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

Percentage Accurate: 89.0% → 99.8%

Time: 21.0s

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: 89.0% 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, 0.7× 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 88.7%

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

   1. +-commutative 88.7%

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

   2. fma-define 88.7%

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

   3. sub-neg 88.7%

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

   4. metadata-eval 88.7%

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

   5. sub-neg 88.7%

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

   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. 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: 99.7% 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 88.7%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.7% 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 88.7%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 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 88.7%

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

3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 5: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+16} \lor \neg \left(-1 + x \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\log y\right) - z \cdot y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -5e+16) (not (<= (+ -1.0 x) 2e+16)))
   (- (* x (log y)) t)
   (- (- (- (log y)) (* z y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5e+16) || !((-1.0 + x) <= 2e+16)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (-log(y) - (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 ((((-1.0d0) + x) <= (-5d+16)) .or. (.not. (((-1.0d0) + x) <= 2d+16))) then
        tmp = (x * log(y)) - t
    else
        tmp = (-log(y) - (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 (((-1.0 + x) <= -5e+16) || !((-1.0 + x) <= 2e+16)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (-Math.log(y) - (z * y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -5e+16) or not ((-1.0 + x) <= 2e+16):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (-math.log(y) - (z * y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -5e+16) || !(Float64(-1.0 + x) <= 2e+16))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(-log(y)) - Float64(z * y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -5e+16) || ~(((-1.0 + x) <= 2e+16)))
		tmp = (x * log(y)) - t;
	else
		tmp = (-log(y) - (z * y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -5e+16], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], 2e+16]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[((-N[Log[y], $MachinePrecision]) - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e16 or 2e16 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0 99.6%

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

   4. Taylor expanded in z around inf 99.6%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. *-commutative 96.5%

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

   7. Simplified 96.5%

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

   if -5e16 < (-.f64 x #s(literal 1 binary64)) < 2e16

   1. Initial program 82.2%

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

      1. add-cube-cbrt 81.4%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y} \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      2. pow3 81.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      3. sub-neg 81.4%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      4. metadata-eval 81.4%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + \color{blue}{-1}\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 81.4%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 97.3%

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

      1. mul-1-neg 97.3%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 97.3%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in z around inf 97.0%

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

      1. associate-*r* 97.0%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) - t \]

      2. mul-1-neg 97.0%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right)} \cdot z\right) - t \]

   10. Simplified 97.0%

       \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right) \cdot z}\right) - t \]

   11. Taylor expanded in x around 0 96.3%

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

       1. mul-1-neg 96.3%

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

       2. unsub-neg 96.3%

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

       3. mul-1-neg 96.3%

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

   13. Simplified 96.3%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+16} \lor \neg \left(-1 + x \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-\log y\right) - z \cdot y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

Python

def code(x, y, z, t):
	return ((math.log(y) * (-1.0 + x)) + (y * (z * (-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(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 = ((log(y) * (-1.0 + x)) + (y * (z * (-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[(y * N[(z * 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 88.7%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around inf 99.3%

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

5. Final simplification 99.3%

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

Alternative 7: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -235000000000 \lor \neg \left(x \leq 3.8 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -235000000000.0) (not (<= x 3.8e+16)))
   (- (* x (log y)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -235000000000.0) || !(x <= 3.8e+16)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - 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 <= (-235000000000.0d0)) .or. (.not. (x <= 3.8d+16))) then
        tmp = (x * log(y)) - t
    else
        tmp = ((y * (1.0d0 - z)) - 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 <= -235000000000.0) || !(x <= 3.8e+16)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -235000000000.0) or not (x <= 3.8e+16):
		tmp = (x * math.log(y)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -235000000000.0) || !(x <= 3.8e+16))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -235000000000.0) || ~((x <= 3.8e+16)))
		tmp = (x * log(y)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -235000000000.0], N[Not[LessEqual[x, 3.8e+16]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.35e11 or 3.8e16 < x

   1. Initial program 96.5%

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

   3. Taylor expanded in y around 0 99.6%

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

   4. Taylor expanded in z around inf 99.6%

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

   5. Taylor expanded in x around inf 96.5%

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

      1. *-commutative 96.5%

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

   7. Simplified 96.5%

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

   if -2.35e11 < x < 3.8e16

   1. Initial program 82.2%

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

      1. add-cube-cbrt 81.4%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y} \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      2. pow3 81.4%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      3. sub-neg 81.4%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      4. metadata-eval 81.4%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + \color{blue}{-1}\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 81.4%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 97.3%

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

      1. mul-1-neg 97.3%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 97.3%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in x around 0 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. mul-1-neg 96.6%

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

      4. sub-neg 96.6%

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

      5. metadata-eval 96.6%

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

      6. unsub-neg 96.6%

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

      7. log-rec 96.6%

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

      8. +-commutative 96.6%

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

   10. Simplified 96.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -235000000000 \lor \neg \left(x \leq 3.8 \cdot 10^{+16}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.8% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (<= (+ -1.0 x) -1.0)))
   (if (or t_1 (not t_1)) (- (* x (log y)) t) (- (- (log y)) t))))

C

double code(double x, double y, double z, double t) {
	int t_1 = (-1.0 + x) <= -1.0;
	double tmp;
	if (t_1 || !t_1) {
		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
    logical :: t_1
    real(8) :: tmp
    t_1 = ((-1.0d0) + x) <= (-1.0d0)
    if (t_1 .or. (.not. t_1)) 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) {
	boolean t_1 = (-1.0 + x) <= -1.0;
	double tmp;
	if (t_1 || !t_1) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (-1.0 + x) <= -1.0
	tmp = 0
	if t_1 or not t_1:
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(-1.0 + x) <= -1.0
	tmp = 0.0
	if (t_1 || !t_1)
		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)
	t_1 = (-1.0 + x) <= -1.0;
	tmp = 0.0;
	if (t_1 || ~(t_1))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0]}, If[Or[t$95$1, N[Not[t$95$1], $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 (-.f64 x #s(literal 1 binary64)) < -1 or -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in z around inf 99.3%

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

   5. Taylor expanded in x around inf 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

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

   1. Initial program 88.7%

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

      1. +-commutative 88.7%

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

      2. fma-define 88.7%

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

      3. sub-neg 88.7%

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

      4. metadata-eval 88.7%

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

      5. sub-neg 88.7%

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

      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 87.1%

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

   6. Taylor expanded in x around 0 54.1%

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

      1. mul-1-neg 54.1%

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

   8. Simplified 54.1%

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

4. Final simplification 67.1%

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

Alternative 9: 99.5% 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 88.7%

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

3. Taylor expanded in y around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 10: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.6e+26)
   (- (* z (- y)) t)
   (if (<= t 3.4e+23)
     (* (log y) (+ -1.0 x))
     (- (* y (* z (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+26) {
		tmp = (z * -y) - t;
	} else if (t <= 3.4e+23) {
		tmp = log(y) * (-1.0 + x);
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.6d+26)) then
        tmp = (z * -y) - t
    else if (t <= 3.4d+23) then
        tmp = log(y) * ((-1.0d0) + x)
    else
        tmp = (y * (z * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e+26) {
		tmp = (z * -y) - t;
	} else if (t <= 3.4e+23) {
		tmp = Math.log(y) * (-1.0 + x);
	} else {
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.6e+26:
		tmp = (z * -y) - t
	elif t <= 3.4e+23:
		tmp = math.log(y) * (-1.0 + x)
	else:
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.6e+26)
		tmp = Float64(Float64(z * Float64(-y)) - t);
	elseif (t <= 3.4e+23)
		tmp = Float64(log(y) * Float64(-1.0 + x));
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.6e+26)
		tmp = (z * -y) - t;
	elseif (t <= 3.4e+23)
		tmp = log(y) * (-1.0 + x);
	else
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.6e+26], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[t, 3.4e+23], N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision], 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. Split input into 3 regimes

2. if t < -2.60000000000000002e26

   1. Initial program 94.2%

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

      1. add-cube-cbrt 94.0%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y} \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      2. pow3 94.0%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      3. sub-neg 94.0%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      4. metadata-eval 94.0%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + \color{blue}{-1}\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 94.0%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 99.8%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in z around inf 99.8%

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

      1. associate-*r* 99.8%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) - t \]

      2. mul-1-neg 99.8%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right)} \cdot z\right) - t \]

   10. Simplified 99.8%

       \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right) \cdot z}\right) - t \]

   11. Taylor expanded in y around inf 92.0%

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

       1. associate-*r* 92.0%

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

       2. mul-1-neg 92.0%

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

   13. Simplified 92.0%

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

   if -2.60000000000000002e26 < t < 3.39999999999999992e23

   1. Initial program 84.2%

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

      1. +-commutative 84.2%

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

      2. fma-define 84.2%

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

      3. sub-neg 84.2%

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

      4. metadata-eval 84.2%

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

      5. sub-neg 84.2%

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

      6. log1p-define 99.7%

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

      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 82.8%

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

   6. Taylor expanded in t around 0 81.0%

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

   if 3.39999999999999992e23 < t

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0 98.1%

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

   4. Taylor expanded in z around inf 98.1%

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

   5. Taylor expanded in z around inf 76.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+80} \lor \neg \left(z \leq 9 \cdot 10^{+86}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.6e+80) (not (<= z 9e+86)))
   (- (* y (* z (+ -1.0 (* y (- (* y -0.3333333333333333) 0.5))))) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.6e+80) || !(z <= 9e+86)) {
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - 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 ((z <= (-6.6d+80)) .or. (.not. (z <= 9d+86))) then
        tmp = (y * (z * ((-1.0d0) + (y * ((y * (-0.3333333333333333d0)) - 0.5d0))))) - 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 ((z <= -6.6e+80) || !(z <= 9e+86)) {
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.6e+80) or not (z <= 9e+86):
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.6e+80) || !(z <= 9e+86))
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5))))) - 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 ((z <= -6.6e+80) || ~((z <= 9e+86)))
		tmp = (y * (z * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5))))) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.6e+80], N[Not[LessEqual[z, 9e+86]], $MachinePrecision]], 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], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.59999999999999982e80 or 8.99999999999999986e86 < z

   1. Initial program 72.2%

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

   3. Taylor expanded in y around 0 98.9%

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

   4. Taylor expanded in z around inf 98.9%

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

   5. Taylor expanded in z around inf 60.1%

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

   if -6.59999999999999982e80 < z < 8.99999999999999986e86

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-define 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. sub-neg 99.2%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.0%

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

   6. Taylor expanded in x around 0 64.9%

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

      1. mul-1-neg 64.9%

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

   8. Simplified 64.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+80} \lor \neg \left(z \leq 9 \cdot 10^{+86}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.2% 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 88.7%

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

3. Taylor expanded in y around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. unsub-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. +-commutative 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 13: 88.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{+285}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 1.05e+285) (- (* (log y) (+ -1.0 x)) t) (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e+285) {
		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 <= 1.05d+285) 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 <= 1.05e+285) {
		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 <= 1.05e+285:
		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 <= 1.05e+285)
		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 <= 1.05e+285)
		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, 1.05e+285], 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 2 regimes

2. if z < 1.05e285

   1. Initial program 91.1%

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

      1. +-commutative 91.1%

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

      2. fma-define 91.1%

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

      3. sub-neg 91.1%

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

      4. metadata-eval 91.1%

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

      5. sub-neg 91.1%

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

      6. log1p-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 89.5%

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

   if 1.05e285 < z

   1. Initial program 2.5%

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

      1. add-cube-cbrt 2.5%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y} \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      2. pow3 2.5%

         \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      3. sub-neg 2.5%

         \[\leadsto \left({\left(\sqrt[3]{\color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

      4. metadata-eval 2.5%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + \color{blue}{-1}\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 2.5%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 99.8%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in z around inf 99.8%

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

      1. associate-*r* 99.8%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) - t \]

      2. mul-1-neg 99.8%

         \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right)} \cdot z\right) - t \]

   10. Simplified 99.8%

       \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right) \cdot z}\right) - t \]

   11. Taylor expanded in y around inf 96.4%

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

       1. associate-*r* 96.4%

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

       2. mul-1-neg 96.4%

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

   13. Simplified 96.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{+285}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around inf 99.3%

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

5. Taylor expanded in y around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. sub-neg 98.7%

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

   3. metadata-eval 98.7%

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

   4. +-commutative 98.7%

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

   5. *-commutative 98.7%

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

   6. mul-1-neg 98.7%

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

   7. unsub-neg 98.7%

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

   8. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 15: 46.5% 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 88.7%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around inf 99.3%

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

5. Taylor expanded in z around inf 46.1%

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

6. Final simplification 46.1%

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

Alternative 16: 46.1% 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 88.7%

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

   1. add-cube-cbrt 87.7%

      \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y} \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}\right) \cdot \sqrt[3]{\left(x - 1\right) \cdot \log y}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   2. pow3 87.7%

      \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x - 1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   3. sub-neg 87.7%

      \[\leadsto \left({\left(\sqrt[3]{\color{blue}{\left(x + \left(-1\right)\right)} \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. metadata-eval 87.7%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + \color{blue}{-1}\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

4. Applied egg-rr 87.7%

   \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3}} + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

5. Taylor expanded in y around 0 97.9%

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

   1. mul-1-neg 97.9%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

7. Simplified 97.9%

   \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

8. Taylor expanded in z around inf 97.7%

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

   1. associate-*r* 97.7%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) - t \]

   2. mul-1-neg 97.7%

      \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right)} \cdot z\right) - t \]

10. Simplified 97.7%

    \[\leadsto \left({\left(\sqrt[3]{\left(x + -1\right) \cdot \log y}\right)}^{3} + \color{blue}{\left(-y\right) \cdot z}\right) - t \]

11. Taylor expanded in y around inf 45.4%

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

    1. associate-*r* 45.4%

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

    2. mul-1-neg 45.4%

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

13. Simplified 45.4%

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

14. Final simplification 45.4%

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

Alternative 17: 35.6% 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 88.7%

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

   1. +-commutative 88.7%

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

   2. fma-define 88.7%

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

   3. sub-neg 88.7%

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

   4. metadata-eval 88.7%

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

   5. sub-neg 88.7%

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

   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 t around inf 34.2%

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

   1. neg-mul-1 34.2%

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

7. Simplified 34.2%

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

Alternative 18: 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 88.7%

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

   1. +-commutative 88.7%

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

   2. fma-define 88.7%

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

   3. sub-neg 88.7%

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

   4. metadata-eval 88.7%

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

   5. sub-neg 88.7%

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

   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 t around inf 34.2%

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

   1. neg-mul-1 34.2%

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

7. Simplified 34.2%

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

   1. expm1-log1p-u 17.1%

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

   2. expm1-undefine 16.9%

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

9. Applied egg-rr 16.9%

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

    1. sub-neg 16.9%

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

    2. log1p-undefine 16.9%

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

    3. rem-exp-log 34.0%

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

    4. unsub-neg 34.0%

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

    5. metadata-eval 34.0%

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

11. Simplified 34.0%

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

12. Taylor expanded in t around 0 2.4%

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

    1. metadata-eval 2.4%

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

14. Applied egg-rr 2.4%

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

Reproduce

herbie shell --seed 2024116 
(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))