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

Percentage Accurate: 89.0% → 99.8%

Time: 14.6s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in x around 0 87.8%

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

   1. +-commutative 87.8%

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

   2. +-commutative 87.8%

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

   3. associate-+r+ 87.8%

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

   4. distribute-rgt-in 87.8%

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

   5. *-commutative 87.8%

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

   6. fma-define 87.8%

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

   7. sub-neg 87.8%

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

   8. log1p-define 99.8%

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

   9. sub-neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. +-commutative 99.8%

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

5. Simplified 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in x around 0 87.8%

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

   1. +-commutative 87.8%

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

   2. +-commutative 87.8%

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

   3. associate-+r+ 87.8%

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

   4. distribute-rgt-in 87.8%

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

   5. *-commutative 87.8%

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

   6. fma-define 87.8%

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

   7. sub-neg 87.8%

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

   8. log1p-define 99.8%

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

   9. sub-neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0 99.5%

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

   1. fma-define 99.6%

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

   2. associate-*r* 99.6%

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

   3. distribute-rgt-out 99.6%

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

   4. sub-neg 99.6%

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

   5. metadata-eval 99.6%

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

   6. +-commutative 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(y, \left(-1 + z\right) \cdot \left(-1 + y \cdot -0.5\right), \log y \cdot \left(-1 + x\right)\right) - t \]
10. Add Preprocessing

Alternative 3: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+28} \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\left(\log y \cdot x - y \cdot \left(-1 + z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -4e+28) (not (<= (+ -1.0 x) -0.5)))
   (- (- (* (log y) x) (* y (+ -1.0 z))) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -4e+28) || !((-1.0 + x) <= -0.5)) {
		tmp = ((log(y) * x) - (y * (-1.0 + z))) - 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 ((((-1.0d0) + x) <= (-4d+28)) .or. (.not. (((-1.0d0) + x) <= (-0.5d0)))) then
        tmp = ((log(y) * x) - (y * ((-1.0d0) + z))) - 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 (((-1.0 + x) <= -4e+28) || !((-1.0 + x) <= -0.5)) {
		tmp = ((Math.log(y) * x) - (y * (-1.0 + z))) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -4e+28) or not ((-1.0 + x) <= -0.5):
		tmp = ((math.log(y) * x) - (y * (-1.0 + z))) - 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(-1.0 + x) <= -4e+28) || !(Float64(-1.0 + x) <= -0.5))
		tmp = Float64(Float64(Float64(log(y) * x) - Float64(y * Float64(-1.0 + z))) - 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 (((-1.0 + x) <= -4e+28) || ~(((-1.0 + x) <= -0.5)))
		tmp = ((log(y) * x) - (y * (-1.0 + z))) - 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[(-1.0 + x), $MachinePrecision], -4e+28], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - N[(y * N[(-1.0 + z), $MachinePrecision]), $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)) < -3.99999999999999983e28 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 92.3%

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

   3. Taylor expanded in x around 0 92.3%

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

      1. +-commutative 92.3%

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

      2. +-commutative 92.3%

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

      3. associate-+r+ 92.3%

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

      4. distribute-rgt-in 92.3%

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

      5. *-commutative 92.3%

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

      6. fma-define 92.3%

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

      7. sub-neg 92.3%

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

      8. log1p-define 99.7%

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

      9. sub-neg 99.7%

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

      10. metadata-eval 99.7%

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

      11. +-commutative 99.7%

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

      12. *-commutative 99.7%

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

      13. +-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. sub-neg 99.0%

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

      3. metadata-eval 99.0%

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

      4. +-commutative 99.0%

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

      5. mul-1-neg 99.0%

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

      6. unsub-neg 99.0%

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

      7. +-commutative 99.0%

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

      8. sub-neg 99.0%

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

      9. metadata-eval 99.0%

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

      10. +-commutative 99.0%

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

   8. Simplified 99.0%

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

   9. Taylor expanded in x around inf 99.0%

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

       1. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   if -3.99999999999999983e28 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. +-commutative 84.0%

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

      3. associate-+r+ 84.0%

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

      4. distribute-rgt-in 84.0%

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

      5. *-commutative 84.0%

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

      6. fma-define 84.0%

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

      7. sub-neg 84.0%

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

      8. log1p-define 99.9%

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

      9. sub-neg 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

      4. +-commutative 99.3%

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

      5. mul-1-neg 99.3%

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

      6. unsub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

      10. +-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 97.9%

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

       1. sub-neg 97.9%

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

       2. neg-mul-1 97.9%

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

       3. mul-1-neg 97.9%

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

       4. +-commutative 97.9%

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

       5. unsub-neg 97.9%

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

       6. mul-1-neg 97.9%

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

       7. sub-neg 97.9%

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

       8. metadata-eval 97.9%

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

       9. +-commutative 97.9%

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

       10. distribute-rgt-neg-in 97.9%

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

       11. distribute-neg-in 97.9%

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

       12. metadata-eval 97.9%

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

       13. sub-neg 97.9%

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

   11. Simplified 97.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+28} \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\left(\log y \cdot x - y \cdot \left(-1 + z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000001 \lor \neg \left(-1 + x \leq 5 \cdot 10^{+29}\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -1.00000001) (not (<= (+ -1.0 x) 5e+29)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- (* y (- 1.0 z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.00000001) || !((-1.0 + x) <= 5e+29)) {
		tmp = (log(y) * (-1.0 + x)) - 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 ((((-1.0d0) + x) <= (-1.00000001d0)) .or. (.not. (((-1.0d0) + x) <= 5d+29))) then
        tmp = (log(y) * ((-1.0d0) + x)) - 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 (((-1.0 + x) <= -1.00000001) || !((-1.0 + x) <= 5e+29)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -1.00000001) or not ((-1.0 + x) <= 5e+29):
		tmp = (math.log(y) * (-1.0 + x)) - 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(-1.0 + x) <= -1.00000001) || !(Float64(-1.0 + x) <= 5e+29))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - 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 (((-1.0 + x) <= -1.00000001) || ~(((-1.0 + x) <= 5e+29)))
		tmp = (log(y) * (-1.0 + x)) - 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[(-1.0 + x), $MachinePrecision], -1.00000001], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], 5e+29]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $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)) < -1.0000000099999999 or 5.0000000000000001e29 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0 92.4%

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

   if -1.0000000099999999 < (-.f64 x #s(literal 1 binary64)) < 5.0000000000000001e29

   1. Initial program 82.8%

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

   3. Taylor expanded in x around 0 82.8%

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

      1. +-commutative 82.8%

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

      2. +-commutative 82.8%

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

      3. associate-+r+ 82.8%

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

      4. distribute-rgt-in 82.8%

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

      5. *-commutative 82.8%

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

      6. fma-define 82.8%

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

      7. sub-neg 82.8%

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

      8. log1p-define 99.9%

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

      9. sub-neg 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

      12. *-commutative 99.9%

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

      13. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

      4. +-commutative 99.6%

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

      5. mul-1-neg 99.6%

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

      6. unsub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

      10. +-commutative 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in x around 0 98.1%

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

       1. sub-neg 98.1%

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

       2. neg-mul-1 98.1%

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

       3. mul-1-neg 98.1%

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

       4. +-commutative 98.1%

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

       5. unsub-neg 98.1%

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

       6. mul-1-neg 98.1%

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

       7. sub-neg 98.1%

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

       8. metadata-eval 98.1%

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

       9. +-commutative 98.1%

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

       10. distribute-rgt-neg-in 98.1%

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

       11. distribute-neg-in 98.1%

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

       12. metadata-eval 98.1%

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

       13. sub-neg 98.1%

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

   11. Simplified 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000001 \lor \neg \left(-1 + x \leq 5 \cdot 10^{+29}\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+176} \lor \neg \left(z \leq 1.55 \cdot 10^{+133}\right):\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.7e+176) (not (<= z 1.55e+133)))
   (- (* (log1p (- y)) z) t)
   (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7e+176) || !(z <= 1.55e+133)) {
		tmp = (log1p(-y) * z) - t;
	} else {
		tmp = (log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.7e+176) || !(z <= 1.55e+133)) {
		tmp = (Math.log1p(-y) * z) - t;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.7e+176) or not (z <= 1.55e+133):
		tmp = (math.log1p(-y) * z) - t
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.7e+176) || !(z <= 1.55e+133))
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.7e+176], N[Not[LessEqual[z, 1.55e+133]], $MachinePrecision]], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000007e176 or 1.55e133 < z

   1. Initial program 53.7%

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

   3. Taylor expanded in z around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. sub-neg 38.8%

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

      3. log1p-define 82.2%

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

   5. Simplified 82.2%

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

   if -1.70000000000000007e176 < z < 1.55e133

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0 98.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+176} \lor \neg \left(z \leq 1.55 \cdot 10^{+133}\right):\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in x around 0 87.8%

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

   1. +-commutative 87.8%

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

   2. +-commutative 87.8%

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

   3. associate-+r+ 87.8%

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

   4. distribute-rgt-in 87.8%

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

   5. *-commutative 87.8%

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

   6. fma-define 87.8%

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

   7. sub-neg 87.8%

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

   8. log1p-define 99.8%

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

   9. sub-neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. unsub-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. +-commutative 99.2%

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

8. Simplified 99.2%

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

9. Final simplification 99.2%

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

Alternative 7: 46.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-y\right) \cdot z - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* (log1p (- y)) z) t))

C

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

Java

public static double code(double x, double y, double z, double t) {
	return (Math.log1p(-y) * z) - t;
}

Python

def code(x, y, z, t):
	return (math.log1p(-y) * z) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(log1p(Float64(-y)) * z) - t)
end

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

Alternative 8: 45.9% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.9%

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

7. Final simplification 48.9%

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

Alternative 9: 42.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00105 \lor \neg \left(t \leq 17500000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -0.00105) (not (<= t 17500000.0))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00105) || !(t <= 17500000.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	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 <= (-0.00105d0)) .or. (.not. (t <= 17500000.0d0))) then
        tmp = -t
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00105) || !(t <= 17500000.0)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -0.00105) or not (t <= 17500000.0):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -0.00105) || !(t <= 17500000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -0.00105) || ~((t <= 17500000.0)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -0.00105], N[Not[LessEqual[t, 17500000.0]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00104999999999999994 or 1.75e7 < t

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf 67.3%

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

      1. neg-mul-1 67.3%

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

   5. Simplified 67.3%

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

   if -0.00104999999999999994 < t < 1.75e7

   1. Initial program 80.7%

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

   3. Taylor expanded in z around inf 4.0%

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

      1. *-commutative 4.0%

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

      2. sub-neg 4.0%

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

      3. log1p-define 21.0%

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

   5. Simplified 21.0%

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

   6. Taylor expanded in y around 0 20.4%

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

      1. mul-1-neg 20.4%

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

      2. distribute-rgt-neg-in 20.4%

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

   8. Simplified 20.4%

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

   9. Taylor expanded in y around inf 20.2%

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

       1. mul-1-neg 20.2%

          \[\leadsto \color{blue}{-y \cdot z} \]

       2. distribute-rgt-neg-in 20.2%

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

   11. Simplified 20.2%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00105 \lor \neg \left(t \leq 17500000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.9%

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

7. Final simplification 48.9%

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

Alternative 11: 45.7% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in x around 0 87.8%

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

   1. +-commutative 87.8%

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

   2. +-commutative 87.8%

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

   3. associate-+r+ 87.8%

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

   4. distribute-rgt-in 87.8%

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

   5. *-commutative 87.8%

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

   6. fma-define 87.8%

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

   7. sub-neg 87.8%

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

   8. log1p-define 99.8%

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

   9. sub-neg 99.8%

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

   10. metadata-eval 99.8%

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

   11. +-commutative 99.8%

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

   12. *-commutative 99.8%

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

   13. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. +-commutative 99.2%

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

   5. mul-1-neg 99.2%

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

   6. unsub-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. +-commutative 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in y around inf 48.6%

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

Alternative 12: 45.5% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.5%

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

   1. mul-1-neg 48.5%

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

   2. distribute-rgt-neg-in 48.5%

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

8. Simplified 48.5%

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

9. Final simplification 48.5%

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

Alternative 13: 34.9% 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 87.8%

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

3. Taylor expanded in t around inf 37.1%

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

   1. neg-mul-1 37.1%

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

5. Simplified 37.1%

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

Reproduce

herbie shell --seed 2024096 
(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))