1.00000002:
		tmp = y + t_1
	elif (-1.0 + x) <= -1.0:
		tmp = (y * (1.0 - z)) - (math.log(y) + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * Float64(-1.0 + x)) - t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -1.00000002)
		tmp = Float64(y + t_1);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(y * Float64(1.0 - z)) - Float64(log(y) + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * (-1.0 + x)) - t;
	tmp = 0.0;
	if ((-1.0 + x) <= -1.00000002)
		tmp = y + t_1;
	elseif ((-1.0 + x) <= -1.0)
		tmp = (y * (1.0 - z)) - (log(y) + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.00000002], N[(y + t$95$1), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.0000000200000001

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. metadata-eval 98.9%

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

      4. fma-define 98.9%

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

      5. mul-1-neg 98.9%

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

      6. fmm-def 98.9%

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

      7. +-commutative 98.9%

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

      8. sub-neg 98.9%

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

      9. metadata-eval 98.9%

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

      10. +-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in z around 0 90.8%

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

      1. associate--r+ 90.8%

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

      2. sub-neg 90.8%

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

      3. sub-neg 90.8%

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

      4. metadata-eval 90.8%

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

      5. mul-1-neg 90.8%

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

      6. remove-double-neg 90.8%

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

   8. Simplified 90.8%

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

   if -1.0000000200000001 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 84.6%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. fma-define 99.5%

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

      5. mul-1-neg 99.5%

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

      6. fmm-def 99.5%

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

      7. +-commutative 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

      10. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.3%

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

      1. neg-mul-1 99.3%

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

      2. associate--r+ 99.3%

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

      3. cancel-sign-sub-inv 99.3%

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

      4. sub-neg 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. +-commutative 99.3%

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

      7. mul-1-neg 99.3%

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

      8. associate-*r* 99.3%

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

      9. +-commutative 99.3%

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

      10. sub-neg 99.3%

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

      11. mul-1-neg 99.3%

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

      12. distribute-rgt-neg-in 99.3%

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

      13. sub-neg 99.3%

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

      14. metadata-eval 99.3%

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

      15. +-commutative 99.3%

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

      16. distribute-neg-in 99.3%

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

      17. metadata-eval 99.3%

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

      18. sub-neg 99.3%

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

      19. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   if -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 98.4%

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

   3. Taylor expanded in y around 0 98.4%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000002:\\ \;\;\;\;y + \left(\log y \cdot \left(-1 + x\right) - t\right)\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) + \log y \cdot \left(-1 + x\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (+ z -1.0) (* y (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5)))))
   (* (log y) (+ -1.0 x)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) + (log(y) * (-1.0 + x))) - 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 = (((z + (-1.0d0)) * (y * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) + (log(y) * ((-1.0d0) + x))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) + (Math.log(y) * (-1.0 + x))) - t;
}

Python

def code(x, y, z, t):
	return (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) + (math.log(y) * (-1.0 + x))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(z + -1.0) * Float64(y * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5))))) + Float64(log(y) * Float64(-1.0 + x))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) + (log(y) * (-1.0 + x))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(z + -1.0), $MachinePrecision] * N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)}\right) - t \]

4. Final simplification 99.8%

   \[\leadsto \left(\left(z + -1\right) \cdot \left(y \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) + \log y \cdot \left(-1 + x\right)\right) - t \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 8: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -240000 \lor \neg \left(t \leq 7.5 \cdot 10^{+40}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -240000.0) (not (<= t 7.5e+40)))
   (- (- t) (* z y))
   (* (log y) (+ -1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -240000.0) || !(t <= 7.5e+40)) {
		tmp = -t - (z * y);
	} else {
		tmp = log(y) * (-1.0 + x);
	}
	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 <= (-240000.0d0)) .or. (.not. (t <= 7.5d+40))) then
        tmp = -t - (z * y)
    else
        tmp = log(y) * ((-1.0d0) + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -240000.0) || !(t <= 7.5e+40)) {
		tmp = -t - (z * y);
	} else {
		tmp = Math.log(y) * (-1.0 + x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -240000.0) or not (t <= 7.5e+40):
		tmp = -t - (z * y)
	else:
		tmp = math.log(y) * (-1.0 + x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -240000.0) || !(t <= 7.5e+40))
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(log(y) * Float64(-1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -240000.0) || ~((t <= 7.5e+40)))
		tmp = -t - (z * y);
	else
		tmp = log(y) * (-1.0 + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -240000.0], N[Not[LessEqual[t, 7.5e+40]], $MachinePrecision]], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4e5 or 7.4999999999999996e40 < t

   1. Initial program 92.6%

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

   3. Taylor expanded in t around -inf 92.5%

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

   4. Taylor expanded in z around inf 73.1%

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

      1. sub-neg 73.1%

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

      2. log1p-undefine 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in y around 0 80.4%

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

   if -2.4e5 < t < 7.4999999999999996e40

   1. Initial program 88.6%

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

   3. Taylor expanded in y around 0 87.3%

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

   4. Taylor expanded in t around 0 86.2%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -240000 \lor \neg \left(t \leq 7.5 \cdot 10^{+40}\right):\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+40} \lor \neg \left(x \leq 1.45 \cdot 10^{+21}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.2e+40) (not (<= x 1.45e+21)))
   (* x (log y))
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+40) || !(x <= 1.45e+21)) {
		tmp = x * log(y);
	} else {
		tmp = -log(y) - 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 ((x <= (-3.2d+40)) .or. (.not. (x <= 1.45d+21))) then
        tmp = x * log(y)
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+40) || !(x <= 1.45e+21)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.2e+40) or not (x <= 1.45e+21):
		tmp = x * math.log(y)
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.2e+40) || !(x <= 1.45e+21))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.2e+40) || ~((x <= 1.45e+21)))
		tmp = x * log(y);
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.2e+40], N[Not[LessEqual[x, 1.45e+21]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999981e40 or 1.45e21 < x

   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. Add Preprocessing

   3. Taylor expanded in y around 0 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}\right) - t \]

   4. Taylor expanded in x around inf 81.4%

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

      1. *-commutative 81.4%

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

   6. Simplified 81.4%

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

   if -3.19999999999999981e40 < x < 1.45e21

   1. Initial program 84.9%

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

   3. Taylor expanded in y around 0 84.1%

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

   4. Taylor expanded in x around 0 79.4%

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

      1. neg-mul-1 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+40} \lor \neg \left(x \leq 1.45 \cdot 10^{+21}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+238}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + \left(\log y \cdot \left(-1 + x\right) - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.8e+238) (- (- t) (* z y)) (+ y (- (* (log y) (+ -1.0 x)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.8e+238) {
		tmp = -t - (z * y);
	} else {
		tmp = y + ((log(y) * (-1.0 + x)) - 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 (z <= (-4.8d+238)) then
        tmp = -t - (z * y)
    else
        tmp = y + ((log(y) * ((-1.0d0) + x)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.8e+238) {
		tmp = -t - (z * y);
	} else {
		tmp = y + ((Math.log(y) * (-1.0 + x)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.8e+238:
		tmp = -t - (z * y)
	else:
		tmp = y + ((math.log(y) * (-1.0 + x)) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.8e+238)
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(y + Float64(Float64(log(y) * Float64(-1.0 + x)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4.8e+238)
		tmp = -t - (z * y);
	else
		tmp = y + ((log(y) * (-1.0 + x)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.8e+238], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(y + N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8e238

   1. Initial program 43.3%

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

   3. Taylor expanded in t around -inf 38.6%

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

   4. Taylor expanded in z around inf 29.6%

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

      1. sub-neg 29.6%

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

      2. log1p-undefine 69.2%

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

   6. Simplified 69.2%

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

   7. Taylor expanded in y around 0 84.6%

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

   if -4.8e238 < z

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. fma-define 99.6%

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

      5. mul-1-neg 99.6%

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

      6. fmm-def 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in z around 0 92.8%

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

      1. associate--r+ 92.8%

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

      2. sub-neg 92.8%

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

      3. sub-neg 92.8%

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

      4. metadata-eval 92.8%

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

      5. mul-1-neg 92.8%

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

      6. remove-double-neg 92.8%

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

   8. Simplified 92.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+238}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;y + \left(\log y \cdot \left(-1 + x\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+96} \lor \neg \left(x \leq 1.1 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e+96) (not (<= x 1.1e+27))) (* x (log y)) (- (- t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+96) || !(x <= 1.1e+27)) {
		tmp = x * log(y);
	} else {
		tmp = -t - (z * 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 <= (-2d+96)) .or. (.not. (x <= 1.1d+27))) then
        tmp = x * log(y)
    else
        tmp = -t - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+96) || !(x <= 1.1e+27)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2e+96) or not (x <= 1.1e+27):
		tmp = x * math.log(y)
	else:
		tmp = -t - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e+96) || !(x <= 1.1e+27))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2e+96) || ~((x <= 1.1e+27)))
		tmp = x * log(y);
	else
		tmp = -t - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e+96], N[Not[LessEqual[x, 1.1e+27]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e96 or 1.0999999999999999e27 < x

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}\right) - t \]

   4. Taylor expanded in x around inf 84.4%

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

      1. *-commutative 84.4%

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

   6. Simplified 84.4%

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

   if -2.0000000000000001e96 < x < 1.0999999999999999e27

   1. Initial program 84.4%

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

   3. Taylor expanded in t around -inf 82.2%

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

   4. Taylor expanded in z around inf 42.2%

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

      1. sub-neg 42.2%

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

      2. log1p-undefine 54.4%

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

   6. Simplified 54.4%

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

   7. Taylor expanded in y around 0 57.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+96} \lor \neg \left(x \leq 1.1 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. fma-define 99.4%

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

   5. mul-1-neg 99.4%

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

   6. fmm-def 99.4%

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

   7. +-commutative 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

   10. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 13: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+238}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+238) (- (- t) (* z y)) (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+238) {
		tmp = -t - (z * y);
	} else {
		tmp = (log(y) * (-1.0 + x)) - 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 (z <= (-3.5d+238)) then
        tmp = -t - (z * y)
    else
        tmp = (log(y) * ((-1.0d0) + x)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+238) {
		tmp = -t - (z * y);
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.5e+238:
		tmp = -t - (z * y)
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+238)
		tmp = Float64(Float64(-t) - Float64(z * y));
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.5e+238)
		tmp = -t - (z * y);
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+238], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000003e238

   1. Initial program 43.3%

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

   3. Taylor expanded in t around -inf 38.6%

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

   4. Taylor expanded in z around inf 29.6%

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

      1. sub-neg 29.6%

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

      2. log1p-undefine 69.2%

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

   6. Simplified 69.2%

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

   7. Taylor expanded in y around 0 84.6%

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

   if -3.50000000000000003e238 < z

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0 92.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+238}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in y around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. fma-define 99.4%

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

   5. mul-1-neg 99.4%

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

   6. fmm-def 99.4%

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

   7. +-commutative 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

   10. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in z around inf 99.3%

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

7. Final simplification 99.3%

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

Alternative 15: 43.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -10500000000 \lor \neg \left(t \leq 3.4 \cdot 10^{-12}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -10500000000.0) (not (<= t 3.4e-12))) (- t) (* y (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -10500000000.0) || !(t <= 3.4e-12)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	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 <= (-10500000000.0d0)) .or. (.not. (t <= 3.4d-12))) then
        tmp = -t
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -10500000000.0) || !(t <= 3.4e-12)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -10500000000.0) or not (t <= 3.4e-12):
		tmp = -t
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -10500000000.0) || !(t <= 3.4e-12))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -10500000000.0) || ~((t <= 3.4e-12)))
		tmp = -t;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -10500000000.0], N[Not[LessEqual[t, 3.4e-12]], $MachinePrecision]], (-t), N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e10 or 3.4000000000000001e-12 < t

   1. Initial program 94.7%

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

   3. Taylor expanded in t around inf 70.5%

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

      1. mul-1-neg 70.5%

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

   5. Simplified 70.5%

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

   if -1.05e10 < t < 3.4000000000000001e-12

   1. Initial program 86.8%

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

   3. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. fma-define 99.0%

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

      5. mul-1-neg 99.0%

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

      6. fmm-def 99.0%

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

      7. +-commutative 99.0%

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

      8. sub-neg 99.0%

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

      9. metadata-eval 99.0%

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

      10. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in y around inf 15.7%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -10500000000 \lor \neg \left(t \leq 3.4 \cdot 10^{-12}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.5% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -62000000000 \lor \neg \left(t \leq 3.4 \cdot 10^{-12}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -62000000000.0) (not (<= t 3.4e-12))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -62000000000.0) || !(t <= 3.4e-12)) {
		tmp = -t;
	} else {
		tmp = z * -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 ((t <= (-62000000000.0d0)) .or. (.not. (t <= 3.4d-12))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -62000000000.0) || !(t <= 3.4e-12)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -62000000000.0) or not (t <= 3.4e-12):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -62000000000.0) || !(t <= 3.4e-12))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -62000000000.0) || ~((t <= 3.4e-12)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -62000000000.0], N[Not[LessEqual[t, 3.4e-12]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.2e10 or 3.4000000000000001e-12 < t

   1. Initial program 94.7%

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

   3. Taylor expanded in t around inf 70.5%

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

      1. mul-1-neg 70.5%

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

   5. Simplified 70.5%

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

   if -6.2e10 < t < 3.4000000000000001e-12

   1. Initial program 86.8%

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

   3. Taylor expanded in y around 0 99.8%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)}\right) - t \]

   4. Taylor expanded in z around inf 15.9%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} \]

   5. Taylor expanded in y around 0 15.1%

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

      1. associate-*r* 15.1%

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

      2. neg-mul-1 15.1%

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

   7. Simplified 15.1%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -62000000000 \lor \neg \left(t \leq 3.4 \cdot 10^{-12}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.8% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

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 - (z * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.2%

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

3. Taylor expanded in t around -inf 75.5%

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

4. Taylor expanded in z around inf 31.8%

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

   1. sub-neg 31.8%

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

   2. log1p-undefine 39.3%

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

6. Simplified 39.3%

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

7. Taylor expanded in y around 0 41.1%

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

8. Final simplification 41.1%

   \[\leadsto \left(-t\right) - z \cdot y \]
9. Add Preprocessing

Alternative 18: 36.0% 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 90.2%

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

3. Taylor expanded in t around inf 31.8%

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

   1. mul-1-neg 31.8%

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

5. Simplified 31.8%

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

6. Final simplification 31.8%

   \[\leadsto -t \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))