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

Percentage Accurate: 89.2% → 99.8%

Time: 21.3s

Alternatives: 20

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: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

   1. +-commutative 88.1%

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

   2. fma-def 88.1%

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

   3. sub-neg 88.1%

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

   4. metadata-eval 88.1%

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

   5. sub-neg 88.1%

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

   6. log1p-def 99.8%

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

   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. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in z around inf 88.0%

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

   1. *-commutative 88.0%

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

   2. sub-neg 88.0%

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

   3. mul-1-neg 88.0%

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

   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. Final simplification 99.7%

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

Alternative 3: 71.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := y \cdot \left(-z\right) - t\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+122}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-228}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (* y (- z)) t)))
   (if (<= x -1.65e+166)
     t_1
     (if (<= x -9e+122)
       t_2
       (if (<= x -5e+19)
         t_1
         (if (<= x -2.9e-228)
           t_2
           (if (<= x 3.9e+31) (- (- (log y)) t) t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (y * -z) - t;
	double tmp;
	if (x <= -1.65e+166) {
		tmp = t_1;
	} else if (x <= -9e+122) {
		tmp = t_2;
	} else if (x <= -5e+19) {
		tmp = t_1;
	} else if (x <= -2.9e-228) {
		tmp = t_2;
	} else if (x <= 3.9e+31) {
		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) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = (y * -z) - t
    if (x <= (-1.65d+166)) then
        tmp = t_1
    else if (x <= (-9d+122)) then
        tmp = t_2
    else if (x <= (-5d+19)) then
        tmp = t_1
    else if (x <= (-2.9d-228)) then
        tmp = t_2
    else if (x <= 3.9d+31) 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 t_2 = (y * -z) - t;
	double tmp;
	if (x <= -1.65e+166) {
		tmp = t_1;
	} else if (x <= -9e+122) {
		tmp = t_2;
	} else if (x <= -5e+19) {
		tmp = t_1;
	} else if (x <= -2.9e-228) {
		tmp = t_2;
	} else if (x <= 3.9e+31) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = (y * -z) - t
	tmp = 0
	if x <= -1.65e+166:
		tmp = t_1
	elif x <= -9e+122:
		tmp = t_2
	elif x <= -5e+19:
		tmp = t_1
	elif x <= -2.9e-228:
		tmp = t_2
	elif x <= 3.9e+31:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(y * Float64(-z)) - t)
	tmp = 0.0
	if (x <= -1.65e+166)
		tmp = t_1;
	elseif (x <= -9e+122)
		tmp = t_2;
	elseif (x <= -5e+19)
		tmp = t_1;
	elseif (x <= -2.9e-228)
		tmp = t_2;
	elseif (x <= 3.9e+31)
		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);
	t_2 = (y * -z) - t;
	tmp = 0.0;
	if (x <= -1.65e+166)
		tmp = t_1;
	elseif (x <= -9e+122)
		tmp = t_2;
	elseif (x <= -5e+19)
		tmp = t_1;
	elseif (x <= -2.9e-228)
		tmp = t_2;
	elseif (x <= 3.9e+31)
		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]}, Block[{t$95$2 = N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.65e+166], t$95$1, If[LessEqual[x, -9e+122], t$95$2, If[LessEqual[x, -5e+19], t$95$1, If[LessEqual[x, -2.9e-228], t$95$2, If[LessEqual[x, 3.9e+31], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6500000000000001e166 or -8.99999999999999995e122 < x < -5e19 or 3.89999999999999999e31 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0 99.0%

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

      1. mul-1-neg 99.0%

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

   5. Simplified 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. sub-neg 99.0%

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

      6. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 76.0%

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

      1. *-commutative 76.0%

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

   10. Simplified 76.0%

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

   if -1.6500000000000001e166 < x < -8.99999999999999995e122 or -5e19 < x < -2.9000000000000001e-228

   1. Initial program 67.2%

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

   3. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in z around inf 89.6%

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

      1. associate-*r* 89.6%

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

      2. neg-mul-1 89.6%

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

   8. Simplified 89.6%

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

   if -2.9000000000000001e-228 < x < 3.89999999999999999e31

   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-def 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-def 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 89.1%

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

      1. fma-neg 89.1%

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

      2. sub-neg 89.1%

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

      3. metadata-eval 89.1%

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

      4. +-commutative 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in x around 0 86.9%

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

      1. neg-mul-1 86.9%

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

   10. Simplified 86.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+166}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+122}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-228}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+19} \lor \neg \left(-1 + x \leq -0.99999999999995\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(\left(--1\right) - z\right) - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -5e+19) (not (<= (+ -1.0 x) -0.99999999999995)))
   (- (- (* x (log y)) (* y (+ z -1.0))) t)
   (- (- (* y (- (- -1.0) z)) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5e+19) || !((-1.0 + x) <= -0.99999999999995)) {
		tmp = ((x * log(y)) - (y * (z + -1.0))) - t;
	} else {
		tmp = ((y * (-(-1.0) - z)) - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-5d+19)) .or. (.not. (((-1.0d0) + x) <= (-0.99999999999995d0)))) then
        tmp = ((x * log(y)) - (y * (z + (-1.0d0)))) - t
    else
        tmp = ((y * (-(-1.0d0) - z)) - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -5e+19) || !((-1.0 + x) <= -0.99999999999995)) {
		tmp = ((x * Math.log(y)) - (y * (z + -1.0))) - t;
	} else {
		tmp = ((y * (-(-1.0) - z)) - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -5e+19) or not ((-1.0 + x) <= -0.99999999999995):
		tmp = ((x * math.log(y)) - (y * (z + -1.0))) - t
	else:
		tmp = ((y * (-(-1.0) - z)) - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -5e+19) || !(Float64(-1.0 + x) <= -0.99999999999995))
		tmp = Float64(Float64(Float64(x * log(y)) - Float64(y * Float64(z + -1.0))) - t);
	else
		tmp = Float64(Float64(Float64(y * Float64(Float64(-(-1.0)) - z)) - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -5e+19) || ~(((-1.0 + x) <= -0.99999999999995)))
		tmp = ((x * log(y)) - (y * (z + -1.0))) - t;
	else
		tmp = ((y * (-(-1.0) - z)) - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -5e+19], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.99999999999995]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(y * N[((--1.0) - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -5e19 or -0.99999999999995004 < (-.f64 x 1)

   1. Initial program 94.8%

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

      1. *-commutative 94.8%

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

      2. sub-neg 94.8%

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

      3. metadata-eval 94.8%

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

      4. flip-+ 53.8%

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

      5. associate-*r/ 53.7%

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

      6. metadata-eval 53.7%

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

      7. fma-neg 53.7%

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

      8. metadata-eval 53.7%

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

      9. sub-neg 53.7%

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

      10. metadata-eval 53.7%

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

      11. +-commutative 53.7%

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

   4. Applied egg-rr 53.7%

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

      1. associate-/l* 53.7%

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

   6. Simplified 53.7%

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

   7. Taylor expanded in x around inf 94.2%

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

   8. Taylor expanded in y around 0 98.6%

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

      1. log-pow 9.1%

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

      2. +-commutative 9.1%

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

      3. mul-1-neg 9.1%

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

      4. unsub-neg 9.1%

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

      5. log-pow 98.6%

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

      6. *-commutative 98.6%

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

      7. sub-neg 98.6%

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

      8. metadata-eval 98.6%

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

      9. +-commutative 98.6%

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

   10. Simplified 98.6%

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

   if -5e19 < (-.f64 x 1) < -0.99999999999995004

   1. Initial program 81.1%

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

   3. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. mul-1-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-neg-out 99.4%

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

      6. unsub-neg 99.4%

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

      7. +-commutative 99.4%

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

   8. Simplified 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+19} \lor \neg \left(-1 + x \leq -0.99999999999995\right):\\ \;\;\;\;\left(x \cdot \log y - y \cdot \left(z + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(\left(--1\right) - z\right) - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -0.99999995:\\ \;\;\;\;\left(y \cdot \left(\left(--1\right) - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -5e+19)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) -0.99999995)
     (- (- (* y (- (- -1.0) z)) (log y)) t)
     (- (+ y (* (log y) (+ -1.0 x))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5e+19) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= -0.99999995) {
		tmp = ((y * (-(-1.0) - z)) - log(y)) - t;
	} else {
		tmp = (y + (log(y) * (-1.0 + x))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((-1.0d0) + x) <= (-5d+19)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= (-0.99999995d0)) then
        tmp = ((y * (-(-1.0d0) - z)) - log(y)) - t
    else
        tmp = (y + (log(y) * ((-1.0d0) + x))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -5e+19) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= -0.99999995) {
		tmp = ((y * (-(-1.0) - z)) - Math.log(y)) - t;
	} else {
		tmp = (y + (Math.log(y) * (-1.0 + x))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -5e+19:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= -0.99999995:
		tmp = ((y * (-(-1.0) - z)) - math.log(y)) - t
	else:
		tmp = (y + (math.log(y) * (-1.0 + x))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5e+19)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= -0.99999995)
		tmp = Float64(Float64(Float64(y * Float64(Float64(-(-1.0)) - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(y + Float64(log(y) * Float64(-1.0 + x))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -5e+19)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= -0.99999995)
		tmp = ((y * (-(-1.0) - z)) - log(y)) - t;
	else
		tmp = (y + (log(y) * (-1.0 + x))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -5e+19], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.99999995], N[(N[(N[(y * N[((--1.0) - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y + N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -5e19

   1. Initial program 94.9%

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

   3. Taylor expanded in z around inf 94.9%

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

      1. *-commutative 94.9%

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

      2. sub-neg 94.9%

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

      3. mul-1-neg 94.9%

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

      4. log1p-def 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 94.9%

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

      1. *-commutative 94.9%

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

   8. Simplified 94.9%

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

   if -5e19 < (-.f64 x 1) < -0.999999949999999971

   1. Initial program 80.5%

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

   3. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 99.4%

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

      1. distribute-lft-out 99.4%

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

      2. mul-1-neg 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. distribute-neg-out 99.4%

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

      6. unsub-neg 99.4%

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

      7. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   if -0.999999949999999971 < (-.f64 x 1)

   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 y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around 0 94.8%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -0.99999995:\\ \;\;\;\;\left(y \cdot \left(\left(--1\right) - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+122}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+19} \lor \neg \left(x \leq 6.5 \cdot 10^{+29}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.65e+166)
     t_1
     (if (<= x -9e+122)
       (- (* y (- z)) t)
       (if (or (<= x -4.5e+19) (not (<= x 6.5e+29)))
         t_1
         (- (* y (- (- -1.0) z)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -1.65e+166) {
		tmp = t_1;
	} else if (x <= -9e+122) {
		tmp = (y * -z) - t;
	} else if ((x <= -4.5e+19) || !(x <= 6.5e+29)) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-1.65d+166)) then
        tmp = t_1
    else if (x <= (-9d+122)) then
        tmp = (y * -z) - t
    else if ((x <= (-4.5d+19)) .or. (.not. (x <= 6.5d+29))) then
        tmp = t_1
    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 t_1 = x * Math.log(y);
	double tmp;
	if (x <= -1.65e+166) {
		tmp = t_1;
	} else if (x <= -9e+122) {
		tmp = (y * -z) - t;
	} else if ((x <= -4.5e+19) || !(x <= 6.5e+29)) {
		tmp = t_1;
	} else {
		tmp = (y * (-(-1.0) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -1.65e+166:
		tmp = t_1
	elif x <= -9e+122:
		tmp = (y * -z) - t
	elif (x <= -4.5e+19) or not (x <= 6.5e+29):
		tmp = t_1
	else:
		tmp = (y * (-(-1.0) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.65e+166)
		tmp = t_1;
	elseif (x <= -9e+122)
		tmp = Float64(Float64(y * Float64(-z)) - t);
	elseif ((x <= -4.5e+19) || !(x <= 6.5e+29))
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(Float64(-(-1.0)) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -1.65e+166)
		tmp = t_1;
	elseif (x <= -9e+122)
		tmp = (y * -z) - t;
	elseif ((x <= -4.5e+19) || ~((x <= 6.5e+29)))
		tmp = t_1;
	else
		tmp = (y * (-(-1.0) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+166], t$95$1, If[LessEqual[x, -9e+122], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[x, -4.5e+19], N[Not[LessEqual[x, 6.5e+29]], $MachinePrecision]], t$95$1, N[(N[(y * N[((--1.0) - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6500000000000001e166 or -8.99999999999999995e122 < x < -4.5e19 or 6.49999999999999971e29 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0 99.0%

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

      1. mul-1-neg 99.0%

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

   5. Simplified 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. sub-neg 99.0%

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

      6. metadata-eval 99.0%

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

   7. Applied egg-rr 99.0%

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

   8. Taylor expanded in x around inf 76.0%

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

      1. *-commutative 76.0%

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

   10. Simplified 76.0%

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

   if -1.6500000000000001e166 < x < -8.99999999999999995e122

   1. Initial program 65.6%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf 92.8%

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

      1. associate-*r* 92.8%

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

      2. neg-mul-1 92.8%

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

   8. Simplified 92.8%

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

   if -4.5e19 < x < 6.49999999999999971e29

   1. Initial program 81.9%

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

   3. Taylor expanded in y around 0 99.4%

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

      1. mul-1-neg 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in y around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. sub-neg 73.4%

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

      3. metadata-eval 73.4%

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

      4. *-commutative 73.4%

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

      5. distribute-rgt-neg-out 73.4%

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

      6. +-commutative 73.4%

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

   8. Simplified 73.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+166}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -9 \cdot 10^{+122}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{+19} \lor \neg \left(x \leq 6.5 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \mathbf{elif}\;t \leq -4.85 \cdot 10^{-203} \lor \neg \left(t \leq -2.6 \cdot 10^{-305}\right) \land t \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;-\log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.15e-111)
   (- (* y (- (- -1.0) z)) t)
   (if (or (<= t -4.85e-203) (and (not (<= t -2.6e-305)) (<= t 1.8e-11)))
     (- (log y))
     (- (* y (- z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.15e-111) {
		tmp = (y * (-(-1.0) - z)) - t;
	} else if ((t <= -4.85e-203) || (!(t <= -2.6e-305) && (t <= 1.8e-11))) {
		tmp = -log(y);
	} else {
		tmp = (y * -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 (t <= (-3.15d-111)) then
        tmp = (y * (-(-1.0d0) - z)) - t
    else if ((t <= (-4.85d-203)) .or. (.not. (t <= (-2.6d-305))) .and. (t <= 1.8d-11)) then
        tmp = -log(y)
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.15e-111) {
		tmp = (y * (-(-1.0) - z)) - t;
	} else if ((t <= -4.85e-203) || (!(t <= -2.6e-305) && (t <= 1.8e-11))) {
		tmp = -Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.15e-111:
		tmp = (y * (-(-1.0) - z)) - t
	elif (t <= -4.85e-203) or (not (t <= -2.6e-305) and (t <= 1.8e-11)):
		tmp = -math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.15e-111)
		tmp = Float64(Float64(y * Float64(Float64(-(-1.0)) - z)) - t);
	elseif ((t <= -4.85e-203) || (!(t <= -2.6e-305) && (t <= 1.8e-11)))
		tmp = Float64(-log(y));
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.15e-111)
		tmp = (y * (-(-1.0) - z)) - t;
	elseif ((t <= -4.85e-203) || (~((t <= -2.6e-305)) && (t <= 1.8e-11)))
		tmp = -log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.15e-111], N[(N[(y * N[((--1.0) - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[t, -4.85e-203], And[N[Not[LessEqual[t, -2.6e-305]], $MachinePrecision], LessEqual[t, 1.8e-11]]], (-N[Log[y], $MachinePrecision]), N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1500000000000002e-111

   1. Initial program 89.6%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around inf 68.2%

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

      1. mul-1-neg 68.2%

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

      2. sub-neg 68.2%

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

      3. metadata-eval 68.2%

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

      4. *-commutative 68.2%

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

      5. distribute-rgt-neg-out 68.2%

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

      6. +-commutative 68.2%

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

   8. Simplified 68.2%

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

   if -3.1500000000000002e-111 < t < -4.8499999999999998e-203 or -2.6000000000000002e-305 < t < 1.79999999999999992e-11

   1. Initial program 89.2%

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

      1. +-commutative 89.2%

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

      2. fma-def 89.2%

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

      3. sub-neg 89.2%

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

      4. metadata-eval 89.2%

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

      5. sub-neg 89.2%

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

      6. log1p-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 y around 0 86.5%

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

      1. fma-neg 86.5%

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

      2. sub-neg 86.5%

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

      3. metadata-eval 86.5%

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

      4. +-commutative 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in x around 0 36.8%

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

      1. neg-mul-1 36.8%

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

   10. Simplified 36.8%

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

   11. Taylor expanded in t around 0 36.8%

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

       1. neg-mul-1 36.8%

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

   13. Simplified 36.8%

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

   if -4.8499999999999998e-203 < t < -2.6000000000000002e-305 or 1.79999999999999992e-11 < t

   1. Initial program 85.3%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around inf 71.6%

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

      1. associate-*r* 71.6%

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

      2. neg-mul-1 71.6%

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

   8. Simplified 71.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.15 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \left(\left(--1\right) - z\right) - t\\ \mathbf{elif}\;t \leq -4.85 \cdot 10^{-203} \lor \neg \left(t \leq -2.6 \cdot 10^{-305}\right) \land t \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;-\log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -95000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -95000000000000.0)
     t_1
     (if (<= x -1e-230)
       (- (* y (- z)) t)
       (if (<= x 9e-14) (- (- (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -95000000000000.0) {
		tmp = t_1;
	} else if (x <= -1e-230) {
		tmp = (y * -z) - t;
	} else if (x <= 9e-14) {
		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)) - t
    if (x <= (-95000000000000.0d0)) then
        tmp = t_1
    else if (x <= (-1d-230)) then
        tmp = (y * -z) - t
    else if (x <= 9d-14) 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)) - t;
	double tmp;
	if (x <= -95000000000000.0) {
		tmp = t_1;
	} else if (x <= -1e-230) {
		tmp = (y * -z) - t;
	} else if (x <= 9e-14) {
		tmp = -Math.log(y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -95000000000000.0:
		tmp = t_1
	elif x <= -1e-230:
		tmp = (y * -z) - t
	elif x <= 9e-14:
		tmp = -math.log(y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -95000000000000.0)
		tmp = t_1;
	elseif (x <= -1e-230)
		tmp = Float64(Float64(y * Float64(-z)) - t);
	elseif (x <= 9e-14)
		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)) - t;
	tmp = 0.0;
	if (x <= -95000000000000.0)
		tmp = t_1;
	elseif (x <= -1e-230)
		tmp = (y * -z) - t;
	elseif (x <= 9e-14)
		tmp = -log(y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -95000000000000.0], t$95$1, If[LessEqual[x, -1e-230], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 9e-14], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5e13 or 8.9999999999999995e-14 < x

   1. Initial program 94.8%

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

   3. Taylor expanded in z around inf 94.8%

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

      1. *-commutative 94.8%

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

      2. sub-neg 94.8%

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

      3. mul-1-neg 94.8%

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

      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 93.6%

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

      1. *-commutative 93.6%

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

   8. Simplified 93.6%

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

   if -9.5e13 < x < -1.00000000000000005e-230

   1. Initial program 67.4%

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

   3. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around inf 89.2%

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

      1. associate-*r* 89.2%

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

      2. neg-mul-1 89.2%

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

   8. Simplified 89.2%

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

   if -1.00000000000000005e-230 < x < 8.9999999999999995e-14

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. fma-def 89.3%

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

      3. sub-neg 89.3%

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

      4. metadata-eval 89.3%

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

      5. sub-neg 89.3%

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

      6. log1p-def 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 88.9%

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

      1. fma-neg 88.9%

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

      2. sub-neg 88.9%

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

      3. metadata-eval 88.9%

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

      4. +-commutative 88.9%

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

   7. Simplified 88.9%

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

   8. Taylor expanded in x around 0 88.8%

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

      1. neg-mul-1 88.8%

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

   10. Simplified 88.8%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95000000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-230}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -32000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-228}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -32000000000.0)
     t_1
     (if (<= x -2.9e-228)
       (- (* y (- z)) t)
       (if (<= x 9e-14) (- (- y (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -32000000000.0) {
		tmp = t_1;
	} else if (x <= -2.9e-228) {
		tmp = (y * -z) - t;
	} else if (x <= 9e-14) {
		tmp = (y - 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)) - t
    if (x <= (-32000000000.0d0)) then
        tmp = t_1
    else if (x <= (-2.9d-228)) then
        tmp = (y * -z) - t
    else if (x <= 9d-14) then
        tmp = (y - 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)) - t;
	double tmp;
	if (x <= -32000000000.0) {
		tmp = t_1;
	} else if (x <= -2.9e-228) {
		tmp = (y * -z) - t;
	} else if (x <= 9e-14) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -32000000000.0:
		tmp = t_1
	elif x <= -2.9e-228:
		tmp = (y * -z) - t
	elif x <= 9e-14:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -32000000000.0)
		tmp = t_1;
	elseif (x <= -2.9e-228)
		tmp = Float64(Float64(y * Float64(-z)) - t);
	elseif (x <= 9e-14)
		tmp = Float64(Float64(y - 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)) - t;
	tmp = 0.0;
	if (x <= -32000000000.0)
		tmp = t_1;
	elseif (x <= -2.9e-228)
		tmp = (y * -z) - t;
	elseif (x <= 9e-14)
		tmp = (y - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -32000000000.0], t$95$1, If[LessEqual[x, -2.9e-228], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 9e-14], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2e10 or 8.9999999999999995e-14 < x

   1. Initial program 94.8%

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

   3. Taylor expanded in z around inf 94.8%

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

      1. *-commutative 94.8%

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

      2. sub-neg 94.8%

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

      3. mul-1-neg 94.8%

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

      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 93.6%

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

      1. *-commutative 93.6%

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

   8. Simplified 93.6%

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

   if -3.2e10 < x < -2.9000000000000001e-228

   1. Initial program 67.4%

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

   3. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in z around inf 89.2%

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

      1. associate-*r* 89.2%

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

      2. neg-mul-1 89.2%

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

   8. Simplified 89.2%

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

   if -2.9000000000000001e-228 < x < 8.9999999999999995e-14

   1. Initial program 89.3%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0 89.3%

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

   7. Taylor expanded in x around 0 89.3%

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

      1. neg-mul-1 89.3%

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

      2. unsub-neg 89.3%

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

   9. Simplified 89.3%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -32000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-228}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -33000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-230}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -33000000000.0)
     t_1
     (if (<= x -3.3e-230)
       (- (* z (log1p (- y))) t)
       (if (<= x 9e-14) (- (- y (log y)) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -33000000000.0) {
		tmp = t_1;
	} else if (x <= -3.3e-230) {
		tmp = (z * log1p(-y)) - t;
	} else if (x <= 9e-14) {
		tmp = (y - log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -33000000000.0) {
		tmp = t_1;
	} else if (x <= -3.3e-230) {
		tmp = (z * Math.log1p(-y)) - t;
	} else if (x <= 9e-14) {
		tmp = (y - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -33000000000.0:
		tmp = t_1
	elif x <= -3.3e-230:
		tmp = (z * math.log1p(-y)) - t
	elif x <= 9e-14:
		tmp = (y - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -33000000000.0)
		tmp = t_1;
	elseif (x <= -3.3e-230)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	elseif (x <= 9e-14)
		tmp = Float64(Float64(y - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -33000000000.0], t$95$1, If[LessEqual[x, -3.3e-230], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 9e-14], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.3e10 or 8.9999999999999995e-14 < x

   1. Initial program 94.8%

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

   3. Taylor expanded in z around inf 94.8%

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

      1. *-commutative 94.8%

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

      2. sub-neg 94.8%

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

      3. mul-1-neg 94.8%

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

      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 93.6%

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

      1. *-commutative 93.6%

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

   8. Simplified 93.6%

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

   if -3.3e10 < x < -3.29999999999999994e-230

   1. Initial program 67.4%

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

   3. Taylor expanded in z around inf 67.4%

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

      1. *-commutative 67.4%

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

      2. sub-neg 67.4%

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

      3. mul-1-neg 67.4%

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

      4. log1p-def 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 z around inf 57.7%

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

      1. sub-neg 57.7%

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

      2. log1p-def 90.7%

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

   8. Simplified 90.7%

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

   if -3.29999999999999994e-230 < x < 8.9999999999999995e-14

   1. Initial program 89.3%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around 0 89.3%

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

   7. Taylor expanded in x around 0 89.3%

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

      1. neg-mul-1 89.3%

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

      2. unsub-neg 89.3%

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

   9. Simplified 89.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -33000000000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-230}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+219}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+250}:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9e+219)
   (- (* y (- z)) t)
   (if (<= z 2.15e+250)
     (- (+ y (* (log y) (+ -1.0 x))) t)
     (- (* z (log1p (- y))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e+219) {
		tmp = (y * -z) - t;
	} else if (z <= 2.15e+250) {
		tmp = (y + (log(y) * (-1.0 + x))) - 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 (z <= -9e+219) {
		tmp = (y * -z) - t;
	} else if (z <= 2.15e+250) {
		tmp = (y + (Math.log(y) * (-1.0 + x))) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9e+219:
		tmp = (y * -z) - t
	elif z <= 2.15e+250:
		tmp = (y + (math.log(y) * (-1.0 + x))) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9e+219)
		tmp = Float64(Float64(y * Float64(-z)) - t);
	elseif (z <= 2.15e+250)
		tmp = Float64(Float64(y + Float64(log(y) * Float64(-1.0 + x))) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9e+219], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 2.15e+250], N[(N[(y + N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.00000000000000047e219

   1. Initial program 37.1%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 85.2%

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

      1. associate-*r* 85.2%

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

      2. neg-mul-1 85.2%

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

   8. Simplified 85.2%

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

   if -9.00000000000000047e219 < z < 2.15e250

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. mul-1-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in z around 0 95.4%

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

   if 2.15e250 < z

   1. Initial program 39.1%

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

   3. Taylor expanded in z around inf 39.1%

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

      1. *-commutative 39.1%

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

      2. sub-neg 39.1%

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

      3. mul-1-neg 39.1%

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

      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 z around inf 39.2%

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

      1. sub-neg 39.2%

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

      2. log1p-def 91.8%

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

   8. Simplified 91.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+219}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{+250}:\\ \;\;\;\;\left(y + \log y \cdot \left(-1 + x\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+220}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+251}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.8e+220)
   (- (* y (- z)) t)
   (if (<= z 3.6e+251)
     (- (* (log y) (+ -1.0 x)) t)
     (- (* z (log1p (- y))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+220) {
		tmp = (y * -z) - t;
	} else if (z <= 3.6e+251) {
		tmp = (log(y) * (-1.0 + x)) - 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 (z <= -3.8e+220) {
		tmp = (y * -z) - t;
	} else if (z <= 3.6e+251) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.8e+220:
		tmp = (y * -z) - t
	elif z <= 3.6e+251:
		tmp = (math.log(y) * (-1.0 + x)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+220)
		tmp = Float64(Float64(y * Float64(-z)) - t);
	elseif (z <= 3.6e+251)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.8e+220], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[z, 3.6e+251], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.79999999999999984e220

   1. Initial program 37.1%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 85.2%

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

      1. associate-*r* 85.2%

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

      2. neg-mul-1 85.2%

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

   8. Simplified 85.2%

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

   if -3.79999999999999984e220 < z < 3.59999999999999997e251

   1. Initial program 95.8%

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

      1. +-commutative 95.8%

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

      2. fma-def 95.8%

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

      3. sub-neg 95.8%

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

      4. metadata-eval 95.8%

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

      5. sub-neg 95.8%

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

      6. log1p-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 y around 0 95.2%

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

   if 3.59999999999999997e251 < z

   1. Initial program 39.1%

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

   3. Taylor expanded in z around inf 39.1%

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

      1. *-commutative 39.1%

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

      2. sub-neg 39.1%

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

      3. mul-1-neg 39.1%

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

      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 z around inf 39.2%

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

      1. sub-neg 39.2%

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

      2. log1p-def 91.8%

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

   8. Simplified 91.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+220}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{+251}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 14: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in z around inf 88.0%

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

   1. *-commutative 88.0%

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

   2. sub-neg 88.0%

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

   3. mul-1-neg 88.0%

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

   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 y around 0 99.1%

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

   1. mul-1-neg 99.1%

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

   2. *-commutative 99.1%

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

   3. distribute-rgt-neg-in 99.1%

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

8. Simplified 99.1%

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

9. Final simplification 99.1%

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

Alternative 15: 43.3% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-16} \lor \neg \left(t \leq 8.5 \cdot 10^{+27}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.2e-16) (not (<= t 8.5e+27))) (- t) (* y (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.2e-16) || !(t <= 8.5e+27)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - 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 <= (-4.2d-16)) .or. (.not. (t <= 8.5d+27))) then
        tmp = -t
    else
        tmp = y * (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.2e-16) || !(t <= 8.5e+27)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.2e-16) or not (t <= 8.5e+27):
		tmp = -t
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.2e-16) || !(t <= 8.5e+27))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.2e-16) || ~((t <= 8.5e+27)))
		tmp = -t;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.2e-16], N[Not[LessEqual[t, 8.5e+27]], $MachinePrecision]], (-t), N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2000000000000002e-16 or 8.5e27 < t

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. fma-def 96.7%

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

      3. sub-neg 96.7%

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

      4. metadata-eval 96.7%

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

      5. sub-neg 96.7%

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

      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 73.8%

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

      1. neg-mul-1 73.8%

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

   7. Simplified 73.8%

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

   if -4.2000000000000002e-16 < t < 8.5e27

   1. Initial program 78.3%

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

   3. Taylor expanded in y around 0 98.6%

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

      1. mul-1-neg 98.6%

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

   5. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-def 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. sub-neg 98.6%

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

      6. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in y around inf 23.5%

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

      1. mul-1-neg 23.5%

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

      2. sub-neg 23.5%

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

      3. metadata-eval 23.5%

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

      4. distribute-rgt-neg-in 23.5%

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

      5. +-commutative 23.5%

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

      6. distribute-neg-in 23.5%

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

      7. metadata-eval 23.5%

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

   10. Simplified 23.5%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-16} \lor \neg \left(t \leq 8.5 \cdot 10^{+27}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-16} \lor \neg \left(t \leq 1.75 \cdot 10^{+21}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.3e-16) (not (<= t 1.75e+21))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.3e-16) || !(t <= 1.75e+21)) {
		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 <= (-3.3d-16)) .or. (.not. (t <= 1.75d+21))) 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 <= -3.3e-16) || !(t <= 1.75e+21)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.3e-16) or not (t <= 1.75e+21):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.3e-16) || !(t <= 1.75e+21))
		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 <= -3.3e-16) || ~((t <= 1.75e+21)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.3e-16], N[Not[LessEqual[t, 1.75e+21]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.29999999999999988e-16 or 1.75e21 < t

   1. Initial program 96.7%

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

      1. +-commutative 96.7%

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

      2. fma-def 96.7%

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

      3. sub-neg 96.7%

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

      4. metadata-eval 96.7%

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

      5. sub-neg 96.7%

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

      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 73.8%

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

      1. neg-mul-1 73.8%

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

   7. Simplified 73.8%

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

   if -3.29999999999999988e-16 < t < 1.75e21

   1. Initial program 78.3%

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

   3. Taylor expanded in y around 0 98.6%

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

      1. mul-1-neg 98.6%

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

   5. Simplified 98.6%

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

      1. +-commutative 98.6%

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

      2. fma-def 98.6%

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

      3. sub-neg 98.6%

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

      4. metadata-eval 98.6%

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

      5. sub-neg 98.6%

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

      6. metadata-eval 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in z around inf 23.2%

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

      1. mul-1-neg 23.2%

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

      2. distribute-rgt-neg-in 23.2%

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

   10. Simplified 23.2%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-16} \lor \neg \left(t \leq 1.75 \cdot 10^{+21}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.9% 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 88.1%

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

3. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in y around inf 52.1%

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

   1. mul-1-neg 52.1%

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

   2. sub-neg 52.1%

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

   3. metadata-eval 52.1%

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

   4. *-commutative 52.1%

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

   5. distribute-rgt-neg-out 52.1%

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

   6. +-commutative 52.1%

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

8. Simplified 52.1%

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

9. Final simplification 52.1%

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

Alternative 18: 46.7% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in z around inf 52.0%

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

   1. associate-*r* 52.0%

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

   2. neg-mul-1 52.0%

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

8. Simplified 52.0%

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

9. Final simplification 52.0%

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

Alternative 19: 36.2% 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 88.1%

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

   1. +-commutative 88.1%

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

   2. fma-def 88.1%

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

   3. sub-neg 88.1%

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

   4. metadata-eval 88.1%

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

   5. sub-neg 88.1%

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

   6. log1p-def 99.8%

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

   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 40.8%

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

   1. neg-mul-1 40.8%

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

7. Simplified 40.8%

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

8. Final simplification 40.8%

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

Alternative 20: 2.8% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in z around 0 87.4%

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

7. Taylor expanded in y around inf 2.8%

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

8. Final simplification 2.8%

   \[\leadsto y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))