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

Percentage Accurate: 89.2% → 99.8%

Time: 15.5s

Alternatives: 16

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: 89.2% 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 85.5%

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

   1. sub-neg 85.5%

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

   2. +-commutative 85.5%

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

   3. associate-+l+ 85.5%

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

   4. fma-define 85.5%

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

   5. sub-neg 85.5%

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

   6. metadata-eval 85.5%

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

   7. sub-neg 85.5%

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

   8. log1p-define 99.9%

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

   9. fma-define 99.9%

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

   10. sub-neg 99.9%

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

   11. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. +-commutative 85.5%

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

   2. fma-define 85.5%

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

   3. sub-neg 85.5%

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

   4. metadata-eval 85.5%

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

   5. sub-neg 85.5%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

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

Alternative 4: 73.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-157}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 0.46:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+117}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.02e+110)
     t_1
     (if (<= x -5.5e-157)
       (- (* y (* z (+ -1.0 (* y -0.5)))) t)
       (if (<= x 0.46)
         (- (- t) (log y))
         (if (<= x 1.65e+117)
           (-
            (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* z y)))) z))
            t)
           t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.02e+110) {
		tmp = t_1;
	} else if (x <= -5.5e-157) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 0.46) {
		tmp = -t - log(y);
	} else if (x <= 1.65e+117) {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - 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 = x * log(y)
    if (x <= (-1.02d+110)) then
        tmp = t_1
    else if (x <= (-5.5d-157)) then
        tmp = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
    else if (x <= 0.46d0) then
        tmp = -t - log(y)
    else if (x <= 1.65d+117) then
        tmp = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (z * y)))) - z)) - 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 = x * Math.log(y);
	double tmp;
	if (x <= -1.02e+110) {
		tmp = t_1;
	} else if (x <= -5.5e-157) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 0.46) {
		tmp = -t - Math.log(y);
	} else if (x <= 1.65e+117) {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.02e+110:
		tmp = t_1
	elif x <= -5.5e-157:
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t
	elif x <= 0.46:
		tmp = -t - math.log(y)
	elif x <= 1.65e+117:
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.02e+110)
		tmp = t_1;
	elseif (x <= -5.5e-157)
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * -0.5)))) - t);
	elseif (x <= 0.46)
		tmp = Float64(Float64(-t) - log(y));
	elseif (x <= 1.65e+117)
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(z * y)))) - z)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -1.02e+110)
		tmp = t_1;
	elseif (x <= -5.5e-157)
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	elseif (x <= 0.46)
		tmp = -t - log(y);
	elseif (x <= 1.65e+117)
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.02e+110], t$95$1, If[LessEqual[x, -5.5e-157], N[(N[(y * N[(z * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 0.46], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+117], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.02e110 or 1.6499999999999999e117 < x

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-define 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. sub-neg 98.6%

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

      6. log1p-define 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-/l* 98.6%

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

      7. sub-neg 98.6%

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

      8. log1p-define 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 83.4%

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

      1. *-commutative 83.4%

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

   10. Simplified 83.4%

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

   if -1.02e110 < x < -5.4999999999999998e-157

   1. Initial program 71.4%

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

      1. +-commutative 71.4%

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

      2. fma-define 71.4%

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

      3. sub-neg 71.4%

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

      4. metadata-eval 71.4%

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

      5. sub-neg 71.4%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 70.7%

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

      1. +-commutative 70.7%

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

      2. mul-1-neg 70.7%

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

      3. unsub-neg 70.7%

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

      4. sub-neg 70.7%

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

      5. metadata-eval 70.7%

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

      6. associate-/l* 65.9%

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

      7. sub-neg 65.9%

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

      8. log1p-define 84.3%

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

      9. +-commutative 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in z around inf 40.2%

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

      1. associate-/l* 40.2%

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

      2. sub-neg 40.2%

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

      3. log1p-undefine 64.0%

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

   10. Simplified 64.0%

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

   11. Taylor expanded in y around 0 69.6%

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

       1. associate-*r* 69.6%

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

       2. distribute-rgt-out 69.6%

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

   13. Simplified 69.6%

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

   if -5.4999999999999998e-157 < x < 0.46000000000000002

   1. Initial program 86.1%

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

      1. +-commutative 86.1%

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

      2. fma-define 86.1%

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

      3. sub-neg 86.1%

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

      4. metadata-eval 86.1%

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

      5. sub-neg 86.1%

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

      6. log1p-define 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. sub-neg 84.9%

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

      5. metadata-eval 84.9%

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

      6. associate-/l* 73.5%

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

      7. sub-neg 73.5%

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

      8. log1p-define 78.7%

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

      9. +-commutative 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in y around 0 84.8%

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

   9. Taylor expanded in x around 0 83.5%

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

       1. mul-1-neg 83.5%

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

   11. Simplified 83.5%

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

   if 0.46000000000000002 < x < 1.6499999999999999e117

   1. Initial program 84.5%

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

      1. +-commutative 84.5%

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

      2. fma-define 84.5%

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

      3. sub-neg 84.5%

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

      4. metadata-eval 84.5%

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

      5. sub-neg 84.5%

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

      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. Taylor expanded in x around inf 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. sub-neg 84.5%

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

      5. metadata-eval 84.5%

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

      6. associate-/l* 84.5%

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

      7. sub-neg 84.5%

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

      8. log1p-define 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in z around inf 58.2%

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

      1. associate-/l* 58.2%

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

      2. sub-neg 58.2%

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

      3. log1p-undefine 73.5%

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

   10. Simplified 73.5%

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

   11. Taylor expanded in y around 0 73.7%

       \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)\right) - t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+110}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-157}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 0.46:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+117}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 99.7%

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

Alternative 6: 84.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-157}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -1.2e+31)
     t_1
     (if (<= x -5.2e-157)
       (- (* y (* z (+ -1.0 (* y -0.5)))) t)
       (if (<= x 1.0) (- (- t) (log y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -1.2e+31) {
		tmp = t_1;
	} else if (x <= -5.2e-157) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 1.0) {
		tmp = -t - log(y);
	} 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 = (x * log(y)) - t
    if (x <= (-1.2d+31)) then
        tmp = t_1
    else if (x <= (-5.2d-157)) then
        tmp = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
    else if (x <= 1.0d0) then
        tmp = -t - log(y)
    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 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -1.2e+31) {
		tmp = t_1;
	} else if (x <= -5.2e-157) {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	} else if (x <= 1.0) {
		tmp = -t - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -1.2e+31:
		tmp = t_1
	elif x <= -5.2e-157:
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t
	elif x <= 1.0:
		tmp = -t - math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -1.2e+31)
		tmp = t_1;
	elseif (x <= -5.2e-157)
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * -0.5)))) - t);
	elseif (x <= 1.0)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -1.2e+31)
		tmp = t_1;
	elseif (x <= -5.2e-157)
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	elseif (x <= 1.0)
		tmp = -t - log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.2e+31], t$95$1, If[LessEqual[x, -5.2e-157], N[(N[(y * N[(z * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.0], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999991e31 or 1 < x

   1. Initial program 92.3%

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

      1. +-commutative 92.3%

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

      2. fma-define 92.3%

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

      3. sub-neg 92.3%

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

      4. metadata-eval 92.3%

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

      5. sub-neg 92.3%

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

      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. Taylor expanded in x around inf 92.3%

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

      1. +-commutative 92.3%

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

      2. mul-1-neg 92.3%

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

      3. unsub-neg 92.3%

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

      4. sub-neg 92.3%

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

      5. metadata-eval 92.3%

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

      6. associate-/l* 92.3%

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

      7. sub-neg 92.3%

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

      8. log1p-define 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 90.2%

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

   if -1.19999999999999991e31 < x < -5.19999999999999977e-157

   1. Initial program 67.5%

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

      1. +-commutative 67.5%

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

      2. fma-define 67.5%

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

      3. sub-neg 67.5%

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

      4. metadata-eval 67.5%

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

      5. sub-neg 67.5%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 66.5%

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

      1. +-commutative 66.5%

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

      2. mul-1-neg 66.5%

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

      3. unsub-neg 66.5%

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

      4. sub-neg 66.5%

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

      5. metadata-eval 66.5%

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

      6. associate-/l* 59.7%

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

      7. sub-neg 59.7%

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

      8. log1p-define 77.7%

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

      9. +-commutative 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in z around inf 41.7%

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

      1. associate-/l* 41.7%

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

      2. sub-neg 41.7%

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

      3. log1p-undefine 67.2%

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

   10. Simplified 67.2%

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

   11. Taylor expanded in y around 0 75.2%

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

       1. associate-*r* 75.2%

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

       2. distribute-rgt-out 75.3%

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

   13. Simplified 75.3%

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

   if -5.19999999999999977e-157 < x < 1

   1. Initial program 86.1%

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

      1. +-commutative 86.1%

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

      2. fma-define 86.1%

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

      3. sub-neg 86.1%

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

      4. metadata-eval 86.1%

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

      5. sub-neg 86.1%

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

      6. log1p-define 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. mul-1-neg 84.9%

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

      3. unsub-neg 84.9%

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

      4. sub-neg 84.9%

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

      5. metadata-eval 84.9%

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

      6. associate-/l* 73.5%

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

      7. sub-neg 73.5%

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

      8. log1p-define 78.7%

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

      9. +-commutative 78.7%

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

   7. Simplified 78.7%

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

   8. Taylor expanded in y around 0 84.8%

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

   9. Taylor expanded in x around 0 83.5%

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

       1. mul-1-neg 83.5%

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

   11. Simplified 83.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-157}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 8: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in y around 0 99.7%

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

4. Taylor expanded in y around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. distribute-rgt-out 99.5%

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

   4. sub-neg 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. *-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 9: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+175}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.5e+137)
   (- (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* z y)))) z)) t)
   (if (<= z 2.65e+175)
     (- (* (+ -1.0 x) (log y)) t)
     (- (* y (* z (+ -1.0 (* y -0.5)))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.5e+137) {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	} else if (z <= 2.65e+175) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - 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 <= (-5.5d+137)) then
        tmp = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (z * y)))) - z)) - t
    else if (z <= 2.65d+175) then
        tmp = (((-1.0d0) + x) * log(y)) - t
    else
        tmp = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.5e+137) {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	} else if (z <= 2.65e+175) {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	} else {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.5e+137:
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t
	elif z <= 2.65e+175:
		tmp = ((-1.0 + x) * math.log(y)) - t
	else:
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.5e+137)
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(z * y)))) - z)) - t);
	elseif (z <= 2.65e+175)
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.5e+137)
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	elseif (z <= 2.65e+175)
		tmp = ((-1.0 + x) * log(y)) - t;
	else
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.5e+137], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 2.65e+175], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5000000000000002e137

   1. Initial program 40.1%

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

      1. +-commutative 40.1%

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

      2. fma-define 40.1%

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

      3. sub-neg 40.1%

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

      4. metadata-eval 40.1%

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

      5. sub-neg 40.1%

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

      6. log1p-define 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 38.0%

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

      1. +-commutative 38.0%

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

      2. mul-1-neg 38.0%

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

      3. unsub-neg 38.0%

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

      4. sub-neg 38.0%

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

      5. metadata-eval 38.0%

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

      6. associate-/l* 22.2%

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

      7. sub-neg 22.2%

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

      8. log1p-define 58.1%

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

      9. +-commutative 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in z around inf 25.9%

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

      1. associate-/l* 25.9%

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

      2. sub-neg 25.9%

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

      3. log1p-undefine 69.0%

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

   10. Simplified 69.0%

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

   11. Taylor expanded in y around 0 88.5%

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

   if -5.5000000000000002e137 < z < 2.65000000000000006e175

   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. Step-by-step derivation

      1. +-commutative 97.2%

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

      2. fma-define 97.2%

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

      3. sub-neg 97.2%

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

      4. metadata-eval 97.2%

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

      5. sub-neg 97.2%

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

      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. Taylor expanded in y around 0 96.4%

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

   if 2.65000000000000006e175 < z

   1. Initial program 47.3%

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

      1. +-commutative 47.3%

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

      2. fma-define 47.3%

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

      3. sub-neg 47.3%

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

      4. metadata-eval 47.3%

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

      5. sub-neg 47.3%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 45.0%

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

      1. +-commutative 45.0%

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

      2. mul-1-neg 45.0%

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

      3. unsub-neg 45.0%

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

      4. sub-neg 45.0%

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

      5. metadata-eval 45.0%

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

      6. associate-/l* 30.1%

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

      7. sub-neg 30.1%

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

      8. log1p-define 58.7%

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

      9. +-commutative 58.7%

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

   7. Simplified 58.7%

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

   8. Taylor expanded in z around inf 31.5%

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

      1. associate-/l* 31.5%

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

      2. sub-neg 31.5%

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

      3. log1p-undefine 68.0%

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

   10. Simplified 68.0%

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

   11. Taylor expanded in y around 0 86.0%

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

       1. associate-*r* 86.0%

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

       2. distribute-rgt-out 86.0%

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

   13. Simplified 86.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+137}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+175}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+110} \lor \neg \left(x \leq 2 \cdot 10^{+117}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.1e+110) (not (<= x 2e+117)))
   (* x (log y))
   (- (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* z y)))) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.1e+110) || !(x <= 2e+117)) {
		tmp = x * log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - 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 <= (-1.1d+110)) .or. (.not. (x <= 2d+117))) then
        tmp = x * log(y)
    else
        tmp = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (z * y)))) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.1e+110) || !(x <= 2e+117)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.1e+110) or not (x <= 2e+117):
		tmp = x * math.log(y)
	else:
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.1e+110) || !(x <= 2e+117))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(z * y)))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.1e+110) || ~((x <= 2e+117)))
		tmp = x * log(y);
	else
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.1e+110], N[Not[LessEqual[x, 2e+117]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999996e110 or 2.0000000000000001e117 < x

   1. Initial program 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-define 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. sub-neg 98.6%

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

      6. log1p-define 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. mul-1-neg 98.6%

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

      3. unsub-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. associate-/l* 98.6%

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

      7. sub-neg 98.6%

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

      8. log1p-define 99.7%

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

      9. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 83.4%

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

      1. *-commutative 83.4%

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

   10. Simplified 83.4%

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

   if -1.09999999999999996e110 < x < 2.0000000000000001e117

   1. Initial program 80.5%

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

      1. +-commutative 80.5%

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

      2. fma-define 80.5%

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

      3. sub-neg 80.5%

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

      4. metadata-eval 80.5%

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

      5. sub-neg 80.5%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 79.7%

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

      1. +-commutative 79.7%

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

      2. mul-1-neg 79.7%

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

      3. unsub-neg 79.7%

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

      4. sub-neg 79.7%

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

      5. metadata-eval 79.7%

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

      6. associate-/l* 72.4%

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

      7. sub-neg 72.4%

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

      8. log1p-define 83.8%

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

      9. +-commutative 83.8%

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

   7. Simplified 83.8%

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

   8. Taylor expanded in z around inf 46.8%

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

      1. associate-/l* 46.8%

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

      2. sub-neg 46.8%

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

      3. log1p-undefine 59.3%

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

   10. Simplified 59.3%

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

   11. Taylor expanded in y around 0 66.5%

       \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)\right) - t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+110} \lor \neg \left(x \leq 2 \cdot 10^{+117}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in y around 0 99.0%

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

   1. mul-1-neg 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 12: 46.3% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. +-commutative 85.5%

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

   2. fma-define 85.5%

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

   3. sub-neg 85.5%

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

   4. metadata-eval 85.5%

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

   5. sub-neg 85.5%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 84.9%

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

   1. +-commutative 84.9%

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

   2. mul-1-neg 84.9%

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

   3. unsub-neg 84.9%

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

   4. sub-neg 84.9%

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

   5. metadata-eval 84.9%

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

   6. associate-/l* 79.6%

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

   7. sub-neg 79.6%

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

   8. log1p-define 88.2%

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

   9. +-commutative 88.2%

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

7. Simplified 88.2%

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

8. Taylor expanded in z around inf 38.3%

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

   1. associate-/l* 38.3%

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

   2. sub-neg 38.3%

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

   3. log1p-undefine 47.6%

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

10. Simplified 47.6%

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

11. Taylor expanded in y around 0 52.8%

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

12. Final simplification 52.8%

    \[\leadsto y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t \]
13. Add Preprocessing

Alternative 13: 46.2% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. +-commutative 85.5%

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

   2. fma-define 85.5%

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

   3. sub-neg 85.5%

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

   4. metadata-eval 85.5%

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

   5. sub-neg 85.5%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 84.9%

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

   1. +-commutative 84.9%

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

   2. mul-1-neg 84.9%

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

   3. unsub-neg 84.9%

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

   4. sub-neg 84.9%

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

   5. metadata-eval 84.9%

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

   6. associate-/l* 79.6%

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

   7. sub-neg 79.6%

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

   8. log1p-define 88.2%

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

   9. +-commutative 88.2%

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

7. Simplified 88.2%

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

8. Taylor expanded in z around inf 38.3%

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

   1. associate-/l* 38.3%

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

   2. sub-neg 38.3%

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

   3. log1p-undefine 47.6%

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

10. Simplified 47.6%

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

11. Taylor expanded in y around 0 52.7%

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

    1. associate-*r* 52.7%

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

    2. distribute-rgt-out 52.7%

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

13. Simplified 52.7%

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

14. Final simplification 52.7%

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

Alternative 14: 45.8% 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 85.5%

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

   1. +-commutative 85.5%

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

   2. fma-define 85.5%

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

   3. sub-neg 85.5%

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

   4. metadata-eval 85.5%

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

   5. sub-neg 85.5%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 84.9%

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

   1. +-commutative 84.9%

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

   2. mul-1-neg 84.9%

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

   3. unsub-neg 84.9%

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

   4. sub-neg 84.9%

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

   5. metadata-eval 84.9%

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

   6. associate-/l* 79.6%

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

   7. sub-neg 79.6%

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

   8. log1p-define 88.2%

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

   9. +-commutative 88.2%

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

7. Simplified 88.2%

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

8. Taylor expanded in z around inf 38.3%

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

   1. associate-/l* 38.3%

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

   2. sub-neg 38.3%

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

   3. log1p-undefine 47.6%

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

10. Simplified 47.6%

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

11. Taylor expanded in y around 0 52.2%

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

    1. mul-1-neg 52.2%

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

    2. distribute-rgt-neg-in 52.2%

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

13. Simplified 52.2%

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

14. Final simplification 52.2%

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

Alternative 15: 37.6% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ 1 - t \end{array} \]

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return 1.0 - t;
}

Python

def code(x, y, z, t):
	return 1.0 - t

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(1.0 - t), $MachinePrecision]

Derivation

1. Initial program 85.5%

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

   1. +-commutative 85.5%

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

   2. fma-define 85.5%

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

   3. sub-neg 85.5%

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

   4. metadata-eval 85.5%

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

   5. sub-neg 85.5%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around inf 37.6%

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

   1. mul-1-neg 37.6%

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

7. Simplified 37.6%

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

   1. expm1-log1p-u 17.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-t\right)\right)} \]

   2. expm1-undefine 17.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]

9. Applied egg-rr 17.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} - 1} \]
10. Step-by-step derivation

    1. sub-neg 17.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-t\right)} + \left(-1\right)} \]

    2. log1p-undefine 17.3%

       \[\leadsto e^{\color{blue}{\log \left(1 + \left(-t\right)\right)}} + \left(-1\right) \]

    3. rem-exp-log 37.4%

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

    4. unsub-neg 37.4%

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

    5. metadata-eval 37.4%

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

11. Simplified 37.4%

    \[\leadsto \color{blue}{\left(1 - t\right) + -1} \]
12. Step-by-step derivation

    1. associate-+l- 37.4%

       \[\leadsto \color{blue}{1 - \left(t - -1\right)} \]

    2. sub-neg 37.4%

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

    3. metadata-eval 37.4%

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

13. Applied egg-rr 37.4%

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

14. Taylor expanded in t around inf 39.7%

    \[\leadsto 1 - \color{blue}{t} \]
15. Add Preprocessing

Alternative 16: 35.4% 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 85.5%

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

   1. +-commutative 85.5%

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

   2. fma-define 85.5%

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

   3. sub-neg 85.5%

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

   4. metadata-eval 85.5%

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

   5. sub-neg 85.5%

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

   6. log1p-define 99.9%

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

   7. sub-neg 99.9%

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

   8. metadata-eval 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around inf 37.6%

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

   1. mul-1-neg 37.6%

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

7. Simplified 37.6%

   \[\leadsto \color{blue}{-t} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))