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

Percentage Accurate: 90.0% → 99.8%

Time: 19.7s

Alternatives: 18

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. sub-neg 90.5%

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

   2. +-commutative 90.5%

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

   3. associate-+l+ 90.5%

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

   4. fma-define 90.5%

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

   5. sub-neg 90.5%

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

   6. metadata-eval 90.5%

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

   7. sub-neg 90.5%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

   1. fma-define 90.5%

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

   2. sub-neg 90.5%

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

   3. metadata-eval 90.5%

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

   4. sub-neg 90.5%

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

   5. metadata-eval 90.5%

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

   6. sub-neg 90.5%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(-1 + x, \log y, \left(z + -1\right) \cdot \mathsf{log1p}\left(-y\right)\right) - t \]
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 90.5%

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

3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 95.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-1 + x\right) \cdot \log y\\ \mathbf{if}\;-1 + x \leq -100000:\\ \;\;\;\;t\_1 - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ -1.0 x) (log y))))
   (if (<= (+ -1.0 x) -100000.0)
     (- t_1 t)
     (if (<= (+ -1.0 x) -1.0)
       (- (- (* y (- 1.0 z)) t) (log y))
       (- (+ y t_1) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * log(y);
	double tmp;
	if ((-1.0 + x) <= -100000.0) {
		tmp = t_1 - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = ((y * (1.0 - z)) - t) - log(y);
	} else {
		tmp = (y + t_1) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) + x) * log(y)
    if (((-1.0d0) + x) <= (-100000.0d0)) then
        tmp = t_1 - t
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = ((y * (1.0d0 - z)) - t) - log(y)
    else
        tmp = (y + t_1) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * Math.log(y);
	double tmp;
	if ((-1.0 + x) <= -100000.0) {
		tmp = t_1 - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = ((y * (1.0 - z)) - t) - Math.log(y);
	} else {
		tmp = (y + t_1) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-1.0 + x) * math.log(y)
	tmp = 0
	if (-1.0 + x) <= -100000.0:
		tmp = t_1 - t
	elif (-1.0 + x) <= -1.0:
		tmp = ((y * (1.0 - z)) - t) - math.log(y)
	else:
		tmp = (y + t_1) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-1.0 + x) * log(y))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -100000.0)
		tmp = Float64(t_1 - t);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - t) - log(y));
	else
		tmp = Float64(Float64(y + t_1) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-1.0 + x) * log(y);
	tmp = 0.0;
	if ((-1.0 + x) <= -100000.0)
		tmp = t_1 - t;
	elseif ((-1.0 + x) <= -1.0)
		tmp = ((y * (1.0 - z)) - t) - log(y);
	else
		tmp = (y + t_1) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -100000.0], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y + t$95$1), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e5

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0 95.2%

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

   if -1e5 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 85.1%

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

      1. flip-- 85.1%

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

      2. metadata-eval 85.1%

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

      3. metadata-eval 85.1%

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

      4. associate-*l/ 85.1%

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

      5. metadata-eval 85.1%

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

      6. fma-neg 85.1%

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

      7. metadata-eval 85.1%

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

      8. +-commutative 85.1%

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in x around 0 85.1%

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

      1. +-commutative 85.1%

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

      2. mul-1-neg 85.1%

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

      3. sub-neg 85.1%

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

      4. sub-neg 85.1%

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

      5. metadata-eval 85.1%

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

      6. sub-neg 85.1%

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

      7. log1p-undefine 99.9%

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

      8. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 99.5%

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

      1. sub-neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. neg-mul-1 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-neg-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

   10. Simplified 99.5%

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

   11. Taylor expanded in t around 0 99.5%

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

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

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. mul-1-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around 0 97.2%

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

      1. sub-neg 97.2%

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

      2. metadata-eval 97.2%

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

      3. +-commutative 97.2%

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

      4. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -100000:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(-1 + x\right) \cdot \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-1 + x\right) \cdot \log y\\ \mathbf{if}\;-1 + x \leq -100000:\\ \;\;\;\;t\_1 - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ -1.0 x) (log y))))
   (if (<= (+ -1.0 x) -100000.0)
     (- t_1 t)
     (if (<= (+ -1.0 x) -1.0)
       (- (- (* y (- 1.0 z)) (log y)) t)
       (- (+ y t_1) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * log(y);
	double tmp;
	if ((-1.0 + x) <= -100000.0) {
		tmp = t_1 - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (y + t_1) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) + x) * log(y)
    if (((-1.0d0) + x) <= (-100000.0d0)) then
        tmp = t_1 - t
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (y + t_1) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (-1.0 + x) * Math.log(y);
	double tmp;
	if ((-1.0 + x) <= -100000.0) {
		tmp = t_1 - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (y + t_1) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-1.0 + x) * math.log(y)
	tmp = 0
	if (-1.0 + x) <= -100000.0:
		tmp = t_1 - t
	elif (-1.0 + x) <= -1.0:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (y + t_1) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-1.0 + x) * log(y))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -100000.0)
		tmp = Float64(t_1 - t);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(y + t_1) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (-1.0 + x) * log(y);
	tmp = 0.0;
	if ((-1.0 + x) <= -100000.0)
		tmp = t_1 - t;
	elseif ((-1.0 + x) <= -1.0)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (y + t_1) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -100000.0], N[(t$95$1 - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y + t$95$1), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e5

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0 95.2%

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

   if -1e5 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 85.1%

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

      1. flip-- 85.1%

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

      2. metadata-eval 85.1%

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

      3. metadata-eval 85.1%

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

      4. associate-*l/ 85.1%

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

      5. metadata-eval 85.1%

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

      6. fma-neg 85.1%

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

      7. metadata-eval 85.1%

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

      8. +-commutative 85.1%

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in x around 0 85.1%

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

      1. +-commutative 85.1%

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

      2. mul-1-neg 85.1%

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

      3. sub-neg 85.1%

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

      4. sub-neg 85.1%

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

      5. metadata-eval 85.1%

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

      6. sub-neg 85.1%

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

      7. log1p-undefine 99.9%

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

      8. +-commutative 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0 99.5%

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

      1. sub-neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. neg-mul-1 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-neg-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

   10. Simplified 99.5%

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

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

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. mul-1-neg 99.7%

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

      5. unsub-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

      9. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in z around 0 97.2%

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

      1. sub-neg 97.2%

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

      2. metadata-eval 97.2%

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

      3. +-commutative 97.2%

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

      4. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -100000:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \left(-1 + x\right) \cdot \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 90.5%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 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 90.5%

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

3. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 8: 88.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-13} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e-13) (not (<= x 1.0)))
   (- (* x (log y)) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-13) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2d-13)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-13) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2e-13) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e-13) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2e-13) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e-13], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-13 or 1 < x

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 94.8%

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

      1. *-commutative 94.8%

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

   9. Simplified 94.8%

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

   if -2.0000000000000001e-13 < x < 1

   1. Initial program 86.2%

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

      1. flip-- 86.2%

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

      2. metadata-eval 86.2%

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

      3. metadata-eval 86.2%

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

      4. associate-*l/ 86.2%

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

      5. metadata-eval 86.2%

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

      6. fma-neg 86.2%

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

      7. metadata-eval 86.2%

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

      8. +-commutative 86.2%

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

   4. Applied egg-rr 86.2%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. mul-1-neg 85.6%

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

      3. sub-neg 85.6%

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

      4. sub-neg 85.6%

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

      5. metadata-eval 85.6%

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

      6. sub-neg 85.6%

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

      7. log1p-undefine 99.3%

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

      8. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y around 0 98.9%

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

      1. sub-neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. neg-mul-1 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. +-commutative 98.9%

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

      6. distribute-neg-in 98.9%

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

      7. metadata-eval 98.9%

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

      8. sub-neg 98.9%

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

   10. Simplified 98.9%

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

   11. Taylor expanded in z around 0 84.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-13} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-13} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e-13) (not (<= x 1.0)))
   (- (* x (log y)) t)
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-13) || !(x <= 1.0)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2d-13)) .or. (.not. (x <= 1.0d0))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e-13) || !(x <= 1.0)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2e-13) or not (x <= 1.0):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e-13) || !(x <= 1.0))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2e-13) || ~((x <= 1.0)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e-13], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-13 or 1 < x

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0 99.6%

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

   4. Taylor expanded in y around 0 99.6%

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

      1. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 94.8%

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

      1. *-commutative 94.8%

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

   9. Simplified 94.8%

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

   if -2.0000000000000001e-13 < x < 1

   1. Initial program 86.2%

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

      1. flip-- 86.2%

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

      2. metadata-eval 86.2%

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

      3. metadata-eval 86.2%

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

      4. associate-*l/ 86.2%

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

      5. metadata-eval 86.2%

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

      6. fma-neg 86.2%

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

      7. metadata-eval 86.2%

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

      8. +-commutative 86.2%

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

   4. Applied egg-rr 86.2%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. mul-1-neg 85.6%

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

      3. sub-neg 85.6%

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

      4. sub-neg 85.6%

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

      5. metadata-eval 85.6%

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

      6. sub-neg 85.6%

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

      7. log1p-undefine 99.3%

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

      8. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in y around 0 98.9%

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

      1. sub-neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. neg-mul-1 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. +-commutative 98.9%

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

      6. distribute-neg-in 98.9%

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

      7. metadata-eval 98.9%

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

      8. sub-neg 98.9%

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

   10. Simplified 98.9%

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

   11. Taylor expanded in y around 0 84.5%

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

       1. mul-1-neg 84.5%

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

   13. Simplified 84.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-13} \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -25500000000 \lor \neg \left(x \leq 1.05 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -25500000000.0) (not (<= x 1.05e+15)))
   (* x (log y))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -25500000000.0) || !(x <= 1.05e+15)) {
		tmp = x * log(y);
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-25500000000.0d0)) .or. (.not. (x <= 1.05d+15))) then
        tmp = x * log(y)
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -25500000000.0) || !(x <= 1.05e+15)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -25500000000.0) or not (x <= 1.05e+15):
		tmp = x * math.log(y)
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -25500000000.0) || !(x <= 1.05e+15))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -25500000000.0) || ~((x <= 1.05e+15)))
		tmp = x * log(y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -25500000000.0], N[Not[LessEqual[x, 1.05e+15]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.55e10 or 1.05e15 < x

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf 76.1%

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

      1. associate--l+ 76.1%

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

      2. sub-neg 76.1%

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

      3. metadata-eval 76.1%

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

      4. associate-/l* 76.0%

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

      5. +-commutative 76.0%

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

      6. associate-/l* 74.8%

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

      7. sub-neg 74.8%

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

      8. metadata-eval 74.8%

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

      9. +-commutative 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in x around inf 76.9%

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

      1. *-commutative 76.9%

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

   8. Simplified 76.9%

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

   if -2.55e10 < x < 1.05e15

   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. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 86.1%

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

      2. metadata-eval 86.1%

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

      3. metadata-eval 86.1%

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

      4. associate-*l/ 86.1%

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

      5. metadata-eval 86.1%

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

      6. fma-neg 86.1%

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

      7. metadata-eval 86.1%

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

      8. +-commutative 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around 0 84.8%

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

      1. +-commutative 84.8%

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

      2. mul-1-neg 84.8%

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

      3. sub-neg 84.8%

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

      4. sub-neg 84.8%

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

      5. metadata-eval 84.8%

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

      6. sub-neg 84.8%

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

      7. log1p-undefine 98.6%

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

      8. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around 0 98.2%

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

      1. sub-neg 98.2%

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

      2. metadata-eval 98.2%

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

      3. neg-mul-1 98.2%

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

      4. distribute-rgt-neg-in 98.2%

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

      5. +-commutative 98.2%

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

      6. distribute-neg-in 98.2%

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

      7. metadata-eval 98.2%

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

      8. sub-neg 98.2%

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

   10. Simplified 98.2%

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

   11. Taylor expanded in y around 0 83.7%

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

       1. mul-1-neg 83.7%

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

   13. Simplified 83.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -25500000000 \lor \neg \left(x \leq 1.05 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11500000000 \lor \neg \left(x \leq 4.6 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -11500000000.0) (not (<= x 4.6e+17)))
   (* x (log y))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -11500000000.0) || !(x <= 4.6e+17)) {
		tmp = x * log(y);
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-11500000000.0d0)) .or. (.not. (x <= 4.6d+17))) then
        tmp = x * log(y)
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -11500000000.0) || !(x <= 4.6e+17)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -11500000000.0) or not (x <= 4.6e+17):
		tmp = x * math.log(y)
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -11500000000.0) || !(x <= 4.6e+17))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -11500000000.0) || ~((x <= 4.6e+17)))
		tmp = x * log(y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -11500000000.0], N[Not[LessEqual[x, 4.6e+17]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e10 or 4.6e17 < x

   1. Initial program 96.1%

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

   3. Taylor expanded in t around inf 76.1%

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

      1. associate--l+ 76.1%

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

      2. sub-neg 76.1%

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

      3. metadata-eval 76.1%

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

      4. associate-/l* 76.0%

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

      5. +-commutative 76.0%

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

      6. associate-/l* 74.8%

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

      7. sub-neg 74.8%

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

      8. metadata-eval 74.8%

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

      9. +-commutative 74.8%

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

   5. Simplified 74.8%

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

   6. Taylor expanded in x around inf 76.9%

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

      1. *-commutative 76.9%

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

   8. Simplified 76.9%

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

   if -1.15e10 < x < 4.6e17

   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. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 86.1%

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

      2. metadata-eval 86.1%

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

      3. metadata-eval 86.1%

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

      4. associate-*l/ 86.1%

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

      5. metadata-eval 86.1%

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

      6. fma-neg 86.1%

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

      7. metadata-eval 86.1%

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

      8. +-commutative 86.1%

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

   4. Applied egg-rr 86.1%

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

   5. Taylor expanded in x around 0 84.8%

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

      1. +-commutative 84.8%

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

      2. mul-1-neg 84.8%

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

      3. sub-neg 84.8%

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

      4. sub-neg 84.8%

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

      5. metadata-eval 84.8%

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

      6. sub-neg 84.8%

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

      7. log1p-undefine 98.6%

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

      8. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in y around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. distribute-neg-in 83.7%

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

      3. unsub-neg 83.7%

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

   10. Simplified 83.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11500000000 \lor \neg \left(x \leq 4.6 \cdot 10^{+17}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+200}:\\ \;\;\;\;\left(y + \left(-1 + x\right) \cdot \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 2.8e+200) (- (+ y (* (+ -1.0 x) (log y))) t) (- (* y (- 1.0 z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.8e+200) {
		tmp = (y + ((-1.0 + x) * log(y))) - t;
	} else {
		tmp = (y * (1.0 - 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 (z <= 2.8d+200) then
        tmp = (y + (((-1.0d0) + x) * log(y))) - t
    else
        tmp = (y * (1.0d0 - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 2.8e+200) {
		tmp = (y + ((-1.0 + x) * Math.log(y))) - t;
	} else {
		tmp = (y * (1.0 - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 2.8e+200:
		tmp = (y + ((-1.0 + x) * math.log(y))) - t
	else:
		tmp = (y * (1.0 - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 2.8e+200)
		tmp = Float64(Float64(y + Float64(Float64(-1.0 + x) * log(y))) - t);
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 2.8e+200)
		tmp = (y + ((-1.0 + x) * log(y))) - t;
	else
		tmp = (y * (1.0 - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 2.8e+200], N[(N[(y + N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.79999999999999985e200

   1. Initial program 96.8%

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

      1. fma-define 96.8%

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

      2. sub-neg 96.8%

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

      3. metadata-eval 96.8%

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

      4. sub-neg 96.8%

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

      5. metadata-eval 96.8%

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

      6. sub-neg 96.8%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. distribute-rgt-neg-in 99.6%

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

      3. sub-neg 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. metadata-eval 99.6%

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

      8. unsub-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around 0 96.4%

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

   if 2.79999999999999985e200 < z

   1. Initial program 37.0%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. mul-1-neg 99.8%

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

      5. unsub-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 79.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+200}:\\ \;\;\;\;\left(y + \left(-1 + x\right) \cdot \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5800000000 \lor \neg \left(x \leq 2.25 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5800000000.0) (not (<= x 2.25e+67)))
   (* x (log y))
   (- (* y (- 1.0 z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5800000000.0) || !(x <= 2.25e+67)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (1.0 - 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 <= (-5800000000.0d0)) .or. (.not. (x <= 2.25d+67))) then
        tmp = x * log(y)
    else
        tmp = (y * (1.0d0 - 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 <= -5800000000.0) || !(x <= 2.25e+67)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (1.0 - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5800000000.0) or not (x <= 2.25e+67):
		tmp = x * math.log(y)
	else:
		tmp = (y * (1.0 - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5800000000.0) || !(x <= 2.25e+67))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5800000000.0) || ~((x <= 2.25e+67)))
		tmp = x * log(y);
	else
		tmp = (y * (1.0 - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5800000000.0], N[Not[LessEqual[x, 2.25e+67]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.8e9 or 2.2499999999999999e67 < x

   1. Initial program 95.7%

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

   3. Taylor expanded in t around inf 74.7%

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

      1. associate--l+ 74.7%

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

      2. sub-neg 74.7%

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

      3. metadata-eval 74.7%

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

      4. associate-/l* 74.6%

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

      5. +-commutative 74.6%

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

      6. associate-/l* 73.3%

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

      7. sub-neg 73.3%

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

      8. metadata-eval 73.3%

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

      9. +-commutative 73.3%

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

   5. Simplified 73.3%

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

   6. Taylor expanded in x around inf 79.4%

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

      1. *-commutative 79.4%

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

   8. Simplified 79.4%

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

   if -5.8e9 < x < 2.2499999999999999e67

   1. Initial program 87.0%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. mul-1-neg 99.5%

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

      5. unsub-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. +-commutative 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in y around inf 60.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5800000000 \lor \neg \left(x \leq 2.25 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

   4. mul-1-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 15: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+200}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.45e+200) (- (* (+ -1.0 x) (log y)) t) (- (* y (- 1.0 z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+200) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = (y * (1.0 - 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 (z <= 1.45d+200) then
        tmp = (((-1.0d0) + x) * log(y)) - t
    else
        tmp = (y * (1.0d0 - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.45e+200) {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	} else {
		tmp = (y * (1.0 - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 1.45e+200:
		tmp = ((-1.0 + x) * math.log(y)) - t
	else:
		tmp = (y * (1.0 - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 1.45e+200)
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(1.0 - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 1.45e+200)
		tmp = ((-1.0 + x) * log(y)) - t;
	else
		tmp = (y * (1.0 - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 1.45e+200], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4499999999999999e200

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0 96.2%

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

   if 1.4499999999999999e200 < z

   1. Initial program 37.0%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

      4. mul-1-neg 99.8%

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

      5. unsub-neg 99.8%

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

      6. +-commutative 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

      9. +-commutative 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 79.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+200}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.7% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

   4. mul-1-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in y around inf 44.5%

   \[\leadsto \color{blue}{y \cdot \left(1 - z\right)} - t \]
7. Add Preprocessing

Alternative 17: 45.5% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

   4. mul-1-neg 99.6%

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

   5. unsub-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

   9. +-commutative 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in z around inf 44.3%

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

   1. associate-*r* 44.3%

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

   2. mul-1-neg 44.3%

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

8. Simplified 44.3%

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

9. Final simplification 44.3%

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

Alternative 18: 36.0% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.5%

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

3. Taylor expanded in t around inf 35.6%

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

   1. mul-1-neg 35.6%

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

5. Simplified 35.6%

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

Reproduce

herbie shell --seed 2024103 
(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))