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

Percentage Accurate: 89.0% → 99.8%

Time: 17.3s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

   1. sub-neg 91.2%

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

   2. +-commutative 91.2%

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

   3. associate-+l+ 91.2%

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

   4. fma-def 91.2%

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

   5. sub-neg 91.2%

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

   6. metadata-eval 91.2%

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

   7. sub-neg 91.2%

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

   8. log1p-def 99.8%

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

   9. sub-neg 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 95.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right)\\ \mathbf{if}\;-1 + x \leq -1.0002:\\ \;\;\;\;\left(y + t_1\right) - t\\ \mathbf{elif}\;-1 + x \leq -0.5:\\ \;\;\;\;\left(-t\right) - \left(\log y + y \cdot \left(z + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (+ -1.0 x))))
   (if (<= (+ -1.0 x) -1.0002)
     (- (+ y t_1) t)
     (if (<= (+ -1.0 x) -0.5)
       (- (- t) (+ (log y) (* y (+ z -1.0))))
       (- t_1 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (-1.0 + x);
	double tmp;
	if ((-1.0 + x) <= -1.0002) {
		tmp = (y + t_1) - t;
	} else if ((-1.0 + x) <= -0.5) {
		tmp = -t - (log(y) + (y * (z + -1.0)));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * ((-1.0d0) + x)
    if (((-1.0d0) + x) <= (-1.0002d0)) then
        tmp = (y + t_1) - t
    else if (((-1.0d0) + x) <= (-0.5d0)) then
        tmp = -t - (log(y) + (y * (z + (-1.0d0))))
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * (-1.0 + x);
	double tmp;
	if ((-1.0 + x) <= -1.0002) {
		tmp = (y + t_1) - t;
	} else if ((-1.0 + x) <= -0.5) {
		tmp = -t - (Math.log(y) + (y * (z + -1.0)));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * (-1.0 + x)
	tmp = 0
	if (-1.0 + x) <= -1.0002:
		tmp = (y + t_1) - t
	elif (-1.0 + x) <= -0.5:
		tmp = -t - (math.log(y) + (y * (z + -1.0)))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(-1.0 + x))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -1.0002)
		tmp = Float64(Float64(y + t_1) - t);
	elseif (Float64(-1.0 + x) <= -0.5)
		tmp = Float64(Float64(-t) - Float64(log(y) + Float64(y * Float64(z + -1.0))));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * (-1.0 + x);
	tmp = 0.0;
	if ((-1.0 + x) <= -1.0002)
		tmp = (y + t_1) - t;
	elseif ((-1.0 + x) <= -0.5)
		tmp = -t - (log(y) + (y * (z + -1.0)));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0002], N[(N[(y + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5], N[((-t) - N[(N[Log[y], $MachinePrecision] + N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -1.0002

   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. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in z around 0 94.2%

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

   if -1.0002 < (-.f64 x 1) < -0.5

   1. Initial program 86.6%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

      4. unsub-neg 99.2%

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

      5. mul-1-neg 99.2%

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

      6. +-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -0.5 < (-.f64 x 1)

   1. Initial program 95.4%

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

   2. Taylor expanded in y around 0 94.4%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.0002:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \mathbf{elif}\;-1 + x \leq -0.5:\\ \;\;\;\;\left(-t\right) - \left(\log y + y \cdot \left(z + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]

Alternative 3: 95.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.0002 \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(\log y + z \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -1.0002) (not (<= (+ -1.0 x) -0.5)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- t) (+ (log y) (* z y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.0002) || !((-1.0 + x) <= -0.5)) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = -t - (log(y) + (z * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-1.0002d0)) .or. (.not. (((-1.0d0) + x) <= (-0.5d0)))) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = -t - (log(y) + (z * y))
    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) <= -1.0002) || !((-1.0 + x) <= -0.5)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = -t - (Math.log(y) + (z * y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -1.0002) or not ((-1.0 + x) <= -0.5):
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = -t - (math.log(y) + (z * y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -1.0002) || !(Float64(-1.0 + x) <= -0.5))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(-t) - Float64(log(y) + Float64(z * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -1.0002) || ~(((-1.0 + x) <= -0.5)))
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = -t - (log(y) + (z * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0002], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[(N[Log[y], $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -1.0002 or -0.5 < (-.f64 x 1)

   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. Taylor expanded in y around 0 94.2%

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

   if -1.0002 < (-.f64 x 1) < -0.5

   1. Initial program 86.6%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

   4. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. expm1-log1p-u 98.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   6. Applied egg-rr 98.8%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   7. Taylor expanded in x around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. log-rec 99.2%

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

      3. +-commutative 99.2%

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

      4. log-rec 99.2%

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

      5. unsub-neg 99.2%

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

      6. mul-1-neg 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

      9. +-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. distribute-rgt-neg-in 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in z around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. distribute-rgt-neg-in 98.7%

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

   12. Simplified 98.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.0002 \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(\log y + z \cdot y\right)\\ \end{array} \]

Alternative 4: 95.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right)\\ \mathbf{if}\;-1 + x \leq -1.0002:\\ \;\;\;\;\left(y + t_1\right) - t\\ \mathbf{elif}\;-1 + x \leq -0.5:\\ \;\;\;\;\left(-t\right) - \left(\log y + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (+ -1.0 x))))
   (if (<= (+ -1.0 x) -1.0002)
     (- (+ y t_1) t)
     (if (<= (+ -1.0 x) -0.5) (- (- t) (+ (log y) (* z y))) (- t_1 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (-1.0 + x);
	double tmp;
	if ((-1.0 + x) <= -1.0002) {
		tmp = (y + t_1) - t;
	} else if ((-1.0 + x) <= -0.5) {
		tmp = -t - (log(y) + (z * y));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * ((-1.0d0) + x)
    if (((-1.0d0) + x) <= (-1.0002d0)) then
        tmp = (y + t_1) - t
    else if (((-1.0d0) + x) <= (-0.5d0)) then
        tmp = -t - (log(y) + (z * y))
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * (-1.0 + x);
	double tmp;
	if ((-1.0 + x) <= -1.0002) {
		tmp = (y + t_1) - t;
	} else if ((-1.0 + x) <= -0.5) {
		tmp = -t - (Math.log(y) + (z * y));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * (-1.0 + x)
	tmp = 0
	if (-1.0 + x) <= -1.0002:
		tmp = (y + t_1) - t
	elif (-1.0 + x) <= -0.5:
		tmp = -t - (math.log(y) + (z * y))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(-1.0 + x))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -1.0002)
		tmp = Float64(Float64(y + t_1) - t);
	elseif (Float64(-1.0 + x) <= -0.5)
		tmp = Float64(Float64(-t) - Float64(log(y) + Float64(z * y)));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * (-1.0 + x);
	tmp = 0.0;
	if ((-1.0 + x) <= -1.0002)
		tmp = (y + t_1) - t;
	elseif ((-1.0 + x) <= -0.5)
		tmp = -t - (log(y) + (z * y));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0002], N[(N[(y + t$95$1), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5], N[((-t) - N[(N[Log[y], $MachinePrecision] + N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -1.0002

   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. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in z around 0 94.2%

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

   if -1.0002 < (-.f64 x 1) < -0.5

   1. Initial program 86.6%

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

   2. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

   4. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. expm1-log1p-u 98.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   6. Applied egg-rr 98.8%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   7. Taylor expanded in x around 0 99.2%

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

      1. mul-1-neg 99.2%

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

      2. log-rec 99.2%

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

      3. +-commutative 99.2%

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

      4. log-rec 99.2%

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

      5. unsub-neg 99.2%

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

      6. mul-1-neg 99.2%

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

      7. sub-neg 99.2%

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

      8. metadata-eval 99.2%

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

      9. +-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. distribute-rgt-neg-in 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in z around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. distribute-rgt-neg-in 98.7%

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

   12. Simplified 98.7%

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

   if -0.5 < (-.f64 x 1)

   1. Initial program 95.4%

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

   2. Taylor expanded in y around 0 94.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.0002:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \mathbf{elif}\;-1 + x \leq -0.5:\\ \;\;\;\;\left(-t\right) - \left(\log y + z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

2. Taylor expanded in y around 0 99.1%

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

   1. mul-1-neg 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

Alternative 6: 55.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{+72}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-302}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= t -3.3e+72)
     (- t)
     (if (<= t -4.4e-101)
       t_1
       (if (<= t 8.2e-302) (- (log y)) (if (<= t 6e+126) t_1 (- t)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (t <= -3.3e+72) {
		tmp = -t;
	} else if (t <= -4.4e-101) {
		tmp = t_1;
	} else if (t <= 8.2e-302) {
		tmp = -log(y);
	} else if (t <= 6e+126) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (t <= (-3.3d+72)) then
        tmp = -t
    else if (t <= (-4.4d-101)) then
        tmp = t_1
    else if (t <= 8.2d-302) then
        tmp = -log(y)
    else if (t <= 6d+126) then
        tmp = t_1
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (t <= -3.3e+72) {
		tmp = -t;
	} else if (t <= -4.4e-101) {
		tmp = t_1;
	} else if (t <= 8.2e-302) {
		tmp = -Math.log(y);
	} else if (t <= 6e+126) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if t <= -3.3e+72:
		tmp = -t
	elif t <= -4.4e-101:
		tmp = t_1
	elif t <= 8.2e-302:
		tmp = -math.log(y)
	elif t <= 6e+126:
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (t <= -3.3e+72)
		tmp = Float64(-t);
	elseif (t <= -4.4e-101)
		tmp = t_1;
	elseif (t <= 8.2e-302)
		tmp = Float64(-log(y));
	elseif (t <= 6e+126)
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (t <= -3.3e+72)
		tmp = -t;
	elseif (t <= -4.4e-101)
		tmp = t_1;
	elseif (t <= 8.2e-302)
		tmp = -log(y);
	elseif (t <= 6e+126)
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.3e+72], (-t), If[LessEqual[t, -4.4e-101], t$95$1, If[LessEqual[t, 8.2e-302], (-N[Log[y], $MachinePrecision]), If[LessEqual[t, 6e+126], t$95$1, (-t)]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.3e72 or 6.0000000000000005e126 < t

   1. Initial program 97.8%

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

   2. Taylor expanded in t around inf 79.6%

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

      1. neg-mul-1 79.6%

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

   4. Simplified 79.6%

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

   if -3.3e72 < t < -4.3999999999999998e-101 or 8.1999999999999996e-302 < t < 6.0000000000000005e126

   1. Initial program 86.9%

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

   2. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in x around inf 57.0%

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

      1. *-commutative 57.0%

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

   7. Simplified 57.0%

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

   if -4.3999999999999998e-101 < t < 8.1999999999999996e-302

   1. Initial program 88.9%

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

   2. Taylor expanded in y around 0 85.8%

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

   3. Taylor expanded in x around 0 55.0%

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

      1. mul-1-neg 55.0%

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

   5. Simplified 55.0%

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

   6. Taylor expanded in t around 0 55.0%

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

      1. mul-1-neg 55.0%

         \[\leadsto \color{blue}{-\log y} \]

   8. Simplified 55.0%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+72}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-101}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-302}:\\ \;\;\;\;-\log y\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+126}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 7: 86.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.1 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 2.1e-14)))
   (- (* x (log y)) t)
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2.1e-14)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 2.1d-14))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2.1e-14)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.0) or not (x <= 2.1e-14):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 2.1e-14))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 2.1e-14)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 2.1e-14]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.0999999999999999e-14 < x

   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. Taylor expanded in y around 0 93.6%

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

      1. *-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. flip-+ 46.7%

         \[\leadsto \color{blue}{\frac{x \cdot x - -1 \cdot -1}{x - -1}} \cdot \log y - t \]

      5. associate-*l/ 46.6%

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

      6. metadata-eval 46.6%

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

      7. fma-neg 46.6%

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

      8. metadata-eval 46.6%

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

      9. sub-neg 46.6%

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

      10. metadata-eval 46.6%

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

   4. Applied egg-rr 46.6%

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

   5. Taylor expanded in x around inf 93.1%

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

      1. *-commutative 93.1%

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

   7. Simplified 93.1%

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

   if -1 < x < 2.0999999999999999e-14

   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. Taylor expanded in y around 0 85.9%

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

   3. Taylor expanded in x around 0 85.4%

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

      1. mul-1-neg 85.4%

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

   5. Simplified 85.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.1 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 8: 86.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.1 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 2.1e-14)))
   (- (* x (log y)) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2.1e-14)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 2.1d-14))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 2.1e-14)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.0) or not (x <= 2.1e-14):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 2.1e-14))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 2.1e-14)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 2.1e-14]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 2.0999999999999999e-14 < x

   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. Taylor expanded in y around 0 93.6%

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

      1. *-commutative 93.6%

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

      2. sub-neg 93.6%

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

      3. metadata-eval 93.6%

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

      4. flip-+ 46.7%

         \[\leadsto \color{blue}{\frac{x \cdot x - -1 \cdot -1}{x - -1}} \cdot \log y - t \]

      5. associate-*l/ 46.6%

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

      6. metadata-eval 46.6%

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

      7. fma-neg 46.6%

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

      8. metadata-eval 46.6%

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

      9. sub-neg 46.6%

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

      10. metadata-eval 46.6%

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

   4. Applied egg-rr 46.6%

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

   5. Taylor expanded in x around inf 93.1%

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

      1. *-commutative 93.1%

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

   7. Simplified 93.1%

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

   if -1 < x < 2.0999999999999999e-14

   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. Taylor expanded in y around 0 99.3%

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

      1. mul-1-neg 99.3%

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

   4. Simplified 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. expm1-log1p-u 98.8%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   6. Applied egg-rr 98.8%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   7. Taylor expanded in x around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. log-rec 98.7%

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

      3. +-commutative 98.7%

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

      4. log-rec 98.7%

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

      5. unsub-neg 98.7%

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

      6. mul-1-neg 98.7%

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

      7. sub-neg 98.7%

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

      8. metadata-eval 98.7%

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

      9. +-commutative 98.7%

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

      10. *-commutative 98.7%

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

      11. distribute-rgt-neg-in 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in z around 0 86.0%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 2.1 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]

Alternative 9: 75.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+48} \lor \neg \left(x \leq 1.65 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.55e+48) (not (<= x 1.65e+14)))
   (* x (log y))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e+48) || !(x <= 1.65e+14)) {
		tmp = x * log(y);
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.55d+48)) .or. (.not. (x <= 1.65d+14))) then
        tmp = x * log(y)
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e+48) || !(x <= 1.65e+14)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.55e+48) or not (x <= 1.65e+14):
		tmp = x * math.log(y)
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.55e+48) || !(x <= 1.65e+14))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.55e+48) || ~((x <= 1.65e+14)))
		tmp = x * log(y);
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.55e+48], N[Not[LessEqual[x, 1.65e+14]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000003e48 or 1.65e14 < x

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

   4. Simplified 98.7%

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

   5. Taylor expanded in x around inf 76.9%

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

      1. *-commutative 76.9%

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

   7. Simplified 76.9%

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

   if -1.55000000000000003e48 < x < 1.65e14

   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. Taylor expanded in y around 0 86.1%

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

   3. Taylor expanded in x around 0 83.1%

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

      1. mul-1-neg 83.1%

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

   5. Simplified 83.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+48} \lor \neg \left(x \leq 1.65 \cdot 10^{+14}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]

Alternative 10: 87.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

2. Taylor expanded in y around 0 89.9%

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

3. Final simplification 89.9%

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

Alternative 11: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00245 \lor \neg \left(t \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -0.00245) (not (<= t 5.2e-9))) (- t) (- (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00245) || !(t <= 5.2e-9)) {
		tmp = -t;
	} else {
		tmp = -log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.00245d0)) .or. (.not. (t <= 5.2d-9))) then
        tmp = -t
    else
        tmp = -log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00245) || !(t <= 5.2e-9)) {
		tmp = -t;
	} else {
		tmp = -Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -0.00245) or not (t <= 5.2e-9):
		tmp = -t
	else:
		tmp = -math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -0.00245) || !(t <= 5.2e-9))
		tmp = Float64(-t);
	else
		tmp = Float64(-log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -0.00245) || ~((t <= 5.2e-9)))
		tmp = -t;
	else
		tmp = -log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -0.00245], N[Not[LessEqual[t, 5.2e-9]], $MachinePrecision]], (-t), (-N[Log[y], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -0.0024499999999999999 or 5.2000000000000002e-9 < t

   1. Initial program 92.8%

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

   2. Taylor expanded in t around inf 61.0%

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

      1. neg-mul-1 61.0%

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

   4. Simplified 61.0%

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

   if -0.0024499999999999999 < t < 5.2000000000000002e-9

   1. Initial program 89.3%

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

   2. Taylor expanded in y around 0 87.7%

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

   3. Taylor expanded in x around 0 43.2%

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

      1. mul-1-neg 43.2%

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

   5. Simplified 43.2%

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

   6. Taylor expanded in t around 0 43.1%

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

      1. mul-1-neg 43.1%

         \[\leadsto \color{blue}{-\log y} \]

   8. Simplified 43.1%

      \[\leadsto \color{blue}{-\log y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00245 \lor \neg \left(t \leq 5.2 \cdot 10^{-9}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-\log y\\ \end{array} \]

Alternative 12: 43.6% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -37000 \lor \neg \left(t \leq 30000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -37000.0) (not (<= t 30000000000000.0))) (- t) (- y (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -37000.0) || !(t <= 30000000000000.0)) {
		tmp = -t;
	} else {
		tmp = y - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-37000.0d0)) .or. (.not. (t <= 30000000000000.0d0))) then
        tmp = -t
    else
        tmp = y - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -37000.0) || !(t <= 30000000000000.0)) {
		tmp = -t;
	} else {
		tmp = y - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -37000.0) or not (t <= 30000000000000.0):
		tmp = -t
	else:
		tmp = y - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -37000.0) || !(t <= 30000000000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(y - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -37000.0) || ~((t <= 30000000000000.0)))
		tmp = -t;
	else
		tmp = y - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -37000.0], N[Not[LessEqual[t, 30000000000000.0]], $MachinePrecision]], (-t), N[(y - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -37000 or 3e13 < t

   1. Initial program 95.1%

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

   2. Taylor expanded in t around inf 64.6%

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

      1. neg-mul-1 64.6%

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

   4. Simplified 64.6%

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

   if -37000 < t < 3e13

   1. Initial program 87.2%

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

   2. Taylor expanded in y around 0 99.0%

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

      1. mul-1-neg 99.0%

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

   4. Simplified 99.0%

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

      1. sub-neg 99.0%

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

      2. metadata-eval 99.0%

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

      3. expm1-log1p-u 73.5%

         \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   6. Applied egg-rr 73.5%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

   7. Taylor expanded in y around inf 15.9%

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

      1. mul-1-neg 15.9%

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

      2. sub-neg 15.9%

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

      3. metadata-eval 15.9%

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

      4. +-commutative 15.9%

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

      5. *-commutative 15.9%

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

      6. distribute-rgt-neg-in 15.9%

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

   9. Simplified 15.9%

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

   10. Taylor expanded in z around 0 15.9%

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

       1. mul-1-neg 15.9%

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

       2. unsub-neg 15.9%

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

   12. Simplified 15.9%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -37000 \lor \neg \left(t \leq 30000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y - z \cdot y\\ \end{array} \]

Alternative 13: 43.3% accurate, 26.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -10200 \lor \neg \left(t \leq 30000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -10200.0) (not (<= t 30000000000000.0))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -10200.0) || !(t <= 30000000000000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-10200.0d0)) .or. (.not. (t <= 30000000000000.0d0))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -10200.0) || !(t <= 30000000000000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -10200.0) or not (t <= 30000000000000.0):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -10200.0) || !(t <= 30000000000000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -10200.0) || ~((t <= 30000000000000.0)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -10200.0], N[Not[LessEqual[t, 30000000000000.0]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -10200 or 3e13 < t

   1. Initial program 95.1%

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

   2. Taylor expanded in t around inf 64.6%

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

      1. neg-mul-1 64.6%

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

   4. Simplified 64.6%

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

   if -10200 < t < 3e13

   1. Initial program 87.2%

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

   2. Taylor expanded in y around 0 99.0%

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

      1. mul-1-neg 99.0%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in z around inf 15.3%

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

      1. mul-1-neg 15.3%

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

      2. *-commutative 15.3%

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

      3. distribute-rgt-neg-in 15.3%

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

   7. Simplified 15.3%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -10200 \lor \neg \left(t \leq 30000000000000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]

Alternative 14: 35.9% 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 91.2%

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

2. Taylor expanded in t around inf 34.3%

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

   1. neg-mul-1 34.3%

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

4. Simplified 34.3%

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

5. Final simplification 34.3%

   \[\leadsto -t \]

Alternative 15: 2.9% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.2%

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

2. Taylor expanded in y around 0 99.1%

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

   1. mul-1-neg 99.1%

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

4. Simplified 99.1%

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

   1. sub-neg 99.1%

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

   2. metadata-eval 99.1%

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

   3. expm1-log1p-u 72.6%

      \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

6. Applied egg-rr 72.6%

   \[\leadsto \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(x + -1\right) \cdot \log y\right)\right)} + \left(z - 1\right) \cdot \left(-y\right)\right) - t \]

7. Taylor expanded in y around inf 11.4%

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

   1. mul-1-neg 11.4%

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

   2. sub-neg 11.4%

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

   3. metadata-eval 11.4%

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

   4. +-commutative 11.4%

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

   5. *-commutative 11.4%

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

   6. distribute-rgt-neg-in 11.4%

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

9. Simplified 11.4%

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

10. Taylor expanded in z around 0 2.8%

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

11. Final simplification 2.8%

    \[\leadsto y \]

Reproduce

herbie shell --seed 2023310 
(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))