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

Percentage Accurate: 89.1% → 99.8%

Time: 17.2s

Alternatives: 19

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: 89.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. fma-define 90.9%

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

   2. sub-neg 90.9%

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

   3. metadata-eval 90.9%

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

   4. sub-neg 90.9%

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

   5. metadata-eval 90.9%

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

   6. sub-neg 90.9%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

Python

def code(x, y, z, t):
	return ((math.log(y) * (x + -1.0)) + ((-1.0 + z) * (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(x + -1.0)) + Float64(Float64(-1.0 + z) * 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) * (x + -1.0)) + ((-1.0 + z) * (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[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + z), $MachinePrecision] * N[(y * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 95.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -15000000000000 \lor \neg \left(x + -1 \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ x -1.0) -15000000000000.0) (not (<= (+ x -1.0) 2e+14)))
   (- (* (log y) (+ x -1.0)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -15000000000000.0) || !((x + -1.0) <= 2e+14)) {
		tmp = (log(y) * (x + -1.0)) - 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 + (-1.0d0)) <= (-15000000000000.0d0)) .or. (.not. ((x + (-1.0d0)) <= 2d+14))) then
        tmp = (log(y) * (x + (-1.0d0))) - 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 + -1.0) <= -15000000000000.0) || !((x + -1.0) <= 2e+14)) {
		tmp = (Math.log(y) * (x + -1.0)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + -1.0) <= -15000000000000.0) or not ((x + -1.0) <= 2e+14):
		tmp = (math.log(y) * (x + -1.0)) - t
	else:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x + -1.0) <= -15000000000000.0) || !(Float64(x + -1.0) <= 2e+14))
		tmp = Float64(Float64(log(y) * Float64(x + -1.0)) - 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 + -1.0) <= -15000000000000.0) || ~(((x + -1.0) <= 2e+14)))
		tmp = (log(y) * (x + -1.0)) - t;
	else
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x + -1.0), $MachinePrecision], -15000000000000.0], N[Not[LessEqual[N[(x + -1.0), $MachinePrecision], 2e+14]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $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 (-.f64 x #s(literal 1 binary64)) < -1.5e13 or 2e14 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 97.5%

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

      1. fma-define 97.5%

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

      2. sub-neg 97.5%

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

      3. metadata-eval 97.5%

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

      4. sub-neg 97.5%

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

      5. metadata-eval 97.5%

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

      6. sub-neg 97.5%

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

      7. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 96.9%

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

   if -1.5e13 < (-.f64 x #s(literal 1 binary64)) < 2e14

   1. Initial program 83.8%

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

      1. sub-neg 83.8%

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

      2. metadata-eval 83.8%

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

      3. add-sqr-sqrt 0.9%

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

      4. pow2 0.9%

         \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 0.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 2.4%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot y\right)}\right) - t \]
   6. Step-by-step derivation

      1. mul-1-neg 2.4%

         \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 2.4%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in x around 0 96.8%

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

      1. neg-mul-1 96.8%

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

      2. +-commutative 96.8%

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

      3. unsub-neg 96.8%

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

      4. mul-1-neg 96.8%

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

      5. sub-neg 96.8%

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

      6. metadata-eval 96.8%

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

      7. distribute-lft-neg-in 96.8%

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

      8. +-commutative 96.8%

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

   10. Simplified 96.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -15000000000000 \lor \neg \left(x + -1 \leq 2 \cdot 10^{+14}\right):\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((log(y) * (x + -1.0)) + ((-1.0 + z) * (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) * (x + (-1.0d0))) + (((-1.0d0) + z) * (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) * (x + -1.0)) + ((-1.0 + z) * (y * (-1.0 + (y * ((y * -0.3333333333333333) - 0.5)))))) - t;
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log(y) * Float64(x + -1.0)) + Float64(Float64(-1.0 + z) * 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) * (x + -1.0)) + ((-1.0 + z) * (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[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + z), $MachinePrecision] * N[(y * N[(-1.0 + N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 6: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in y around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 7: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -12000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -12000000000000.0)
     t_1
     (if (<= x -1.05e-30)
       (- (* z (log1p (- y))) t)
       (if (<= x 1.0) (- (+ (log y) t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -12000000000000.0) {
		tmp = t_1;
	} else if (x <= -1.05e-30) {
		tmp = (z * log1p(-y)) - t;
	} else if (x <= 1.0) {
		tmp = -(log(y) + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -12000000000000.0) {
		tmp = t_1;
	} else if (x <= -1.05e-30) {
		tmp = (z * Math.log1p(-y)) - t;
	} else if (x <= 1.0) {
		tmp = -(Math.log(y) + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -12000000000000.0:
		tmp = t_1
	elif x <= -1.05e-30:
		tmp = (z * math.log1p(-y)) - t
	elif x <= 1.0:
		tmp = -(math.log(y) + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -12000000000000.0)
		tmp = t_1;
	elseif (x <= -1.05e-30)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	elseif (x <= 1.0)
		tmp = Float64(-Float64(log(y) + t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -12000000000000.0], t$95$1, If[LessEqual[x, -1.05e-30], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.0], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e13 or 1 < x

   1. Initial program 96.2%

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

      1. fma-define 96.2%

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

      2. sub-neg 96.2%

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

      3. metadata-eval 96.2%

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

      4. sub-neg 96.2%

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

      5. metadata-eval 96.2%

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

      6. sub-neg 96.2%

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

      7. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in x around inf 95.4%

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

      1. *-commutative 95.4%

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

   8. Simplified 95.4%

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

   if -1.2e13 < x < -1.0500000000000001e-30

   1. Initial program 55.0%

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

      1. flip-- 55.0%

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

      2. metadata-eval 55.0%

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

      3. metadata-eval 55.0%

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

      4. associate-*l/ 55.0%

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

      5. metadata-eval 55.0%

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

      6. fmm-def 55.0%

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

      7. metadata-eval 55.0%

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

      8. +-commutative 55.0%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in z around inf 37.7%

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

      1. sub-neg 37.7%

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

      2. log1p-undefine 81.2%

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

   7. Simplified 81.2%

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

   if -1.0500000000000001e-30 < x < 1

   1. Initial program 87.7%

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

      1. fma-define 87.7%

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

      2. sub-neg 87.7%

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

      3. metadata-eval 87.7%

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

      4. sub-neg 87.7%

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

      5. metadata-eval 87.7%

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

      6. sub-neg 87.7%

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

      7. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 86.2%

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

   6. Taylor expanded in x around 0 85.2%

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

      1. mul-1-neg 85.2%

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

   8. Simplified 85.2%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12000000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in y around 0 99.2%

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

4. Final simplification 99.2%

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

Alternative 9: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

3. Taylor expanded in y around 0 99.2%

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

4. Taylor expanded in z around -inf 99.0%

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

   1. mul-1-neg 99.0%

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

   2. associate-*r* 99.0%

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

   3. distribute-rgt-neg-in 99.0%

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

   4. *-commutative 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 10: 86.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+19} \lor \neg \left(t \leq 4.4 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.85e+19) (not (<= t 4.4e-19)))
   (- (* x (log y)) t)
   (* (log y) (+ x -1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.85e+19) || !(t <= 4.4e-19)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = log(y) * (x + -1.0);
	}
	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 <= (-1.85d+19)) .or. (.not. (t <= 4.4d-19))) then
        tmp = (x * log(y)) - t
    else
        tmp = log(y) * (x + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.85e+19) || !(t <= 4.4e-19)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = Math.log(y) * (x + -1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.85e+19) or not (t <= 4.4e-19):
		tmp = (x * math.log(y)) - t
	else:
		tmp = math.log(y) * (x + -1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.85e+19) || !(t <= 4.4e-19))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(log(y) * Float64(x + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.85e+19) || ~((t <= 4.4e-19)))
		tmp = (x * log(y)) - t;
	else
		tmp = log(y) * (x + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.85e+19], N[Not[LessEqual[t, 4.4e-19]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.85e19 or 4.3999999999999997e-19 < t

   1. Initial program 94.8%

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

      1. fma-define 94.8%

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

      2. sub-neg 94.8%

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

      3. metadata-eval 94.8%

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

      4. sub-neg 94.8%

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

      5. metadata-eval 94.8%

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

      6. sub-neg 94.8%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.8%

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

   6. Taylor expanded in x around inf 93.8%

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

      1. *-commutative 93.8%

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

   8. Simplified 93.8%

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

   if -1.85e19 < t < 4.3999999999999997e-19

   1. Initial program 87.0%

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

      1. fma-define 87.0%

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

      2. sub-neg 87.0%

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

      3. metadata-eval 87.0%

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

      4. sub-neg 87.0%

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

      5. metadata-eval 87.0%

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

      6. sub-neg 87.0%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 84.9%

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

   6. Taylor expanded in t around 0 83.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+19} \lor \neg \left(t \leq 4.4 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+66}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.2e-7)
   (- (+ (log y) t))
   (if (<= t 6.2e+66) (* (log y) (+ x -1.0)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.2e-7) {
		tmp = -(log(y) + t);
	} else if (t <= 6.2e+66) {
		tmp = log(y) * (x + -1.0);
	} else {
		tmp = -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 <= (-6.2d-7)) then
        tmp = -(log(y) + t)
    else if (t <= 6.2d+66) then
        tmp = log(y) * (x + (-1.0d0))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.2e-7) {
		tmp = -(Math.log(y) + t);
	} else if (t <= 6.2e+66) {
		tmp = Math.log(y) * (x + -1.0);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.2e-7:
		tmp = -(math.log(y) + t)
	elif t <= 6.2e+66:
		tmp = math.log(y) * (x + -1.0)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.2e-7)
		tmp = Float64(-Float64(log(y) + t));
	elseif (t <= 6.2e+66)
		tmp = Float64(log(y) * Float64(x + -1.0));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.2e-7)
		tmp = -(log(y) + t);
	elseif (t <= 6.2e+66)
		tmp = log(y) * (x + -1.0);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.2e-7], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), If[LessEqual[t, 6.2e+66], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], (-t)]]

Derivation

1. Split input into 3 regimes

2. if t < -6.1999999999999999e-7

   1. Initial program 93.5%

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

      1. fma-define 93.5%

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

      2. sub-neg 93.5%

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

      3. metadata-eval 93.5%

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

      4. sub-neg 93.5%

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

      5. metadata-eval 93.5%

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

      6. sub-neg 93.5%

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

      7. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 92.1%

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

   6. Taylor expanded in x around 0 78.8%

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

      1. mul-1-neg 78.8%

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

   8. Simplified 78.8%

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

   if -6.1999999999999999e-7 < t < 6.20000000000000037e66

   1. Initial program 87.3%

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

      1. fma-define 87.3%

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

      2. sub-neg 87.3%

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

      3. metadata-eval 87.3%

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

      4. sub-neg 87.3%

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

      5. metadata-eval 87.3%

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

      6. sub-neg 87.3%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 85.4%

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

   6. Taylor expanded in t around 0 84.7%

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

   if 6.20000000000000037e66 < t

   1. Initial program 97.2%

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

      1. fma-define 97.2%

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

      2. sub-neg 97.2%

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

      3. metadata-eval 97.2%

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

      4. sub-neg 97.2%

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

      5. metadata-eval 97.2%

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

      6. sub-neg 97.2%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 79.7%

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

      1. mul-1-neg 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-7}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+66}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+264}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(x + -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.3e+264)
   (- (* z (log1p (- y))) t)
   (- (+ y (* (log y) (+ x -1.0))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+264) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (y + (log(y) * (x + -1.0))) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+264) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (y + (Math.log(y) * (x + -1.0))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.3e+264:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (y + (math.log(y) * (x + -1.0))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e+264)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(y + Float64(log(y) * Float64(x + -1.0))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.3e+264], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e264

   1. Initial program 31.1%

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

      1. flip-- 19.7%

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

      2. metadata-eval 19.7%

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

      3. metadata-eval 19.7%

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

      4. associate-*l/ 19.7%

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

      5. metadata-eval 19.7%

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

      6. fmm-def 19.7%

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

      7. metadata-eval 19.7%

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

      8. +-commutative 19.7%

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

   4. Applied egg-rr 19.7%

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

   5. Taylor expanded in z around inf 18.2%

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

      1. sub-neg 18.2%

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

      2. log1p-undefine 87.3%

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

   7. Simplified 87.3%

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

   if -1.3e264 < z

   1. Initial program 92.9%

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

      1. sub-neg 92.9%

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

      2. metadata-eval 92.9%

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

      3. add-sqr-sqrt 25.5%

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

      4. pow2 25.5%

         \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 25.5%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 25.9%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot y\right)}\right) - t \]
   6. Step-by-step derivation

      1. mul-1-neg 25.9%

         \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 25.9%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in z around 0 91.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+264}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(x + -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. fma-define 90.9%

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

   2. sub-neg 90.9%

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

   3. metadata-eval 90.9%

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

   4. sub-neg 90.9%

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

   5. metadata-eval 90.9%

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

   6. sub-neg 90.9%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around 0 98.8%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
6. 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. mul-1-neg 98.8%

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

   5. unsub-neg 98.8%

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

   6. +-commutative 98.8%

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

   7. sub-neg 98.8%

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

   8. metadata-eval 98.8%

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

   9. +-commutative 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 14: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+259}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.45e+259) (- (* z (log1p (- y))) t) (- (* (log y) (+ x -1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.45e+259) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (log(y) * (x + -1.0)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.45e+259) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (Math.log(y) * (x + -1.0)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.45e+259:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (math.log(y) * (x + -1.0)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.45e+259)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(x + -1.0)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.45e+259], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e259

   1. Initial program 31.1%

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

      1. flip-- 19.7%

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

      2. metadata-eval 19.7%

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

      3. metadata-eval 19.7%

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

      4. associate-*l/ 19.7%

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

      5. metadata-eval 19.7%

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

      6. fmm-def 19.7%

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

      7. metadata-eval 19.7%

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

      8. +-commutative 19.7%

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

   4. Applied egg-rr 19.7%

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

   5. Taylor expanded in z around inf 18.2%

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

      1. sub-neg 18.2%

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

      2. log1p-undefine 87.3%

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

   7. Simplified 87.3%

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

   if -1.45e259 < z

   1. Initial program 92.9%

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

      1. fma-define 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. sub-neg 92.9%

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

      5. metadata-eval 92.9%

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

      6. sub-neg 92.9%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 91.4%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+259}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

   1. fma-define 90.9%

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

   2. sub-neg 90.9%

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

   3. metadata-eval 90.9%

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

   4. sub-neg 90.9%

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

   5. metadata-eval 90.9%

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

   6. sub-neg 90.9%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around inf 99.5%

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

6. Taylor expanded in y around 0 98.7%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + \log y \cdot \left(x - 1\right)\right)} - t \]
7. 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%

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

   5. mul-1-neg 98.7%

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

   6. unsub-neg 98.7%

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

   7. *-commutative 98.7%

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

   8. +-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 16: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+260}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.8e+260) (* y (- 1.0 z)) (- (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+260) {
		tmp = y * (1.0 - z);
	} 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 <= (-3.8d+260)) then
        tmp = y * (1.0d0 - z)
    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 <= -3.8e+260) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -(Math.log(y) + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.8e+260:
		tmp = y * (1.0 - z)
	else:
		tmp = -(math.log(y) + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+260)
		tmp = Float64(y * Float64(1.0 - z));
	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 <= -3.8e+260)
		tmp = y * (1.0 - z);
	else
		tmp = -(log(y) + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.8e+260], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999997e260

   1. Initial program 31.1%

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

      1. sub-neg 31.1%

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

      2. metadata-eval 31.1%

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

      3. add-sqr-sqrt 15.9%

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

      4. pow2 15.9%

         \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 15.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 36.6%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot y\right)}\right) - t \]
   6. Step-by-step derivation

      1. mul-1-neg 36.6%

         \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 36.6%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in y around inf 75.1%

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

      1. mul-1-neg 75.1%

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

      2. sub-neg 75.1%

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

      3. metadata-eval 75.1%

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

      4. distribute-lft-neg-in 75.1%

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

      5. +-commutative 75.1%

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

   10. Simplified 75.1%

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

   if -3.7999999999999997e260 < z

   1. Initial program 92.9%

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

      1. fma-define 92.9%

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

      2. sub-neg 92.9%

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

      3. metadata-eval 92.9%

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

      4. sub-neg 92.9%

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

      5. metadata-eval 92.9%

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

      6. sub-neg 92.9%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 91.4%

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

   6. Taylor expanded in x around 0 53.6%

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

      1. mul-1-neg 53.6%

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

   8. Simplified 53.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+260}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-\left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -9.2e+14)
   (- t)
   (if (<= t 7e+56) (* y (- 1.0 z)) (+ -1.0 (- 1.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.2e+14) {
		tmp = -t;
	} else if (t <= 7e+56) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -1.0 + (1.0 - 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 <= (-9.2d+14)) then
        tmp = -t
    else if (t <= 7d+56) then
        tmp = y * (1.0d0 - z)
    else
        tmp = (-1.0d0) + (1.0d0 - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.2e+14) {
		tmp = -t;
	} else if (t <= 7e+56) {
		tmp = y * (1.0 - z);
	} else {
		tmp = -1.0 + (1.0 - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -9.2e+14:
		tmp = -t
	elif t <= 7e+56:
		tmp = y * (1.0 - z)
	else:
		tmp = -1.0 + (1.0 - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9.2e+14)
		tmp = Float64(-t);
	elseif (t <= 7e+56)
		tmp = Float64(y * Float64(1.0 - z));
	else
		tmp = Float64(-1.0 + Float64(1.0 - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9.2e+14)
		tmp = -t;
	elseif (t <= 7e+56)
		tmp = y * (1.0 - z);
	else
		tmp = -1.0 + (1.0 - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -9.2e+14], (-t), If[LessEqual[t, 7e+56], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(1.0 - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.2e14

   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. fma-define 94.7%

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

      2. sub-neg 94.7%

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

      3. metadata-eval 94.7%

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

      4. sub-neg 94.7%

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

      5. metadata-eval 94.7%

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

      6. sub-neg 94.7%

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

      7. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 79.1%

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

      1. mul-1-neg 79.1%

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

   7. Simplified 79.1%

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

   if -9.2e14 < t < 6.99999999999999999e56

   1. Initial program 87.4%

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

      1. sub-neg 87.4%

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

      2. metadata-eval 87.4%

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

      3. add-sqr-sqrt 27.9%

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

      4. pow2 27.9%

         \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   4. Applied egg-rr 27.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{x + -1}\right)}^{2}} \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]

   5. Taylor expanded in y around 0 29.3%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-1 \cdot y\right)}\right) - t \]
   6. Step-by-step derivation

      1. mul-1-neg 29.3%

         \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   7. Simplified 29.3%

      \[\leadsto \left({\left(\sqrt{x + -1}\right)}^{2} \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(-y\right)}\right) - t \]

   8. Taylor expanded in y around inf 15.1%

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

      1. mul-1-neg 15.1%

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

      2. sub-neg 15.1%

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

      3. metadata-eval 15.1%

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

      4. distribute-lft-neg-in 15.1%

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

      5. +-commutative 15.1%

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

   10. Simplified 15.1%

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

   if 6.99999999999999999e56 < t

   1. Initial program 95.5%

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

      1. fma-define 95.5%

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

      2. sub-neg 95.5%

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

      3. metadata-eval 95.5%

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

      4. sub-neg 95.5%

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

      5. metadata-eval 95.5%

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

      6. sub-neg 95.5%

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

      7. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in t around inf 76.2%

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

      1. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

      1. expm1-log1p-u 0.0%

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

      2. expm1-undefine 0.0%

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

   9. Applied egg-rr 0.0%

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

       1. sub-neg 0.0%

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

       2. log1p-undefine 0.0%

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

       3. rem-exp-log 76.2%

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

       4. unsub-neg 76.2%

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

       5. metadata-eval 76.2%

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

   11. Simplified 76.2%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+56}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.3% 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 90.9%

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

   1. fma-define 90.9%

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

   2. sub-neg 90.9%

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

   3. metadata-eval 90.9%

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

   4. sub-neg 90.9%

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

   5. metadata-eval 90.9%

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

   6. sub-neg 90.9%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 37.2%

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

   1. mul-1-neg 37.2%

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

7. Simplified 37.2%

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

Alternative 19: 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 90.9%

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

   1. fma-define 90.9%

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

   2. sub-neg 90.9%

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

   3. metadata-eval 90.9%

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

   4. sub-neg 90.9%

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

   5. metadata-eval 90.9%

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

   6. sub-neg 90.9%

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

   7. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 37.2%

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

   1. mul-1-neg 37.2%

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

7. Simplified 37.2%

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

   1. expm1-log1p-u 17.7%

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

   2. expm1-undefine 17.6%

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

9. Applied egg-rr 17.6%

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

    1. sub-neg 17.6%

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

    2. log1p-undefine 17.6%

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

    3. rem-exp-log 37.0%

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

    4. unsub-neg 37.0%

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

    5. metadata-eval 37.0%

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

11. Simplified 37.0%

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

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 2024180 
(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))