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

Percentage Accurate: 89.4% → 99.8%

Time: 19.2s

Alternatives: 9

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: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. +-commutative 89.3%

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

   2. fma-def 89.3%

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

   3. sub-neg 89.3%

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

   4. metadata-eval 89.3%

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

   5. sub-neg 89.3%

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

   6. log1p-def 99.7%

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

   7. sub-neg 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 88.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1 \lor \neg \left(-1 + x \leq 2000000000\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(\log y + y \cdot \left(z + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -1.0) (not (<= (+ -1.0 x) 2000000000.0)))
   (- (* (log y) (+ -1.0 x)) t)
   (- (- t) (+ (log y) (* y (+ z -1.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -1.0) || !((-1.0 + x) <= 2000000000.0)) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else {
		tmp = -t - (log(y) + (y * (z + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-1.0d0)) .or. (.not. (((-1.0d0) + x) <= 2000000000.0d0))) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else
        tmp = -t - (log(y) + (y * (z + (-1.0d0))))
    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) <= -1.0) || !((-1.0 + x) <= 2000000000.0)) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = -t - (Math.log(y) + (y * (z + -1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -1.0) or not ((-1.0 + x) <= 2000000000.0):
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = -t - (math.log(y) + (y * (z + -1.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -1.0) || !(Float64(-1.0 + x) <= 2000000000.0))
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(-t) - Float64(log(y) + Float64(y * Float64(z + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -1.0) || ~(((-1.0 + x) <= 2000000000.0)))
		tmp = (log(y) * (-1.0 + x)) - t;
	else
		tmp = -t - (log(y) + (y * (z + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], 2000000000.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[(N[Log[y], $MachinePrecision] + N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -1 or 2e9 < (-.f64 x 1)

   1. Initial program 89.5%

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

   3. Taylor expanded in y around 0 88.6%

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

   if -1 < (-.f64 x 1) < 2e9

   1. Initial program 78.4%

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

   3. Taylor expanded in x around 0 78.4%

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

      1. +-commutative 78.4%

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

      2. fma-def 78.4%

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

      3. sub-neg 78.4%

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

      4. mul-1-neg 78.4%

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

      5. log1p-def 100.0%

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

      6. mul-1-neg 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

      9. +-commutative 100.0%

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

      10. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. log-rec 100.0%

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

      3. +-commutative 100.0%

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

      4. log-rec 100.0%

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

      5. unsub-neg 100.0%

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

      6. mul-1-neg 100.0%

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

      7. *-commutative 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1 \lor \neg \left(-1 + x \leq 2000000000\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \left(\log y + y \cdot \left(z + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 5.8 \cdot 10^{-40}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 5.8e-40)))
   (- (* x (log y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 5.8e-40)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 5.8d-40))) then
        tmp = (x * log(y)) - t
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 5.8e-40)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.0) or not (x <= 5.8e-40):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 5.8e-40))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 5.8e-40)))
		tmp = (x * log(y)) - t;
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 5.8e-40]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 5.7999999999999998e-40 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0 94.0%

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

      1. *-commutative 94.0%

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

      2. add-cube-cbrt 92.9%

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

      3. pow3 92.9%

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

      4. sub-neg 92.9%

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

      5. metadata-eval 92.9%

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

   5. Applied egg-rr 92.9%

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

      1. unpow3 92.9%

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

      2. add-cube-cbrt 94.0%

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

      3. *-commutative 94.0%

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

      4. flip-+ 52.7%

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

      5. unpow2 52.7%

         \[\leadsto \log y \cdot \frac{\color{blue}{{x}^{2}} - -1 \cdot -1}{x - -1} - t \]

      6. metadata-eval 52.7%

         \[\leadsto \log y \cdot \frac{{x}^{2} - \color{blue}{1}}{x - -1} - t \]

      7. sub-neg 52.7%

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

      8. metadata-eval 52.7%

         \[\leadsto \log y \cdot \frac{{x}^{2} - 1}{x + \color{blue}{1}} - t \]

      9. +-commutative 52.7%

         \[\leadsto \log y \cdot \frac{{x}^{2} - 1}{\color{blue}{1 + x}} - t \]

      10. clear-num 52.6%

          \[\leadsto \log y \cdot \color{blue}{\frac{1}{\frac{1 + x}{{x}^{2} - 1}}} - t \]

      11. div-inv 52.7%

          \[\leadsto \color{blue}{\frac{\log y}{\frac{1 + x}{{x}^{2} - 1}}} - t \]

      12. clear-num 52.7%

          \[\leadsto \frac{\log y}{\color{blue}{\frac{1}{\frac{{x}^{2} - 1}{1 + x}}}} - t \]

      13. unpow2 52.7%

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

      14. metadata-eval 52.7%

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

      15. +-commutative 52.7%

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

      16. metadata-eval 52.7%

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

      17. sub-neg 52.7%

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

      18. flip-+ 94.0%

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

   7. Applied egg-rr 94.0%

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

   8. Taylor expanded in x around inf 93.9%

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

      1. *-commutative 93.9%

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

   10. Simplified 93.9%

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

   if -1 < x < 5.7999999999999998e-40

   1. Initial program 84.1%

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

   3. Taylor expanded in y around 0 83.0%

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

   4. Taylor expanded in x around 0 82.4%

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

      1. mul-1-neg 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 5.8 \cdot 10^{-40}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+55} \lor \neg \left(x \leq 4.6 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.25e+55) (not (<= x 4.6e+19)))
   (* x (log y))
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e+55) || !(x <= 4.6e+19)) {
		tmp = x * log(y);
	} else {
		tmp = -log(y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.25d+55)) .or. (.not. (x <= 4.6d+19))) then
        tmp = x * log(y)
    else
        tmp = -log(y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e+55) || !(x <= 4.6e+19)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -Math.log(y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.25e+55) or not (x <= 4.6e+19):
		tmp = x * math.log(y)
	else:
		tmp = -math.log(y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.25e+55) || !(x <= 4.6e+19))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.25e+55) || ~((x <= 4.6e+19)))
		tmp = x * log(y);
	else
		tmp = -log(y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.25e+55], N[Not[LessEqual[x, 4.6e+19]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.24999999999999999e55 or 4.6e19 < x

   1. Initial program 96.5%

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

      1. flip3-- 96.5%

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

      2. log-div 96.5%

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

      3. metadata-eval 96.5%

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

      4. pow3 96.5%

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

      5. sub-neg 96.5%

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

      6. distribute-rgt-neg-out 96.5%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 96.5%

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

      9. sqr-neg 96.5%

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

      10. sqrt-unprod 96.5%

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

      11. add-sqr-sqrt 96.5%

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

      12. log1p-udef 96.5%

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

      13. metadata-eval 96.5%

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

      14. log1p-udef 99.6%

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

      15. *-un-lft-identity 99.6%

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

   4. Applied egg-rr 99.6%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   5. Taylor expanded in x around inf 78.4%

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

      1. *-commutative 78.4%

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

   7. Simplified 78.4%

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

   if -2.24999999999999999e55 < x < 4.6e19

   1. Initial program 84.5%

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

   3. Taylor expanded in y around 0 83.5%

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

   4. Taylor expanded in x around 0 80.8%

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

      1. mul-1-neg 80.8%

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

   6. Simplified 80.8%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{+55} \lor \neg \left(x \leq 4.6 \cdot 10^{+19}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+41} \lor \neg \left(t \leq 1.45 \cdot 10^{+15}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4e+41) (not (<= t 1.45e+15))) (- t) (* x (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4e+41) || !(t <= 1.45e+15)) {
		tmp = -t;
	} else {
		tmp = x * 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 <= (-4d+41)) .or. (.not. (t <= 1.45d+15))) then
        tmp = -t
    else
        tmp = x * 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 <= -4e+41) || !(t <= 1.45e+15)) {
		tmp = -t;
	} else {
		tmp = x * Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4e+41) or not (t <= 1.45e+15):
		tmp = -t
	else:
		tmp = x * math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4e+41) || !(t <= 1.45e+15))
		tmp = Float64(-t);
	else
		tmp = Float64(x * log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4e+41) || ~((t <= 1.45e+15)))
		tmp = -t;
	else
		tmp = x * log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4e+41], N[Not[LessEqual[t, 1.45e+15]], $MachinePrecision]], (-t), N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.00000000000000002e41 or 1.45e15 < t

   1. Initial program 98.5%

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

   3. Taylor expanded in t around inf 81.8%

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

      1. neg-mul-1 81.8%

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

   5. Simplified 81.8%

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

   if -4.00000000000000002e41 < t < 1.45e15

   1. Initial program 82.1%

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

      1. flip3-- 82.1%

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

      2. log-div 82.1%

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

      3. metadata-eval 82.1%

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

      4. pow3 82.1%

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

      5. sub-neg 82.1%

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

      6. distribute-rgt-neg-out 82.1%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 81.7%

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

      9. sqr-neg 81.7%

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

      10. sqrt-unprod 81.7%

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

      11. add-sqr-sqrt 81.7%

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

      12. log1p-udef 81.6%

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

      13. metadata-eval 81.6%

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

      14. log1p-udef 99.2%

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

      15. *-un-lft-identity 99.2%

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

   4. Applied egg-rr 99.2%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\mathsf{log1p}\left({y}^{3}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(y, y, y\right)\right)\right)}\right) - t \]

   5. Taylor expanded in x around inf 48.0%

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

      1. *-commutative 48.0%

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

   7. Simplified 48.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{+41} \lor \neg \left(t \leq 1.45 \cdot 10^{+15}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

   2. distribute-rgt-neg-in 99.3%

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

   3. mul-1-neg 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

   6. +-commutative 99.3%

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

   7. distribute-lft-in 99.3%

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

   8. metadata-eval 99.3%

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

   9. neg-mul-1 99.3%

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

   10. unsub-neg 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 7: 88.4% 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 89.3%

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

3. Taylor expanded in y around 0 88.4%

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

4. Final simplification 88.4%

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

Alternative 8: 42.4% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+34} \lor \neg \left(t \leq 270\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -7.2e+34) (not (<= t 270.0))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.2e+34) || !(t <= 270.0)) {
		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 <= (-7.2d+34)) .or. (.not. (t <= 270.0d0))) 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 <= -7.2e+34) || !(t <= 270.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -7.2e+34) or not (t <= 270.0):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -7.2e+34) || !(t <= 270.0))
		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 <= -7.2e+34) || ~((t <= 270.0)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -7.2e+34], N[Not[LessEqual[t, 270.0]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.2000000000000001e34 or 270 < t

   1. Initial program 98.4%

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

   3. Taylor expanded in t around inf 79.1%

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

      1. neg-mul-1 79.1%

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

   5. Simplified 79.1%

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

   if -7.2000000000000001e34 < t < 270

   1. Initial program 81.4%

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

   3. Taylor expanded in z around inf 2.9%

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

      1. *-commutative 2.9%

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

      2. sub-neg 2.9%

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

      3. mul-1-neg 2.9%

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

      4. log1p-def 20.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} \cdot z \]

      5. mul-1-neg 20.9%

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

   5. Simplified 20.9%

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

   6. Taylor expanded in y around 0 20.8%

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

      1. mul-1-neg 20.8%

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

      2. distribute-rgt-neg-in 20.8%

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

   8. Simplified 20.8%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+34} \lor \neg \left(t \leq 270\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.5% 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 89.3%

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

3. Taylor expanded in t around inf 38.4%

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

   1. neg-mul-1 38.4%

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

5. Simplified 38.4%

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

6. Final simplification 38.4%

   \[\leadsto -t \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(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))