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

Percentage Accurate: 89.1% → 99.6%

Time: 16.9s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log1p(-y) * z) + ((x + -1.0) * Math.log(y))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log1p(-y) * z) + ((x + -1.0) * math.log(y))) - t

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in z around inf 88.4%

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

   1. *-commutative 88.4%

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

   2. sub-neg 88.4%

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

   3. mul-1-neg 88.4%

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

   4. log1p-def 99.8%

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

   5. mul-1-neg 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

3. Taylor expanded in 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. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. distribute-rgt-neg-in 99.3%

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

   5. mul-1-neg 99.3%

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

   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. Step-by-step derivation

   1. associate--l+ 99.3%

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

   2. +-commutative 99.3%

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

   3. fma-neg 99.4%

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

   4. sub-neg 99.4%

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

   5. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 95.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + -1 \leq -4000000000 \lor \neg \left(x + -1 \leq -1\right):\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(-z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ x -1.0) -4000000000.0) (not (<= (+ x -1.0) -1.0)))
   (- (* (+ x -1.0) (log y)) t)
   (- (- (* y (- z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -4000000000.0) || !((x + -1.0) <= -1.0)) {
		tmp = ((x + -1.0) * log(y)) - t;
	} else {
		tmp = ((y * -z) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x + (-1.0d0)) <= (-4000000000.0d0)) .or. (.not. ((x + (-1.0d0)) <= (-1.0d0)))) then
        tmp = ((x + (-1.0d0)) * log(y)) - t
    else
        tmp = ((y * -z) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x + -1.0) <= -4000000000.0) || !((x + -1.0) <= -1.0)) {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	} else {
		tmp = ((y * -z) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x + -1.0) <= -4000000000.0) or not ((x + -1.0) <= -1.0):
		tmp = ((x + -1.0) * math.log(y)) - t
	else:
		tmp = ((y * -z) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x + -1.0) <= -4000000000.0) || !(Float64(x + -1.0) <= -1.0))
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(-z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x + -1.0) <= -4000000000.0) || ~(((x + -1.0) <= -1.0)))
		tmp = ((x + -1.0) * log(y)) - t;
	else
		tmp = ((y * -z) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x + -1.0), $MachinePrecision], -4000000000.0], N[Not[LessEqual[N[(x + -1.0), $MachinePrecision], -1.0]], $MachinePrecision]], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * (-z)), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -4e9 or -1 < (-.f64 x 1)

   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-def 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-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. Taylor expanded in y around 0 96.4%

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

   if -4e9 < (-.f64 x 1) < -1

   1. Initial program 79.9%

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

   3. Taylor expanded in z around inf 79.9%

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

      1. *-commutative 79.9%

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

      2. sub-neg 79.9%

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

      3. mul-1-neg 79.9%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 79.8%

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

      1. neg-mul-1 79.8%

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

      2. +-commutative 79.8%

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

      3. unsub-neg 79.8%

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

      4. sub-neg 79.8%

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

      5. log1p-def 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around 0 98.4%

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

       1. neg-mul-1 98.4%

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

       2. +-commutative 98.4%

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

       3. unsub-neg 98.4%

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

       4. mul-1-neg 98.4%

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

       5. distribute-rgt-neg-in 98.4%

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

   11. Simplified 98.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -4000000000 \lor \neg \left(x + -1 \leq -1\right):\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(-z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) - \log y\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- t) (log y))) (t_2 (- (* x (log y)) t)))
   (if (<= x -62.0)
     t_2
     (if (<= x 4.05e-113)
       t_1
       (if (<= x 9e-109) (* y (- z)) (if (<= x 1.72e-13) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -t - log(y);
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (x <= -62.0) {
		tmp = t_2;
	} else if (x <= 4.05e-113) {
		tmp = t_1;
	} else if (x <= 9e-109) {
		tmp = y * -z;
	} else if (x <= 1.72e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = -t - log(y)
    t_2 = (x * log(y)) - t
    if (x <= (-62.0d0)) then
        tmp = t_2
    else if (x <= 4.05d-113) then
        tmp = t_1
    else if (x <= 9d-109) then
        tmp = y * -z
    else if (x <= 1.72d-13) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -t - Math.log(y);
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -62.0) {
		tmp = t_2;
	} else if (x <= 4.05e-113) {
		tmp = t_1;
	} else if (x <= 9e-109) {
		tmp = y * -z;
	} else if (x <= 1.72e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -t - math.log(y)
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if x <= -62.0:
		tmp = t_2
	elif x <= 4.05e-113:
		tmp = t_1
	elif x <= 9e-109:
		tmp = y * -z
	elif x <= 1.72e-13:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-t) - log(y))
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -62.0)
		tmp = t_2;
	elseif (x <= 4.05e-113)
		tmp = t_1;
	elseif (x <= 9e-109)
		tmp = Float64(y * Float64(-z));
	elseif (x <= 1.72e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -t - log(y);
	t_2 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = t_2;
	elseif (x <= 4.05e-113)
		tmp = t_1;
	elseif (x <= 9e-109)
		tmp = y * -z;
	elseif (x <= 1.72e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -62.0], t$95$2, If[LessEqual[x, 4.05e-113], t$95$1, If[LessEqual[x, 9e-109], N[(y * (-z)), $MachinePrecision], If[LessEqual[x, 1.72e-13], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62 or 1.71999999999999999e-13 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf 96.0%

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

      1. *-commutative 96.0%

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

      2. sub-neg 96.0%

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

      3. mul-1-neg 96.0%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if -62 < x < 4.05000000000000022e-113 or 9.0000000000000002e-109 < x < 1.71999999999999999e-13

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf 82.5%

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

      1. *-commutative 82.5%

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

      2. sub-neg 82.5%

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

      3. mul-1-neg 82.5%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 82.5%

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

      1. neg-mul-1 82.5%

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

      2. +-commutative 82.5%

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

      3. unsub-neg 82.5%

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

      4. sub-neg 82.5%

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

      5. log1p-def 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0 82.0%

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

       1. neg-mul-1 82.0%

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

   11. Simplified 82.0%

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

   if 4.05000000000000022e-113 < x < 9.0000000000000002e-109

   1. Initial program 28.2%

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

   3. Taylor expanded in z around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. sub-neg 28.2%

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

      3. mul-1-neg 28.2%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 83.9%

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

      1. +-commutative 83.9%

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

      2. sub-neg 83.9%

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

      3. metadata-eval 83.9%

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

      4. *-commutative 83.9%

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

      5. mul-1-neg 83.9%

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

      6. unsub-neg 83.9%

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

      7. *-commutative 83.9%

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

      8. +-commutative 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in y around inf 83.9%

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

       1. associate-*r* 83.9%

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

       2. neg-mul-1 83.9%

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

       3. *-commutative 83.9%

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

   11. Simplified 83.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{-113}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-13}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) - \log y\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-z\right) - \log y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- t) (log y))) (t_2 (- (* x (log y)) t)))
   (if (<= x -62.0)
     t_2
     (if (<= x 4.1e-155)
       t_1
       (if (<= x 9e-109)
         (- (* y (- z)) (log y))
         (if (<= x 1.72e-13) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -t - log(y);
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (x <= -62.0) {
		tmp = t_2;
	} else if (x <= 4.1e-155) {
		tmp = t_1;
	} else if (x <= 9e-109) {
		tmp = (y * -z) - log(y);
	} else if (x <= 1.72e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = -t - log(y)
    t_2 = (x * log(y)) - t
    if (x <= (-62.0d0)) then
        tmp = t_2
    else if (x <= 4.1d-155) then
        tmp = t_1
    else if (x <= 9d-109) then
        tmp = (y * -z) - log(y)
    else if (x <= 1.72d-13) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -t - Math.log(y);
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -62.0) {
		tmp = t_2;
	} else if (x <= 4.1e-155) {
		tmp = t_1;
	} else if (x <= 9e-109) {
		tmp = (y * -z) - Math.log(y);
	} else if (x <= 1.72e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -t - math.log(y)
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if x <= -62.0:
		tmp = t_2
	elif x <= 4.1e-155:
		tmp = t_1
	elif x <= 9e-109:
		tmp = (y * -z) - math.log(y)
	elif x <= 1.72e-13:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-t) - log(y))
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -62.0)
		tmp = t_2;
	elseif (x <= 4.1e-155)
		tmp = t_1;
	elseif (x <= 9e-109)
		tmp = Float64(Float64(y * Float64(-z)) - log(y));
	elseif (x <= 1.72e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -t - log(y);
	t_2 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -62.0)
		tmp = t_2;
	elseif (x <= 4.1e-155)
		tmp = t_1;
	elseif (x <= 9e-109)
		tmp = (y * -z) - log(y);
	elseif (x <= 1.72e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -62.0], t$95$2, If[LessEqual[x, 4.1e-155], t$95$1, If[LessEqual[x, 9e-109], N[(N[(y * (-z)), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.72e-13], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62 or 1.71999999999999999e-13 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf 96.0%

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

      1. *-commutative 96.0%

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

      2. sub-neg 96.0%

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

      3. mul-1-neg 96.0%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if -62 < x < 4.0999999999999998e-155 or 9.0000000000000002e-109 < x < 1.71999999999999999e-13

   1. Initial program 82.2%

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

   3. Taylor expanded in z around inf 82.2%

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

      1. *-commutative 82.2%

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

      2. sub-neg 82.2%

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

      3. mul-1-neg 82.2%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 82.2%

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

      1. neg-mul-1 82.2%

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

      2. +-commutative 82.2%

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

      3. unsub-neg 82.2%

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

      4. sub-neg 82.2%

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

      5. log1p-def 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0 81.7%

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

       1. neg-mul-1 81.7%

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

   11. Simplified 81.7%

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

   if 4.0999999999999998e-155 < x < 9.0000000000000002e-109

   1. Initial program 48.7%

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

   3. Taylor expanded in z around inf 48.7%

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

      1. *-commutative 48.7%

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

      2. sub-neg 48.7%

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

      3. mul-1-neg 48.7%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. +-commutative 48.7%

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

      3. unsub-neg 48.7%

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

      4. sub-neg 48.7%

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

      5. log1p-def 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around 0 88.5%

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

       1. distribute-lft-out 88.5%

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

       2. mul-1-neg 88.5%

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

       3. +-commutative 88.5%

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

       4. fma-def 88.5%

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

   11. Simplified 88.5%

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

   12. Taylor expanded in t around 0 88.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{-155}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-109}:\\ \;\;\;\;y \cdot \left(-z\right) - \log y\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-13}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-t\right) - \log y\\ t_2 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -62:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- t) (log y))) (t_2 (- (* x (log y)) t)))
   (if (<= x -62.0)
     t_2
     (if (<= x 1.65e-114)
       t_1
       (if (<= x 4.9e-108)
         (- (* (log1p (- y)) z) t)
         (if (<= x 1.72e-13) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -t - log(y);
	double t_2 = (x * log(y)) - t;
	double tmp;
	if (x <= -62.0) {
		tmp = t_2;
	} else if (x <= 1.65e-114) {
		tmp = t_1;
	} else if (x <= 4.9e-108) {
		tmp = (log1p(-y) * z) - t;
	} else if (x <= 1.72e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -t - Math.log(y);
	double t_2 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -62.0) {
		tmp = t_2;
	} else if (x <= 1.65e-114) {
		tmp = t_1;
	} else if (x <= 4.9e-108) {
		tmp = (Math.log1p(-y) * z) - t;
	} else if (x <= 1.72e-13) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -t - math.log(y)
	t_2 = (x * math.log(y)) - t
	tmp = 0
	if x <= -62.0:
		tmp = t_2
	elif x <= 1.65e-114:
		tmp = t_1
	elif x <= 4.9e-108:
		tmp = (math.log1p(-y) * z) - t
	elif x <= 1.72e-13:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-t) - log(y))
	t_2 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -62.0)
		tmp = t_2;
	elseif (x <= 1.65e-114)
		tmp = t_1;
	elseif (x <= 4.9e-108)
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	elseif (x <= 1.72e-13)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -62.0], t$95$2, If[LessEqual[x, 1.65e-114], t$95$1, If[LessEqual[x, 4.9e-108], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.72e-13], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -62 or 1.71999999999999999e-13 < x

   1. Initial program 96.0%

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

   3. Taylor expanded in z around inf 96.0%

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

      1. *-commutative 96.0%

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

      2. sub-neg 96.0%

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

      3. mul-1-neg 96.0%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if -62 < x < 1.65000000000000017e-114 or 4.8999999999999998e-108 < x < 1.71999999999999999e-13

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf 82.5%

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

      1. *-commutative 82.5%

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

      2. sub-neg 82.5%

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

      3. mul-1-neg 82.5%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 82.5%

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

      1. neg-mul-1 82.5%

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

      2. +-commutative 82.5%

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

      3. unsub-neg 82.5%

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

      4. sub-neg 82.5%

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

      5. log1p-def 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around 0 82.0%

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

       1. neg-mul-1 82.0%

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

   11. Simplified 82.0%

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

   if 1.65000000000000017e-114 < x < 4.8999999999999998e-108

   1. Initial program 28.2%

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

   3. Taylor expanded in z around inf 28.2%

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

      1. *-commutative 28.2%

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

      2. sub-neg 28.2%

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

      3. mul-1-neg 28.2%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 28.2%

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

      1. neg-mul-1 28.2%

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

      2. +-commutative 28.2%

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

      3. unsub-neg 28.2%

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

      4. sub-neg 28.2%

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

      5. log1p-def 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around inf 28.4%

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

       1. sub-neg 28.4%

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

       2. log1p-def 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -62:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-114}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{-13}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% 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 88.4%

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

3. Taylor expanded in z around inf 88.4%

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

   1. *-commutative 88.4%

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

   2. sub-neg 88.4%

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

   3. mul-1-neg 88.4%

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

   4. log1p-def 99.8%

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

   5. mul-1-neg 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. sub-neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. *-commutative 99.3%

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

   5. mul-1-neg 99.3%

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

   6. unsub-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. +-commutative 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 8: 88.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{+171}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.15e+171) (- (* (+ x -1.0) (log y)) t) (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.15e+171) {
		tmp = ((x + -1.0) * log(y)) - t;
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.15e+171) {
		tmp = ((x + -1.0) * Math.log(y)) - t;
	} else {
		tmp = (Math.log1p(-y) * z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.15e+171:
		tmp = ((x + -1.0) * math.log(y)) - t
	else:
		tmp = (math.log1p(-y) * z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.15e+171)
		tmp = Float64(Float64(Float64(x + -1.0) * log(y)) - t);
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.15e+171], N[(N[(N[(x + -1.0), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.1500000000000002e171

   1. Initial program 93.3%

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

      1. +-commutative 93.3%

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

      2. fma-def 93.3%

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

      3. sub-neg 93.3%

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

      4. metadata-eval 93.3%

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

      5. sub-neg 93.3%

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

      6. log1p-def 99.9%

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

      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 y around 0 92.8%

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

   if 3.1500000000000002e171 < z

   1. Initial program 40.5%

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

   3. Taylor expanded in z around inf 40.5%

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

      1. *-commutative 40.5%

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

      2. sub-neg 40.5%

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

      3. mul-1-neg 40.5%

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

      4. log1p-def 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around 0 22.0%

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

      1. neg-mul-1 22.0%

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

      2. +-commutative 22.0%

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

      3. unsub-neg 22.0%

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

      4. sub-neg 22.0%

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

      5. log1p-def 81.3%

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

   8. Simplified 81.3%

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

   9. Taylor expanded in z around inf 22.3%

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

       1. sub-neg 22.3%

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

       2. log1p-def 76.4%

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

   11. Simplified 76.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{+171}:\\ \;\;\;\;\left(x + -1\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+241} \lor \neg \left(z \leq 1.8 \cdot 10^{+174}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e+241) (not (<= z 1.8e+174)))
   (* y (- z))
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+241) || !(z <= 1.8e+174)) {
		tmp = y * -z;
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.25d+241)) .or. (.not. (z <= 1.8d+174))) then
        tmp = y * -z
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e+241) || !(z <= 1.8e+174)) {
		tmp = y * -z;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e+241) or not (z <= 1.8e+174):
		tmp = y * -z
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.25e+241) || !(z <= 1.8e+174))
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.25e+241) || ~((z <= 1.8e+174)))
		tmp = y * -z;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.25e+241], N[Not[LessEqual[z, 1.8e+174]], $MachinePrecision]], N[(y * (-z)), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000006e241 or 1.8000000000000001e174 < z

   1. Initial program 46.7%

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

   3. Taylor expanded in z around inf 46.7%

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

      1. *-commutative 46.7%

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

      2. sub-neg 46.7%

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

      3. mul-1-neg 46.7%

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

      4. log1p-def 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. sub-neg 98.6%

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

      3. metadata-eval 98.6%

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

      4. *-commutative 98.6%

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

      5. mul-1-neg 98.6%

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

      6. unsub-neg 98.6%

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

      7. *-commutative 98.6%

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

      8. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in y around inf 50.6%

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

       1. associate-*r* 50.6%

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

       2. neg-mul-1 50.6%

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

       3. *-commutative 50.6%

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

   11. Simplified 50.6%

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

   if -1.25000000000000006e241 < z < 1.8000000000000001e174

   1. Initial program 96.3%

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

   3. Taylor expanded in z around inf 96.3%

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

      1. *-commutative 96.3%

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

      2. sub-neg 96.3%

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

      3. mul-1-neg 96.3%

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

      4. log1p-def 99.9%

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

      5. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 60.6%

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

      1. neg-mul-1 60.6%

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

      2. +-commutative 60.6%

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

      3. unsub-neg 60.6%

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

      4. sub-neg 60.6%

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

      5. log1p-def 63.8%

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

   8. Simplified 63.8%

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

   9. Taylor expanded in z around 0 60.1%

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

       1. neg-mul-1 60.1%

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

   11. Simplified 60.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+241} \lor \neg \left(z \leq 1.8 \cdot 10^{+174}\right):\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.9% accurate, 26.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -80000000000000 \lor \neg \left(t \leq 0.3\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -80000000000000.0) (not (<= t 0.3))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -80000000000000.0) || !(t <= 0.3)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-80000000000000.0d0)) .or. (.not. (t <= 0.3d0))) then
        tmp = -t
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -80000000000000.0) || !(t <= 0.3)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -80000000000000.0) or not (t <= 0.3):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -80000000000000.0) || !(t <= 0.3))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -80000000000000.0) || ~((t <= 0.3)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -80000000000000.0], N[Not[LessEqual[t, 0.3]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8e13 or 0.299999999999999989 < t

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. fma-def 94.1%

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

      3. sub-neg 94.1%

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

      4. metadata-eval 94.1%

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

      5. sub-neg 94.1%

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

      6. log1p-def 99.9%

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

      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 t around inf 70.1%

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

      1. neg-mul-1 70.1%

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

   7. Simplified 70.1%

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

   if -8e13 < t < 0.299999999999999989

   1. Initial program 82.0%

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

   3. Taylor expanded in z around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. sub-neg 82.0%

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

      3. mul-1-neg 82.0%

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

      4. log1p-def 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 99.5%

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

      1. +-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. mul-1-neg 99.5%

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

      6. unsub-neg 99.5%

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

      7. *-commutative 99.5%

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

      8. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in y around inf 18.8%

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

       1. associate-*r* 18.8%

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

       2. neg-mul-1 18.8%

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

       3. *-commutative 18.8%

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

   11. Simplified 18.8%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -80000000000000 \lor \neg \left(t \leq 0.3\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.6% 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 88.4%

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

   1. +-commutative 88.4%

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

   2. fma-def 88.4%

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

   3. sub-neg 88.4%

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

   4. metadata-eval 88.4%

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

   5. sub-neg 88.4%

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

   6. log1p-def 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 38.4%

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

   1. neg-mul-1 38.4%

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

7. Simplified 38.4%

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

8. Final simplification 38.4%

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

Reproduce

herbie shell --seed 2024013 
(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))