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

Percentage Accurate: 89.5% → 99.8%

Time: 16.2s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. +-commutative 86.8%

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

   2. fma-define 86.8%

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

   3. sub-neg 86.8%

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

   4. metadata-eval 86.8%

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

   5. sub-neg 86.8%

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

   6. log1p-define 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((log(y) * ((-1.0d0) + x)) + (y * ((1.0d0 - z) + (y * (((z + (-1.0d0)) * (-0.5d0)) + (y * (((z + (-1.0d0)) * (-0.3333333333333333d0)) + ((-0.25d0) * (y * (z + (-1.0d0))))))))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (log y) (+ -1.0 x))
   (*
    (+ 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 ((log(y) * (-1.0 + x)) + ((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 = ((log(y) * ((-1.0d0) + x)) + ((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 ((Math.log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)))))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log(y) * (-1.0 + x)) + ((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(log(y) * Float64(-1.0 + x)) + 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 = ((log(y) * (-1.0 + x)) + ((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[Log[y], $MachinePrecision] * N[(-1.0 + x), $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 86.8%

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

3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 95.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.000002:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -1.000002)
   (- (* (log y) (+ -1.0 x)) t)
   (if (<= (+ -1.0 x) 4e+15)
     (- (- (* y (- 1.0 z)) (log y)) t)
     (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -1.000002) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 4e+15) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (x * 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 (((-1.0d0) + x) <= (-1.000002d0)) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else if (((-1.0d0) + x) <= 4d+15) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (x * 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 ((-1.0 + x) <= -1.000002) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 4e+15) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (x * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -1.000002:
		tmp = (math.log(y) * (-1.0 + x)) - t
	elif (-1.0 + x) <= 4e+15:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (x * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -1.000002)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	elseif (Float64(-1.0 + x) <= 4e+15)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -1.000002)
		tmp = (log(y) * (-1.0 + x)) - t;
	elseif ((-1.0 + x) <= 4e+15)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (x * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.000002], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 4e+15], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1.00000200000000006

   1. Initial program 94.7%

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

      1. +-commutative 94.7%

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

      2. fma-define 94.7%

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

      3. sub-neg 94.7%

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

      4. metadata-eval 94.7%

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

      5. sub-neg 94.7%

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

      6. log1p-define 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 93.5%

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

   if -1.00000200000000006 < (-.f64 x #s(literal 1 binary64)) < 4e15

   1. Initial program 76.4%

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

   3. Taylor expanded in y around 0 99.5%

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

   4. Taylor expanded in x around 0 98.4%

      \[\leadsto \left(\color{blue}{-1 \cdot \log y} + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)\right) - t \]
   5. Step-by-step derivation

      1. mul-1-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in y around 0 97.4%

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

      1. associate-*r* 97.4%

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

      2. mul-1-neg 97.4%

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

      3. sub-neg 97.4%

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

      4. metadata-eval 97.4%

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

      5. +-commutative 97.4%

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

   9. Simplified 97.4%

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

   if 4e15 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around -inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. distribute-lft-out 99.8%

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

      4. associate-*r* 99.8%

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

      5. *-commutative 99.8%

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

      6. neg-mul-1 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

      9. *-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.000002:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 4 \cdot 10^{+15}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* (log y) (+ -1.0 x))
   (* (+ 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 ((log(y) * (-1.0 + x)) + ((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 = ((log(y) * ((-1.0d0) + x)) + ((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 ((Math.log(y) * (-1.0 + x)) + ((z + -1.0) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))))) - t;
}

Python

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] + N[(N[(z + -1.0), $MachinePrecision] * 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 86.8%

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

3. Taylor expanded in y around 0 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 7: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+158}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+239}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+158)
   (-
    (*
     y
     (-
      (*
       y
       (+ (* z -0.5) (* y (+ (* z -0.3333333333333333) (* -0.25 (* z y))))))
      z))
    t)
   (if (<= z 5e+239) (- (* (log y) (+ -1.0 x)) t) (- (* z (- y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+158) {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	} else if (z <= 5e+239) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = (z * -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 (z <= (-3.5d+158)) then
        tmp = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((-0.25d0) * (z * y)))))) - z)) - t
    else if (z <= 5d+239) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = (z * -y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+158) {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	} else if (z <= 5e+239) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.5e+158:
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t
	elif z <= 5e+239:
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+158)
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(y * Float64(Float64(z * -0.3333333333333333) + Float64(-0.25 * Float64(z * y)))))) - z)) - t);
	elseif (z <= 5e+239)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.5e+158)
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	elseif (z <= 5e+239)
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+158], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(y * N[(N[(z * -0.3333333333333333), $MachinePrecision] + N[(-0.25 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 5e+239], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5000000000000001e158

   1. Initial program 48.4%

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

   3. Taylor expanded in x around -inf 48.4%

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

      1. associate-*r* 48.4%

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

      2. *-commutative 48.4%

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

      3. distribute-lft-out 48.4%

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

      4. associate-*r* 48.4%

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

      5. *-commutative 48.4%

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

      6. neg-mul-1 48.4%

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

      7. distribute-rgt-neg-in 48.4%

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

      8. metadata-eval 48.4%

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

      9. *-rgt-identity 48.4%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in z around inf 30.3%

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

      1. sub-neg 30.3%

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

      2. log1p-undefine 81.3%

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

   8. Simplified 81.3%

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

   9. Taylor expanded in y around 0 81.3%

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

   if -3.5000000000000001e158 < z < 5.00000000000000007e239

   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. Step-by-step derivation

      1. +-commutative 95.7%

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

      2. fma-define 95.7%

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

      3. sub-neg 95.7%

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

      4. metadata-eval 95.7%

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

      5. sub-neg 95.7%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 94.3%

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

   if 5.00000000000000007e239 < z

   1. Initial program 42.8%

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

   3. Taylor expanded in x around -inf 42.8%

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

      1. associate-*r* 42.8%

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

      2. *-commutative 42.8%

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

      3. distribute-lft-out 42.8%

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

      4. associate-*r* 42.8%

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

      5. *-commutative 42.8%

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

      6. neg-mul-1 42.8%

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

      7. distribute-rgt-neg-in 42.8%

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

      8. metadata-eval 42.8%

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

      9. *-rgt-identity 42.8%

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

   5. Simplified 91.5%

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

   6. Taylor expanded in z around inf 27.1%

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

      1. sub-neg 27.1%

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

      2. log1p-undefine 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in y around 0 80.7%

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

       1. mul-1-neg 80.7%

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

       2. distribute-rgt-neg-in 80.7%

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

   11. Simplified 80.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+158}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+239}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in y around 0 99.6%

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

4. Taylor expanded in y around 0 99.3%

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

   1. *-commutative 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in z around inf 99.1%

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

8. Final simplification 99.1%

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

Alternative 9: 87.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1100000 \lor \neg \left(x \leq 2.4 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1100000.0) (not (<= x 2.4e+15)))
   (- (* x (log y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1100000.0) || !(x <= 2.4e+15)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1100000.0d0)) .or. (.not. (x <= 2.4d+15))) then
        tmp = (x * log(y)) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1100000.0) || !(x <= 2.4e+15)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1100000.0) or not (x <= 2.4e+15):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1100000.0) || !(x <= 2.4e+15))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1100000.0) || ~((x <= 2.4e+15)))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1100000.0], N[Not[LessEqual[x, 2.4e+15]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1e6 or 2.4e15 < x

   1. Initial program 97.8%

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

   3. Taylor expanded in x around -inf 97.8%

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

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. distribute-lft-out 97.8%

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

      4. associate-*r* 97.8%

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

      5. *-commutative 97.8%

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

      6. neg-mul-1 97.8%

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

      7. distribute-rgt-neg-in 97.8%

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

      8. metadata-eval 97.8%

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

      9. *-rgt-identity 97.8%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 96.6%

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

      1. *-commutative 96.6%

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

   8. Simplified 96.6%

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

   if -1.1e6 < x < 2.4e15

   1. Initial program 76.4%

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

   3. Taylor expanded in y around 0 99.5%

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

   4. Taylor expanded in x around 0 97.5%

      \[\leadsto \left(\color{blue}{-1 \cdot \log y} + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)\right) - t \]
   5. Step-by-step derivation

      1. mul-1-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in y around 0 72.3%

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

      1. neg-mul-1 72.3%

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

   9. Simplified 72.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1100000 \lor \neg \left(x \leq 2.4 \cdot 10^{+15}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+184}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.5e+98)
   (-
    (*
     y
     (-
      (*
       y
       (+ (* z -0.5) (* y (+ (* z -0.3333333333333333) (* -0.25 (* z y))))))
      z))
    t)
   (if (<= z 2.45e+184) (- (- (log y)) t) (- (* z (- y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e+98) {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	} else if (z <= 2.45e+184) {
		tmp = -log(y) - t;
	} else {
		tmp = (z * -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 (z <= (-1.5d+98)) then
        tmp = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((-0.25d0) * (z * y)))))) - z)) - t
    else if (z <= 2.45d+184) then
        tmp = -log(y) - t
    else
        tmp = (z * -y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.5e+98) {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	} else if (z <= 2.45e+184) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.5e+98:
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t
	elif z <= 2.45e+184:
		tmp = -math.log(y) - t
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.5e+98)
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(y * Float64(Float64(z * -0.3333333333333333) + Float64(-0.25 * Float64(z * y)))))) - z)) - t);
	elseif (z <= 2.45e+184)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.5e+98)
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	elseif (z <= 2.45e+184)
		tmp = -log(y) - t;
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.5e+98], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(y * N[(N[(z * -0.3333333333333333), $MachinePrecision] + N[(-0.25 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 2.45e+184], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.5000000000000001e98

   1. Initial program 64.4%

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

   3. Taylor expanded in x around -inf 62.7%

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

      1. associate-*r* 62.7%

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

      2. *-commutative 62.7%

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

      3. distribute-lft-out 62.7%

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

      4. associate-*r* 62.7%

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

      5. *-commutative 62.7%

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

      6. neg-mul-1 62.7%

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

      7. distribute-rgt-neg-in 62.7%

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

      8. metadata-eval 62.7%

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

      9. *-rgt-identity 62.7%

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

   5. Simplified 85.7%

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

   6. Taylor expanded in z around inf 38.3%

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

      1. sub-neg 38.3%

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

      2. log1p-undefine 73.4%

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

   8. Simplified 73.4%

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

   9. Taylor expanded in y around 0 72.7%

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

   if -1.5000000000000001e98 < z < 2.45000000000000015e184

   1. Initial program 97.8%

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

   3. Taylor expanded in y around 0 99.8%

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

   4. Taylor expanded in x around 0 58.7%

      \[\leadsto \left(\color{blue}{-1 \cdot \log y} + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)\right) - t \]
   5. Step-by-step derivation

      1. mul-1-neg 58.7%

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

   6. Simplified 58.7%

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

   7. Taylor expanded in y around 0 56.2%

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

      1. neg-mul-1 56.2%

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

   9. Simplified 56.2%

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

   if 2.45000000000000015e184 < z

   1. Initial program 59.8%

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

   3. Taylor expanded in x around -inf 59.8%

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

      1. associate-*r* 59.8%

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

      2. *-commutative 59.8%

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

      3. distribute-lft-out 59.8%

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

      4. associate-*r* 59.8%

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

      5. *-commutative 59.8%

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

      6. neg-mul-1 59.8%

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

      7. distribute-rgt-neg-in 59.8%

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

      8. metadata-eval 59.8%

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

      9. *-rgt-identity 59.8%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in z around inf 30.5%

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

      1. sub-neg 30.5%

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

      2. log1p-undefine 68.8%

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

   8. Simplified 68.8%

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

   9. Taylor expanded in y around 0 68.8%

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

       1. mul-1-neg 68.8%

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

       2. distribute-rgt-neg-in 68.8%

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

   11. Simplified 68.8%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+184}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. sub-neg 98.9%

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

   3. metadata-eval 98.9%

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

   4. fma-define 98.9%

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

   5. mul-1-neg 98.9%

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

   6. fmm-undef 98.9%

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

   7. +-commutative 98.9%

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

   8. sub-neg 98.9%

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

   9. metadata-eval 98.9%

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

   10. +-commutative 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

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

Alternative 12: 46.8% accurate, 9.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((-0.25d0) * (z * y)))))) - z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around -inf 85.6%

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

   1. associate-*r* 85.6%

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

   2. *-commutative 85.6%

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

   3. distribute-lft-out 85.6%

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

   4. associate-*r* 85.6%

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

   5. *-commutative 85.6%

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

   6. neg-mul-1 85.6%

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

   7. distribute-rgt-neg-in 85.6%

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

   8. metadata-eval 85.6%

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

   9. *-rgt-identity 85.6%

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

5. Simplified 94.5%

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

6. Taylor expanded in z around inf 38.3%

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

   1. sub-neg 38.3%

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

   2. log1p-undefine 50.7%

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

8. Simplified 50.7%

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

9. Taylor expanded in y around 0 50.5%

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

10. Final simplification 50.5%

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

Alternative 13: 46.7% accurate, 14.3× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (y * (z * (-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 = (y * (z * ((-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 (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in y around 0 99.6%

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

4. Taylor expanded in x around 0 64.4%

   \[\leadsto \left(\color{blue}{-1 \cdot \log y} + \left(z - 1\right) \cdot \left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)\right) - t \]
5. Step-by-step derivation

   1. mul-1-neg 64.4%

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

6. Simplified 64.4%

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

7. Taylor expanded in z around inf 50.4%

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

8. Final simplification 50.4%

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

Alternative 14: 46.6% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around -inf 85.6%

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

   1. associate-*r* 85.6%

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

   2. *-commutative 85.6%

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

   3. distribute-lft-out 85.6%

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

   4. associate-*r* 85.6%

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

   5. *-commutative 85.6%

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

   6. neg-mul-1 85.6%

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

   7. distribute-rgt-neg-in 85.6%

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

   8. metadata-eval 85.6%

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

   9. *-rgt-identity 85.6%

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

5. Simplified 94.5%

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

6. Taylor expanded in z around inf 38.3%

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

   1. sub-neg 38.3%

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

   2. log1p-undefine 50.7%

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

8. Simplified 50.7%

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

9. Taylor expanded in y around 0 50.2%

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

    1. associate-*r* 50.2%

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

    2. distribute-rgt-out 50.2%

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

11. Simplified 50.2%

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

12. Final simplification 50.2%

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

Alternative 15: 46.3% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around -inf 85.6%

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

   1. associate-*r* 85.6%

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

   2. *-commutative 85.6%

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

   3. distribute-lft-out 85.6%

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

   4. associate-*r* 85.6%

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

   5. *-commutative 85.6%

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

   6. neg-mul-1 85.6%

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

   7. distribute-rgt-neg-in 85.6%

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

   8. metadata-eval 85.6%

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

   9. *-rgt-identity 85.6%

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

5. Simplified 94.5%

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

6. Taylor expanded in z around inf 38.3%

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

   1. sub-neg 38.3%

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

   2. log1p-undefine 50.7%

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

8. Simplified 50.7%

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

9. Taylor expanded in y around 0 49.8%

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

    1. mul-1-neg 49.8%

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

    2. distribute-rgt-neg-in 49.8%

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

11. Simplified 49.8%

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

12. Final simplification 49.8%

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

Alternative 16: 36.3% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

   1. +-commutative 86.8%

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

   2. fma-define 86.8%

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

   3. sub-neg 86.8%

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

   4. metadata-eval 86.8%

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

   5. sub-neg 86.8%

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

   6. log1p-define 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 37.0%

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

   1. neg-mul-1 37.0%

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

7. Simplified 37.0%

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

Alternative 17: 2.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 0.0)

C

double code(double x, double y, double z, double t) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0;
}

Python

def code(x, y, z, t):
	return 0.0

Julia

function code(x, y, z, t)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_] := 0.0

Derivation

1. Initial program 86.8%

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

   1. +-commutative 86.8%

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

   2. fma-define 86.8%

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

   3. sub-neg 86.8%

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

   4. metadata-eval 86.8%

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

   5. sub-neg 86.8%

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

   6. log1p-define 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 37.0%

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

   1. neg-mul-1 37.0%

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

7. Simplified 37.0%

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

   1. expm1-log1p-u 13.8%

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

   2. expm1-undefine 13.7%

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

9. Applied egg-rr 13.7%

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

    1. sub-neg 13.7%

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

    2. log1p-undefine 13.7%

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

    3. rem-exp-log 36.8%

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

    4. unsub-neg 36.8%

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

    5. metadata-eval 36.8%

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

11. Simplified 36.8%

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

12. Taylor expanded in t around 0 2.3%

    \[\leadsto \color{blue}{1} + -1 \]
13. Step-by-step derivation

    1. metadata-eval 2.3%

       \[\leadsto \color{blue}{0} \]

14. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{0} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(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))