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

Percentage Accurate: 88.7% → 99.3%

Time: 13.8s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.7% 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: 99.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in y around 0 99.8%

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

4. Taylor expanded in z around inf 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 97.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+16} \lor \neg \left(x + -1 \leq -0.99999999995\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-y\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ x -1.0) -5e+16) (not (<= (+ x -1.0) -0.99999999995)))
   (- (- (* x (log y)) (* y z)) t)
   (- (- (* z (- y)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -5e+16) || !((x + -1.0) <= -0.99999999995)) {
		tmp = ((x * log(y)) - (y * z)) - t;
	} else {
		tmp = ((z * -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)) <= (-5d+16)) .or. (.not. ((x + (-1.0d0)) <= (-0.99999999995d0)))) then
        tmp = ((x * log(y)) - (y * z)) - t
    else
        tmp = ((z * -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) <= -5e+16) || !((x + -1.0) <= -0.99999999995)) {
		tmp = ((x * Math.log(y)) - (y * z)) - t;
	} else {
		tmp = ((z * -y) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + -1.0) <= -5e+16) or not ((x + -1.0) <= -0.99999999995):
		tmp = ((x * math.log(y)) - (y * z)) - t
	else:
		tmp = ((z * -y) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x + -1.0) <= -5e+16) || !(Float64(x + -1.0) <= -0.99999999995))
		tmp = Float64(Float64(Float64(x * log(y)) - Float64(y * z)) - t);
	else
		tmp = Float64(Float64(Float64(z * 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) <= -5e+16) || ~(((x + -1.0) <= -0.99999999995)))
		tmp = ((x * log(y)) - (y * z)) - t;
	else
		tmp = ((z * -y) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x + -1.0), $MachinePrecision], -5e+16], N[Not[LessEqual[N[(x + -1.0), $MachinePrecision], -0.99999999995]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(z * (-y)), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e16 or -0.99999999995 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0 99.7%

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

   4. Taylor expanded in z around inf 99.7%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. mul-1-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

   10. Simplified 99.6%

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

   if -5e16 < (-.f64 x #s(literal 1 binary64)) < -0.99999999995

   1. Initial program 82.3%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in z around inf 100.0%

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

   5. Taylor expanded in y around 0 99.7%

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

      1. mul-1-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 98.8%

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

      1. neg-mul-1 98.8%

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

      2. log-rec 98.8%

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

      3. +-commutative 98.8%

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

      4. log-rec 98.8%

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

      5. unsub-neg 98.8%

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

      6. mul-1-neg 98.8%

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

      7. *-commutative 98.8%

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

      8. distribute-rgt-neg-in 98.8%

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

   10. Simplified 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+16} \lor \neg \left(x + -1 \leq -0.99999999995\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot z\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-y\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+59} \lor \neg \left(x + -1 \leq -0.5\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-y\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ x -1.0) -5e+59) (not (<= (+ x -1.0) -0.5)))
   (- (* x (log y)) t)
   (- (- (* z (- y)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -5e+59) || !((x + -1.0) <= -0.5)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = ((z * -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)) <= (-5d+59)) .or. (.not. ((x + (-1.0d0)) <= (-0.5d0)))) then
        tmp = (x * log(y)) - t
    else
        tmp = ((z * -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) <= -5e+59) || !((x + -1.0) <= -0.5)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = ((z * -y) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + -1.0) <= -5e+59) or not ((x + -1.0) <= -0.5):
		tmp = (x * math.log(y)) - t
	else:
		tmp = ((z * -y) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x + -1.0) <= -5e+59) || !(Float64(x + -1.0) <= -0.5))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(z * 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) <= -5e+59) || ~(((x + -1.0) <= -0.5)))
		tmp = (x * log(y)) - t;
	else
		tmp = ((z * -y) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x + -1.0), $MachinePrecision], -5e+59], N[Not[LessEqual[N[(x + -1.0), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(z * (-y)), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.9999999999999997e59 or -0.5 < (-.f64 x #s(literal 1 binary64))

   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. Step-by-step derivation

      1. +-commutative 95.4%

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

      2. fma-define 95.4%

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

      3. sub-neg 95.4%

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

      4. metadata-eval 95.4%

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

      5. sub-neg 95.4%

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

      6. log1p-define 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 94.9%

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

   6. Taylor expanded in x around inf 94.9%

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

   if -4.9999999999999997e59 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 80.9%

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

   3. Taylor expanded in y around 0 100.0%

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

   4. Taylor expanded in z around inf 100.0%

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

   5. Taylor expanded in y around 0 99.7%

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

      1. mul-1-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 97.1%

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

      1. neg-mul-1 97.1%

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

      2. log-rec 97.1%

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

      3. +-commutative 97.1%

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

      4. log-rec 97.1%

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

      5. unsub-neg 97.1%

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

      6. mul-1-neg 97.1%

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

      7. *-commutative 97.1%

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

      8. distribute-rgt-neg-in 97.1%

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

   10. Simplified 97.1%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+59} \lor \neg \left(x + -1 \leq -0.5\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \left(-y\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -0.0011:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -6.5e+170)
     t_1
     (if (<= x -0.0011)
       (- (* y (- (- -1.0) z)) t)
       (if (<= x 4.2e+75) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -6.5e+170) {
		tmp = t_1;
	} else if (x <= -0.0011) {
		tmp = (y * (-(-1.0) - z)) - t;
	} else if (x <= 4.2e+75) {
		tmp = -log(y) - t;
	} else {
		tmp = t_1;
	}
	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 (x <= (-6.5d+170)) then
        tmp = t_1
    else if (x <= (-0.0011d0)) then
        tmp = (y * (-(-1.0d0) - z)) - t
    else if (x <= 4.2d+75) then
        tmp = -log(y) - t
    else
        tmp = t_1
    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 (x <= -6.5e+170) {
		tmp = t_1;
	} else if (x <= -0.0011) {
		tmp = (y * (-(-1.0) - z)) - t;
	} else if (x <= 4.2e+75) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -6.5e+170:
		tmp = t_1
	elif x <= -0.0011:
		tmp = (y * (-(-1.0) - z)) - t
	elif x <= 4.2e+75:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -6.5e+170)
		tmp = t_1;
	elseif (x <= -0.0011)
		tmp = Float64(Float64(y * Float64(Float64(-(-1.0)) - z)) - t);
	elseif (x <= 4.2e+75)
		tmp = Float64(Float64(-log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -6.5e+170)
		tmp = t_1;
	elseif (x <= -0.0011)
		tmp = (y * (-(-1.0) - z)) - t;
	elseif (x <= 4.2e+75)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+170], t$95$1, If[LessEqual[x, -0.0011], N[(N[(y * N[((--1.0) - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 4.2e+75], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5e170 or 4.19999999999999997e75 < x

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. fma-define 96.6%

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

      3. sub-neg 96.6%

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

      4. metadata-eval 96.6%

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

      5. sub-neg 96.6%

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

      6. log1p-define 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 63.4%

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

      1. sub-neg 63.4%

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

      2. log1p-define 66.4%

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

      3. +-commutative 66.4%

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

      4. mul-1-neg 66.4%

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

      5. unsub-neg 66.4%

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

      6. sub-neg 66.4%

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

      7. metadata-eval 66.4%

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

      8. associate-/l* 66.3%

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

      9. +-commutative 66.3%

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

      10. sub-neg 66.3%

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

      11. log1p-define 66.3%

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

   7. Simplified 66.3%

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

   8. Taylor expanded in x around inf 80.3%

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

      1. *-commutative 80.3%

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

   10. Simplified 80.3%

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

   if -6.5e170 < x < -0.00110000000000000007

   1. Initial program 80.1%

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

      1. add-cbrt-cube 79.8%

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

      2. pow3 79.8%

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

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y around inf 63.1%

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

      1. associate-*r* 63.1%

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

      2. neg-mul-1 63.1%

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

      3. *-commutative 63.1%

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

      4. sub-neg 63.1%

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

      5. metadata-eval 63.1%

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

      6. +-commutative 63.1%

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

   10. Simplified 63.1%

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

   if -0.00110000000000000007 < x < 4.19999999999999997e75

   1. Initial program 84.3%

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

      1. +-commutative 84.3%

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

      2. fma-define 84.3%

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

      3. sub-neg 84.3%

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

      4. metadata-eval 84.3%

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

      5. sub-neg 84.3%

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

      6. log1p-define 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 83.7%

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

   6. Taylor expanded in x around 0 80.3%

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

      1. mul-1-neg 80.3%

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

   8. Simplified 80.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+170}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -0.0011:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -230000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -230000.0) (not (<= x 1.0)))
   (- (* x (log y)) t)
   (- (- (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -230000.0) || !(x <= 1.0)) {
		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 <= (-230000.0d0)) .or. (.not. (x <= 1.0d0))) 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 <= -230000.0) || !(x <= 1.0)) {
		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 <= -230000.0) or not (x <= 1.0):
		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 <= -230000.0) || !(x <= 1.0))
		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 <= -230000.0) || ~((x <= 1.0)))
		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, -230000.0], N[Not[LessEqual[x, 1.0]], $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 < -2.3e5 or 1 < x

   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. Step-by-step derivation

      1. +-commutative 92.8%

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

      2. fma-define 92.8%

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

      3. sub-neg 92.8%

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

      4. metadata-eval 92.8%

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

      5. sub-neg 92.8%

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

      6. log1p-define 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 92.2%

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

   6. Taylor expanded in x around inf 92.2%

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

   if -2.3e5 < x < 1

   1. Initial program 82.4%

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

      1. +-commutative 82.4%

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

      2. fma-define 82.4%

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

      3. sub-neg 82.4%

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

      4. metadata-eval 82.4%

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

      5. sub-neg 82.4%

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

      6. log1p-define 100.0%

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

      7. sub-neg 100.0%

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

      8. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 81.7%

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

   6. Taylor expanded in x around 0 80.8%

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

      1. mul-1-neg 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -230000 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{+26} \lor \neg \left(t \leq 1.45 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -6.1e+26) (not (<= t 1.45e+48)))
   (- (* y (* z (+ (* y -0.5) -1.0))) t)
   (* (+ x -1.0) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.1e+26) || !(t <= 1.45e+48)) {
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	} else {
		tmp = (x + -1.0) * 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 <= (-6.1d+26)) .or. (.not. (t <= 1.45d+48))) then
        tmp = (y * (z * ((y * (-0.5d0)) + (-1.0d0)))) - t
    else
        tmp = (x + (-1.0d0)) * 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 <= -6.1e+26) || !(t <= 1.45e+48)) {
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	} else {
		tmp = (x + -1.0) * Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -6.1e+26) or not (t <= 1.45e+48):
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t
	else:
		tmp = (x + -1.0) * math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -6.1e+26) || !(t <= 1.45e+48))
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * -0.5) + -1.0))) - t);
	else
		tmp = Float64(Float64(x + -1.0) * log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -6.1e+26) || ~((t <= 1.45e+48)))
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	else
		tmp = (x + -1.0) * log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -6.1e+26], N[Not[LessEqual[t, 1.45e+48]], $MachinePrecision]], N[(N[(y * N[(z * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.1000000000000003e26 or 1.4499999999999999e48 < t

   1. Initial program 97.6%

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

   3. Taylor expanded in y around 0 99.9%

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

   4. Taylor expanded in z around inf 99.9%

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

   5. Taylor expanded in z around inf 81.7%

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

   if -6.1000000000000003e26 < t < 1.4499999999999999e48

   1. Initial program 79.5%

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

      1. +-commutative 79.5%

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

      2. fma-define 79.5%

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

      3. sub-neg 79.5%

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

      4. metadata-eval 79.5%

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

      5. sub-neg 79.5%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 78.8%

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

   6. Taylor expanded in t around 0 77.5%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.1 \cdot 10^{+26} \lor \neg \left(t \leq 1.45 \cdot 10^{+48}\right):\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+170} \lor \neg \left(x \leq 3.2 \cdot 10^{+77}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.2e+170) (not (<= x 3.2e+77)))
   (* x (log y))
   (- (* y (- (- -1.0) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.2e+170) || !(x <= 3.2e+77)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (-(-1.0) - z)) - 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 <= (-8.2d+170)) .or. (.not. (x <= 3.2d+77))) then
        tmp = x * log(y)
    else
        tmp = (y * (-(-1.0d0) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.2e+170) || !(x <= 3.2e+77)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (-(-1.0) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8.2e+170) or not (x <= 3.2e+77):
		tmp = x * math.log(y)
	else:
		tmp = (y * (-(-1.0) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.2e+170) || !(x <= 3.2e+77))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(-(-1.0)) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8.2e+170) || ~((x <= 3.2e+77)))
		tmp = x * log(y);
	else
		tmp = (y * (-(-1.0) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8.2e+170], N[Not[LessEqual[x, 3.2e+77]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[((--1.0) - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.2000000000000001e170 or 3.2000000000000002e77 < x

   1. Initial program 96.6%

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

      1. +-commutative 96.6%

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

      2. fma-define 96.6%

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

      3. sub-neg 96.6%

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

      4. metadata-eval 96.6%

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

      5. sub-neg 96.6%

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

      6. log1p-define 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 63.4%

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

      1. sub-neg 63.4%

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

      2. log1p-define 66.4%

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

      3. +-commutative 66.4%

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

      4. mul-1-neg 66.4%

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

      5. unsub-neg 66.4%

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

      6. sub-neg 66.4%

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

      7. metadata-eval 66.4%

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

      8. associate-/l* 66.3%

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

      9. +-commutative 66.3%

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

      10. sub-neg 66.3%

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

      11. log1p-define 66.3%

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

   7. Simplified 66.3%

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

   8. Taylor expanded in x around inf 80.3%

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

      1. *-commutative 80.3%

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

   10. Simplified 80.3%

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

   if -8.2000000000000001e170 < x < 3.2000000000000002e77

   1. Initial program 83.5%

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

      1. add-cbrt-cube 83.3%

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

      2. pow3 83.3%

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

   4. Applied egg-rr 83.3%

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

   5. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y around inf 62.2%

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

      1. associate-*r* 62.2%

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

      2. neg-mul-1 62.2%

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

      3. *-commutative 62.2%

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

      4. sub-neg 62.2%

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

      5. metadata-eval 62.2%

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

      6. +-commutative 62.2%

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

   10. Simplified 62.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+170} \lor \neg \left(x \leq 3.2 \cdot 10^{+77}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-28}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 8.2e-28) (- (* (+ x -1.0) (log y)) t) (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.2e-28) {
		tmp = ((x + -1.0) * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 8.2e-28) {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 8.2e-28:
		tmp = ((x + -1.0) * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 8.2e-28)
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 8.2e-28], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.2000000000000005e-28

   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. Step-by-step derivation

      1. +-commutative 89.5%

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

      2. fma-define 89.5%

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

      3. sub-neg 89.5%

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

      4. metadata-eval 89.5%

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

      5. sub-neg 89.5%

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

      6. log1p-define 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 89.5%

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

   if 8.2000000000000005e-28 < y

   1. Initial program 71.2%

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

      1. +-commutative 71.2%

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

      2. fma-define 71.2%

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

      3. sub-neg 71.2%

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

      4. metadata-eval 71.2%

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

      5. sub-neg 71.2%

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

      6. log1p-define 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 57.3%

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

      1. sub-neg 57.3%

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

      2. log1p-define 86.0%

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

      3. +-commutative 86.0%

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

      4. mul-1-neg 86.0%

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

      5. unsub-neg 86.0%

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

      6. sub-neg 86.0%

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

      7. metadata-eval 86.0%

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

      8. associate-/l* 86.0%

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

      9. +-commutative 86.0%

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

      10. sub-neg 86.0%

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

      11. log1p-define 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in z around inf 51.4%

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

      1. sub-neg 51.4%

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

      2. log1p-undefine 80.4%

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

   10. Simplified 80.4%

       \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-28}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in y around 0 99.8%

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

4. Taylor expanded in z around inf 99.8%

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

5. Taylor expanded in y around 0 99.6%

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

   1. mul-1-neg 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 10: 46.3% accurate, 26.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. add-cbrt-cube 87.3%

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

   2. pow3 87.3%

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

4. Applied egg-rr 87.3%

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

5. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in y around inf 48.9%

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

   1. associate-*r* 48.9%

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

   2. neg-mul-1 48.9%

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

   3. *-commutative 48.9%

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

   4. sub-neg 48.9%

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

   5. metadata-eval 48.9%

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

   6. +-commutative 48.9%

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

10. Simplified 48.9%

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

11. Final simplification 48.9%

    \[\leadsto y \cdot \left(\left(--1\right) - z\right) - t \]
12. Add Preprocessing

Alternative 11: 46.1% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

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 - (y * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. add-cbrt-cube 87.3%

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

   2. pow3 87.3%

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

4. Applied egg-rr 87.3%

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

5. Taylor expanded in y around 0 99.4%

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

   1. mul-1-neg 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in z around inf 48.7%

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

   1. associate-*r* 48.7%

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

   2. neg-mul-1 48.7%

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

10. Simplified 48.7%

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

11. Final simplification 48.7%

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

Alternative 12: 35.4% accurate, 107.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-define 87.6%

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

   3. sub-neg 87.6%

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

   4. metadata-eval 87.6%

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

   5. sub-neg 87.6%

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

   6. log1p-define 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 37.2%

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

   1. neg-mul-1 37.2%

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

7. Simplified 37.2%

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

Alternative 13: 2.3% accurate, 215.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 0.0)

C

double code(double x, double y, double z, double t) {
	return 0.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0;
}

Python

def code(x, y, z, t):
	return 0.0

Julia

function code(x, y, z, t)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_] := 0.0

Derivation

1. Initial program 87.6%

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

   1. +-commutative 87.6%

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

   2. fma-define 87.6%

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

   3. sub-neg 87.6%

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

   4. metadata-eval 87.6%

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

   5. sub-neg 87.6%

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

   6. log1p-define 99.8%

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

   7. sub-neg 99.8%

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

   8. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in t around inf 37.2%

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

   1. neg-mul-1 37.2%

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

7. Simplified 37.2%

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

   1. expm1-log1p-u 18.4%

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

   2. expm1-undefine 18.3%

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

9. Applied egg-rr 18.3%

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

    1. sub-neg 18.3%

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

    2. log1p-undefine 18.3%

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

    3. rem-exp-log 37.1%

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

    4. unsub-neg 37.1%

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

    5. metadata-eval 37.1%

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

11. Simplified 37.1%

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

12. Taylor expanded in t around 0 2.3%

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

    1. metadata-eval 2.3%

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

14. Applied egg-rr 2.3%

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

Reproduce

herbie shell --seed 2024163 
(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))