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

Percentage Accurate: 89.8% → 99.8%

Time: 19.0s

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: 89.8% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

3. Taylor expanded in z around 0 88.1%

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

   1. distribute-rgt-out 88.1%

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

   2. sub-neg 88.1%

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

   3. mul-1-neg 88.1%

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

   4. log1p-def 99.8%

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

   5. mul-1-neg 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(1.0 - z), $MachinePrecision] + N[(N[(-1.0 + x), $MachinePrecision] * N[Log[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 0 88.1%

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

   1. distribute-rgt-out 88.1%

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

   2. sub-neg 88.1%

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

   3. mul-1-neg 88.1%

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

   4. log1p-def 99.8%

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

   5. mul-1-neg 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in y around 0 98.8%

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

   1. sub-neg 98.8%

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

   2. metadata-eval 98.8%

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

   3. +-commutative 98.8%

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

   4. neg-mul-1 98.8%

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

   5. distribute-rgt-neg-in 98.8%

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

   6. distribute-neg-in 98.8%

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

   7. metadata-eval 98.8%

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

   8. sub-neg 98.8%

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

   9. fma-def 98.8%

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

   10. sub-neg 98.8%

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

   11. metadata-eval 98.8%

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

   12. +-commutative 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 4: 95.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -4e+30)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) -1.0)
     (- (* y (- 1.0 z)) (+ (log y) t))
     (- (* (log y) (+ -1.0 x)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -4e+30) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = (y * (1.0 - z)) - (log(y) + t);
	} else {
		tmp = (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) <= (-4d+30)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = (y * (1.0d0 - z)) - (log(y) + t)
    else
        tmp = (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) <= -4e+30) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = (y * (1.0 - z)) - (Math.log(y) + t);
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -4e+30:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= -1.0:
		tmp = (y * (1.0 - z)) - (math.log(y) + t)
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -4e+30)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(y * Float64(1.0 - z)) - Float64(log(y) + t));
	else
		tmp = Float64(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) <= -4e+30)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= -1.0)
		tmp = (y * (1.0 - z)) - (log(y) + t);
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -4e+30], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -4.0000000000000001e30

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0 95.2%

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

      1. distribute-rgt-out 95.2%

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

      2. sub-neg 95.2%

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

      3. mul-1-neg 95.2%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if -4.0000000000000001e30 < (-.f64 x 1) < -1

   1. Initial program 79.3%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. sub-neg 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-commutative 98.3%

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

      5. mul-1-neg 98.3%

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

      6. unsub-neg 98.3%

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

      7. *-commutative 98.3%

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

      8. +-commutative 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

      11. +-commutative 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in y around 0 97.2%

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

   10. Taylor expanded in t around 0 97.2%

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

       1. neg-mul-1 97.2%

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

       2. neg-mul-1 97.2%

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

       3. sub-neg 97.2%

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

       4. metadata-eval 97.2%

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

       5. distribute-neg-in 97.2%

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

       6. distribute-rgt-neg-in 97.2%

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

       7. sub-neg 97.2%

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

       8. associate-+r- 97.2%

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

       9. neg-mul-1 97.2%

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

       10. remove-double-neg 97.2%

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

       11. mul-1-neg 97.2%

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

       12. log-rec 97.2%

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

       13. neg-mul-1 97.2%

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

       14. distribute-lft-in 97.2%

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

       15. unsub-neg 97.2%

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

       16. distribute-rgt-neg-in 97.2%

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

       17. distribute-neg-in 97.2%

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

       18. metadata-eval 97.2%

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

   12. Simplified 97.2%

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

   if -1 < (-.f64 x 1)

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

      2. fma-def 97.8%

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

      3. sub-neg 97.8%

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

      4. metadata-eval 97.8%

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

      5. sub-neg 97.8%

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

      6. log1p-def 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.8%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;y \cdot \left(1 - z\right) - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;y \cdot \left(1 - z\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -1.0)
     t_1
     (if (<= x 5.2e-70)
       (- y (+ (log y) t))
       (if (<= x 3.2e-15) (- (* y (- 1.0 z)) (log y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= 5.2e-70) {
		tmp = y - (log(y) + t);
	} else if (x <= 3.2e-15) {
		tmp = (y * (1.0 - z)) - log(y);
	} 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 <= (-1.0d0)) then
        tmp = t_1
    else if (x <= 5.2d-70) then
        tmp = y - (log(y) + t)
    else if (x <= 3.2d-15) then
        tmp = (y * (1.0d0 - z)) - log(y)
    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 <= -1.0) {
		tmp = t_1;
	} else if (x <= 5.2e-70) {
		tmp = y - (Math.log(y) + t);
	} else if (x <= 3.2e-15) {
		tmp = (y * (1.0 - z)) - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -1.0:
		tmp = t_1
	elif x <= 5.2e-70:
		tmp = y - (math.log(y) + t)
	elif x <= 3.2e-15:
		tmp = (y * (1.0 - z)) - math.log(y)
	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 <= -1.0)
		tmp = t_1;
	elseif (x <= 5.2e-70)
		tmp = Float64(y - Float64(log(y) + t));
	elseif (x <= 3.2e-15)
		tmp = Float64(Float64(y * Float64(1.0 - z)) - log(y));
	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 <= -1.0)
		tmp = t_1;
	elseif (x <= 5.2e-70)
		tmp = y - (log(y) + t);
	elseif (x <= 3.2e-15)
		tmp = (y * (1.0 - z)) - log(y);
	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, -1.0], t$95$1, If[LessEqual[x, 5.2e-70], N[(y - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e-15], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 3.1999999999999999e-15 < 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 0 94.8%

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

      1. distribute-rgt-out 94.8%

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

      2. sub-neg 94.8%

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

      3. mul-1-neg 94.8%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 94.0%

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

      1. *-commutative 94.0%

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

   8. Simplified 94.0%

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

   if -1 < x < 5.20000000000000004e-70

   1. Initial program 82.0%

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

   3. Taylor expanded in y around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. sub-neg 98.0%

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

      3. metadata-eval 98.0%

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

      4. *-commutative 98.0%

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

      5. mul-1-neg 98.0%

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

      6. unsub-neg 98.0%

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

      7. *-commutative 98.0%

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

      8. +-commutative 98.0%

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

      9. sub-neg 98.0%

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

      10. metadata-eval 98.0%

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

      11. +-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in x around 0 97.7%

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

      1. mul-1-neg 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in y around 0 97.7%

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

   10. Taylor expanded in z around 0 79.2%

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

       1. mul-1-neg 79.2%

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

       2. +-commutative 79.2%

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

       3. unsub-neg 79.2%

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

   12. Simplified 79.2%

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

   if 5.20000000000000004e-70 < x < 3.1999999999999999e-15

   1. Initial program 63.2%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

      6. unsub-neg 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around 0 99.9%

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

   10. Taylor expanded in t around 0 99.9%

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

       1. neg-mul-1 99.9%

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

       2. +-commutative 99.9%

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

       3. sub-neg 99.9%

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

       4. metadata-eval 99.9%

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

       5. distribute-neg-in 99.9%

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

       6. distribute-rgt-neg-in 99.9%

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

       7. unsub-neg 99.9%

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

       8. distribute-rgt-neg-in 99.9%

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

       9. distribute-neg-in 99.9%

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

       10. metadata-eval 99.9%

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

       11. sub-neg 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-70}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-15}:\\ \;\;\;\;y \cdot \left(1 - z\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(y \cdot \left(-z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -4e+30)
   (- (* x (log y)) t)
   (if (<= (+ -1.0 x) -1.0)
     (- (- (* y (- z)) (log y)) t)
     (- (* (log y) (+ -1.0 x)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -4e+30) {
		tmp = (x * log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = ((y * -z) - log(y)) - t;
	} else {
		tmp = (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) <= (-4d+30)) then
        tmp = (x * log(y)) - t
    else if (((-1.0d0) + x) <= (-1.0d0)) then
        tmp = ((y * -z) - log(y)) - t
    else
        tmp = (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) <= -4e+30) {
		tmp = (x * Math.log(y)) - t;
	} else if ((-1.0 + x) <= -1.0) {
		tmp = ((y * -z) - Math.log(y)) - t;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -4e+30:
		tmp = (x * math.log(y)) - t
	elif (-1.0 + x) <= -1.0:
		tmp = ((y * -z) - math.log(y)) - t
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -4e+30)
		tmp = Float64(Float64(x * log(y)) - t);
	elseif (Float64(-1.0 + x) <= -1.0)
		tmp = Float64(Float64(Float64(y * Float64(-z)) - log(y)) - t);
	else
		tmp = Float64(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) <= -4e+30)
		tmp = (x * log(y)) - t;
	elseif ((-1.0 + x) <= -1.0)
		tmp = ((y * -z) - log(y)) - t;
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -4e+30], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.0], N[(N[(N[(y * (-z)), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -4.0000000000000001e30

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0 95.2%

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

      1. distribute-rgt-out 95.2%

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

      2. sub-neg 95.2%

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

      3. mul-1-neg 95.2%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 93.9%

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

      1. *-commutative 93.9%

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

   8. Simplified 93.9%

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

   if -4.0000000000000001e30 < (-.f64 x 1) < -1

   1. Initial program 79.3%

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

   3. Taylor expanded in y around 0 98.3%

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

      1. +-commutative 98.3%

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

      2. sub-neg 98.3%

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

      3. metadata-eval 98.3%

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

      4. *-commutative 98.3%

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

      5. mul-1-neg 98.3%

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

      6. unsub-neg 98.3%

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

      7. *-commutative 98.3%

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

      8. +-commutative 98.3%

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

      9. sub-neg 98.3%

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

      10. metadata-eval 98.3%

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

      11. +-commutative 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in z around inf 96.9%

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

   if -1 < (-.f64 x 1)

   1. Initial program 97.8%

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

      1. +-commutative 97.8%

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

      2. fma-def 97.8%

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

      3. sub-neg 97.8%

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

      4. metadata-eval 97.8%

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

      5. sub-neg 97.8%

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

      6. log1p-def 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 97.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+30}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;-1 + x \leq -1:\\ \;\;\;\;\left(y \cdot \left(-z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \left(\left(-z\right) - -1\right) - t\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+25}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3e+49)
     t_1
     (if (<= x -1.3e-25)
       (- (* y (- (- z) -1.0)) t)
       (if (<= x 7.6e+25) (- y (+ (log y) t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3e+49) {
		tmp = t_1;
	} else if (x <= -1.3e-25) {
		tmp = (y * (-z - -1.0)) - t;
	} else if (x <= 7.6e+25) {
		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)
    if (x <= (-3d+49)) then
        tmp = t_1
    else if (x <= (-1.3d-25)) then
        tmp = (y * (-z - (-1.0d0))) - t
    else if (x <= 7.6d+25) 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);
	double tmp;
	if (x <= -3e+49) {
		tmp = t_1;
	} else if (x <= -1.3e-25) {
		tmp = (y * (-z - -1.0)) - t;
	} else if (x <= 7.6e+25) {
		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)
	tmp = 0
	if x <= -3e+49:
		tmp = t_1
	elif x <= -1.3e-25:
		tmp = (y * (-z - -1.0)) - t
	elif x <= 7.6e+25:
		tmp = y - (math.log(y) + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3e+49)
		tmp = t_1;
	elseif (x <= -1.3e-25)
		tmp = Float64(Float64(y * Float64(Float64(-z) - -1.0)) - t);
	elseif (x <= 7.6e+25)
		tmp = Float64(y - Float64(log(y) + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -3e+49)
		tmp = t_1;
	elseif (x <= -1.3e-25)
		tmp = (y * (-z - -1.0)) - t;
	elseif (x <= 7.6e+25)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+49], t$95$1, If[LessEqual[x, -1.3e-25], N[(N[(y * N[((-z) - -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 7.6e+25], N[(y - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0000000000000002e49 or 7.6000000000000001e25 < x

   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. Add Preprocessing

   3. Taylor expanded in z around 0 96.7%

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

      1. distribute-rgt-out 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. log1p-def 99.6%

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

      5. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. neg-mul-1 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. distribute-neg-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. sub-neg 99.1%

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

      9. fma-def 99.1%

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

      10. sub-neg 99.1%

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

      11. metadata-eval 99.1%

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

      12. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in x around inf 82.3%

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

       1. *-commutative 82.3%

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

   11. Simplified 82.3%

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

   if -3.0000000000000002e49 < x < -1.3e-25

   1. Initial program 72.5%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

      6. unsub-neg 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 85.6%

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

      1. mul-1-neg 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in y around -inf 79.3%

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

   if -1.3e-25 < x < 7.6000000000000001e25

   1. Initial program 82.6%

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

   3. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. sub-neg 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. mul-1-neg 98.2%

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

      6. unsub-neg 98.2%

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

      7. *-commutative 98.2%

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

      8. +-commutative 98.2%

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

      9. sub-neg 98.2%

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

      10. metadata-eval 98.2%

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

      11. +-commutative 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 97.4%

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

      1. mul-1-neg 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in y around 0 97.4%

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

   10. Taylor expanded in z around 0 79.5%

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

       1. mul-1-neg 79.5%

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

       2. +-commutative 79.5%

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

       3. unsub-neg 79.5%

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

   12. Simplified 79.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+49}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \left(\left(-z\right) - -1\right) - t\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+25}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \left(\left(-z\right) - -1\right) - t\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5.6e+47)
     t_1
     (if (<= x -1.3e-25)
       (- (* y (- (- z) -1.0)) t)
       (if (<= x 3.6e+25) (- (- t) (log y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -5.6e+47) {
		tmp = t_1;
	} else if (x <= -1.3e-25) {
		tmp = (y * (-z - -1.0)) - t;
	} else if (x <= 3.6e+25) {
		tmp = -t - log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-5.6d+47)) then
        tmp = t_1
    else if (x <= (-1.3d-25)) then
        tmp = (y * (-z - (-1.0d0))) - t
    else if (x <= 3.6d+25) then
        tmp = -t - log(y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -5.6e+47) {
		tmp = t_1;
	} else if (x <= -1.3e-25) {
		tmp = (y * (-z - -1.0)) - t;
	} else if (x <= 3.6e+25) {
		tmp = -t - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -5.6e+47:
		tmp = t_1
	elif x <= -1.3e-25:
		tmp = (y * (-z - -1.0)) - t
	elif x <= 3.6e+25:
		tmp = -t - math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -5.6e+47)
		tmp = t_1;
	elseif (x <= -1.3e-25)
		tmp = Float64(Float64(y * Float64(Float64(-z) - -1.0)) - t);
	elseif (x <= 3.6e+25)
		tmp = Float64(Float64(-t) - log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -5.6e+47)
		tmp = t_1;
	elseif (x <= -1.3e-25)
		tmp = (y * (-z - -1.0)) - t;
	elseif (x <= 3.6e+25)
		tmp = -t - log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e+47], t$95$1, If[LessEqual[x, -1.3e-25], N[(N[(y * N[((-z) - -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 3.6e+25], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.59999999999999976e47 or 3.60000000000000015e25 < x

   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. Add Preprocessing

   3. Taylor expanded in z around 0 96.7%

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

      1. distribute-rgt-out 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. log1p-def 99.6%

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

      5. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. neg-mul-1 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. distribute-neg-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. sub-neg 99.1%

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

      9. fma-def 99.1%

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

      10. sub-neg 99.1%

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

      11. metadata-eval 99.1%

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

      12. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in x around inf 82.3%

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

       1. *-commutative 82.3%

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

   11. Simplified 82.3%

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

   if -5.59999999999999976e47 < x < -1.3e-25

   1. Initial program 72.5%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. *-commutative 99.9%

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

      5. mul-1-neg 99.9%

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

      6. unsub-neg 99.9%

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

      7. *-commutative 99.9%

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

      8. +-commutative 99.9%

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

      9. sub-neg 99.9%

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

      10. metadata-eval 99.9%

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

      11. +-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 85.6%

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

      1. mul-1-neg 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in y around -inf 79.3%

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

   if -1.3e-25 < x < 3.60000000000000015e25

   1. Initial program 82.6%

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

   3. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. sub-neg 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. mul-1-neg 98.2%

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

      6. unsub-neg 98.2%

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

      7. *-commutative 98.2%

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

      8. +-commutative 98.2%

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

      9. sub-neg 98.2%

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

      10. metadata-eval 98.2%

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

      11. +-commutative 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 97.4%

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

      1. mul-1-neg 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in y around 0 79.2%

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

       1. mul-1-neg 79.2%

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

   11. Simplified 79.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-25}:\\ \;\;\;\;y \cdot \left(\left(-z\right) - -1\right) - t\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right)\\ \mathbf{if}\;t \leq -6600000 \lor \neg \left(t \leq 1.05 \cdot 10^{-33}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (+ -1.0 x))))
   (if (or (<= t -6600000.0) (not (<= t 1.05e-33)))
     (- t_1 t)
     (- t_1 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (-1.0 + x);
	double tmp;
	if ((t <= -6600000.0) || !(t <= 1.05e-33)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * ((-1.0d0) + x)
    if ((t <= (-6600000.0d0)) .or. (.not. (t <= 1.05d-33))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * (-1.0 + x);
	double tmp;
	if ((t <= -6600000.0) || !(t <= 1.05e-33)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * (-1.0 + x)
	tmp = 0
	if (t <= -6600000.0) or not (t <= 1.05e-33):
		tmp = t_1 - t
	else:
		tmp = t_1 - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(-1.0 + x))
	tmp = 0.0
	if ((t <= -6600000.0) || !(t <= 1.05e-33))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * (-1.0 + x);
	tmp = 0.0;
	if ((t <= -6600000.0) || ~((t <= 1.05e-33)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -6600000.0], N[Not[LessEqual[t, 1.05e-33]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.6e6 or 1.05e-33 < t

   1. Initial program 96.4%

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

      1. +-commutative 96.4%

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

      2. fma-def 96.4%

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

      3. sub-neg 96.4%

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

      4. metadata-eval 96.4%

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

      5. sub-neg 96.4%

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

      6. log1p-def 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 96.4%

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

   if -6.6e6 < t < 1.05e-33

   1. Initial program 81.5%

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

   3. Taylor expanded in y around 0 97.9%

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

      1. +-commutative 97.9%

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

      2. sub-neg 97.9%

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

      3. metadata-eval 97.9%

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

      4. *-commutative 97.9%

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

      5. mul-1-neg 97.9%

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

      6. unsub-neg 97.9%

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

      7. *-commutative 97.9%

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

      8. +-commutative 97.9%

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

      9. sub-neg 97.9%

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

      10. metadata-eval 97.9%

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

      11. +-commutative 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in z around inf 97.7%

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

   7. Taylor expanded in t around 0 97.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6600000 \lor \neg \left(t \leq 1.05 \cdot 10^{-33}\right):\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4.8 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 4.8e-14)))
   (- (* x (log y)) t)
   (- y (+ (log y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 4.8e-14)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = y - (log(y) + t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 4.8d-14))) then
        tmp = (x * log(y)) - t
    else
        tmp = y - (log(y) + t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.0) || !(x <= 4.8e-14)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = y - (Math.log(y) + t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.0) or not (x <= 4.8e-14):
		tmp = (x * math.log(y)) - t
	else:
		tmp = y - (math.log(y) + t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 4.8e-14))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(y - Float64(log(y) + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 4.8e-14)))
		tmp = (x * log(y)) - t;
	else
		tmp = y - (log(y) + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 4.8e-14]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(y - N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.8e-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 0 94.8%

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

      1. distribute-rgt-out 94.8%

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

      2. sub-neg 94.8%

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

      3. mul-1-neg 94.8%

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

      4. log1p-def 99.7%

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

      5. mul-1-neg 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 94.0%

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

      1. *-commutative 94.0%

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

   8. Simplified 94.0%

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

   if -1 < x < 4.8e-14

   1. Initial program 80.3%

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

   3. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. sub-neg 98.2%

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

      3. metadata-eval 98.2%

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

      4. *-commutative 98.2%

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

      5. mul-1-neg 98.2%

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

      6. unsub-neg 98.2%

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

      7. *-commutative 98.2%

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

      8. +-commutative 98.2%

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

      9. sub-neg 98.2%

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

      10. metadata-eval 98.2%

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

      11. +-commutative 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0 97.9%

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

      1. mul-1-neg 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in y around 0 97.9%

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

   10. Taylor expanded in z around 0 77.7%

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

       1. mul-1-neg 77.7%

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

       2. +-commutative 77.7%

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

       3. unsub-neg 77.7%

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

   12. Simplified 77.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 4.8 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y - \left(\log y + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ z -1.0) -2e+300) (* y (- z)) (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z + -1.0) <= -2e+300) {
		tmp = y * -z;
	} else {
		tmp = (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 ((z + (-1.0d0)) <= (-2d+300)) then
        tmp = y * -z
    else
        tmp = (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 ((z + -1.0) <= -2e+300) {
		tmp = y * -z;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z + -1.0) <= -2e+300:
		tmp = y * -z
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z + -1.0) <= -2e+300)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(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 ((z + -1.0) <= -2e+300)
		tmp = y * -z;
	else
		tmp = (log(y) * (-1.0 + x)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z + -1.0), $MachinePrecision], -2e+300], N[(y * (-z)), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z 1) < -2.0000000000000001e300

   1. Initial program 4.5%

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

   3. Taylor expanded in z around 0 4.5%

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

      1. distribute-rgt-out 4.5%

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

      2. sub-neg 4.5%

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

      3. mul-1-neg 4.5%

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

      4. log1p-def 100.0%

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

      5. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. metadata-eval 100.0%

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

      3. +-commutative 100.0%

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

      4. neg-mul-1 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. sub-neg 100.0%

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

      9. fma-def 100.0%

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

      10. sub-neg 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in z around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

   if -2.0000000000000001e300 < (-.f64 z 1)

   1. Initial program 89.7%

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

      1. +-commutative 89.7%

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

      2. fma-def 89.7%

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

      3. sub-neg 89.7%

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

      4. metadata-eval 89.7%

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

      5. sub-neg 89.7%

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

      6. log1p-def 99.8%

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

      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 88.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z + -1 \leq -2 \cdot 10^{+300}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+47} \lor \neg \left(x \leq 1.25 \cdot 10^{+23}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) - -1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.32e+47) (not (<= x 1.25e+23)))
   (* x (log y))
   (- (* y (- (- z) -1.0)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.32e+47) || !(x <= 1.25e+23)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (-z - -1.0)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.32d+47)) .or. (.not. (x <= 1.25d+23))) then
        tmp = x * log(y)
    else
        tmp = (y * (-z - (-1.0d0))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.32e+47) || !(x <= 1.25e+23)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (-z - -1.0)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.32e+47) or not (x <= 1.25e+23):
		tmp = x * math.log(y)
	else:
		tmp = (y * (-z - -1.0)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.32e+47) || !(x <= 1.25e+23))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(-z) - -1.0)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.32e+47) || ~((x <= 1.25e+23)))
		tmp = x * log(y);
	else
		tmp = (y * (-z - -1.0)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.32e+47], N[Not[LessEqual[x, 1.25e+23]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[((-z) - -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.31999999999999992e47 or 1.25e23 < x

   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. Add Preprocessing

   3. Taylor expanded in z around 0 96.7%

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

      1. distribute-rgt-out 96.7%

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

      2. sub-neg 96.7%

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

      3. mul-1-neg 96.7%

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

      4. log1p-def 99.6%

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

      5. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around 0 99.1%

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

      1. sub-neg 99.1%

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

      2. metadata-eval 99.1%

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

      3. +-commutative 99.1%

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

      4. neg-mul-1 99.1%

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

      5. distribute-rgt-neg-in 99.1%

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

      6. distribute-neg-in 99.1%

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

      7. metadata-eval 99.1%

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

      8. sub-neg 99.1%

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

      9. fma-def 99.1%

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

      10. sub-neg 99.1%

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

      11. metadata-eval 99.1%

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

      12. +-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in x around inf 82.3%

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

       1. *-commutative 82.3%

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

   11. Simplified 82.3%

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

   if -1.31999999999999992e47 < x < 1.25e23

   1. Initial program 81.0%

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

   3. Taylor expanded in y around 0 98.5%

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

      1. +-commutative 98.5%

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

      2. sub-neg 98.5%

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

      3. metadata-eval 98.5%

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

      4. *-commutative 98.5%

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

      5. mul-1-neg 98.5%

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

      6. unsub-neg 98.5%

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

      7. *-commutative 98.5%

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

      8. +-commutative 98.5%

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

      9. sub-neg 98.5%

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

      10. metadata-eval 98.5%

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

      11. +-commutative 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in x around 0 95.4%

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

      1. mul-1-neg 95.4%

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

   8. Simplified 95.4%

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

   9. Taylor expanded in y around -inf 62.5%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+47} \lor \neg \left(x \leq 1.25 \cdot 10^{+23}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(-z\right) - -1\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 98.8%

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

   1. +-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. unsub-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. +-commutative 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 14: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] - N[(y * N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $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 98.8%

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

   1. +-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. unsub-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. +-commutative 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in t around 0 98.8%

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

7. Final simplification 98.8%

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

Alternative 15: 99.0% 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 y around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. unsub-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. +-commutative 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in z around inf 98.6%

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

7. Final simplification 98.6%

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

Alternative 16: 43.2% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2900000 \lor \neg \left(t \leq 145000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2900000.0) (not (<= t 145000.0))) (- t) (* y (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2900000.0) || !(t <= 145000.0)) {
		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 <= (-2900000.0d0)) .or. (.not. (t <= 145000.0d0))) 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 <= -2900000.0) || !(t <= 145000.0)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2900000.0) or not (t <= 145000.0):
		tmp = -t
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2900000.0) || !(t <= 145000.0))
		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 <= -2900000.0) || ~((t <= 145000.0)))
		tmp = -t;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2900000.0], N[Not[LessEqual[t, 145000.0]], $MachinePrecision]], (-t), N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e6 or 145000 < t

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

      2. fma-def 96.0%

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

      3. sub-neg 96.0%

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

      4. metadata-eval 96.0%

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

      5. sub-neg 96.0%

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

      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 75.1%

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

      1. neg-mul-1 75.1%

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

   7. Simplified 75.1%

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

   if -2.9e6 < t < 145000

   1. Initial program 82.9%

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

   3. Taylor expanded in z around 0 82.9%

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

      1. distribute-rgt-out 82.9%

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

      2. sub-neg 82.9%

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

      3. mul-1-neg 82.9%

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

      4. log1p-def 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 98.1%

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

      1. sub-neg 98.1%

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

      2. metadata-eval 98.1%

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

      3. +-commutative 98.1%

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

      4. neg-mul-1 98.1%

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

      5. distribute-rgt-neg-in 98.1%

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

      6. distribute-neg-in 98.1%

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

      7. metadata-eval 98.1%

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

      8. sub-neg 98.1%

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

      9. fma-def 98.1%

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

      10. sub-neg 98.1%

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

      11. metadata-eval 98.1%

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

      12. +-commutative 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in y around inf 19.3%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2900000 \lor \neg \left(t \leq 145000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 42.9% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9200000 \lor \neg \left(t \leq 3350000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -9200000.0) (not (<= t 3350000.0))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -9200000.0) || !(t <= 3350000.0)) {
		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 <= (-9200000.0d0)) .or. (.not. (t <= 3350000.0d0))) 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 <= -9200000.0) || !(t <= 3350000.0)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -9200000.0) or not (t <= 3350000.0):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -9200000.0) || !(t <= 3350000.0))
		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 <= -9200000.0) || ~((t <= 3350000.0)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -9200000.0], N[Not[LessEqual[t, 3350000.0]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.2e6 or 3.35e6 < t

   1. Initial program 96.0%

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

      1. +-commutative 96.0%

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

      2. fma-def 96.0%

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

      3. sub-neg 96.0%

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

      4. metadata-eval 96.0%

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

      5. sub-neg 96.0%

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

      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 75.1%

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

      1. neg-mul-1 75.1%

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

   7. Simplified 75.1%

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

   if -9.2e6 < t < 3.35e6

   1. Initial program 82.9%

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

   3. Taylor expanded in z around 0 82.9%

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

      1. distribute-rgt-out 82.9%

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

      2. sub-neg 82.9%

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

      3. mul-1-neg 82.9%

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

      4. log1p-def 99.8%

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

      5. mul-1-neg 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 98.1%

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

      1. sub-neg 98.1%

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

      2. metadata-eval 98.1%

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

      3. +-commutative 98.1%

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

      4. neg-mul-1 98.1%

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

      5. distribute-rgt-neg-in 98.1%

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

      6. distribute-neg-in 98.1%

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

      7. metadata-eval 98.1%

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

      8. sub-neg 98.1%

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

      9. fma-def 98.1%

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

      10. sub-neg 98.1%

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

      11. metadata-eval 98.1%

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

      12. +-commutative 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in z around inf 18.8%

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

       1. associate-*r* 18.8%

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

       2. mul-1-neg 18.8%

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

   11. Simplified 18.8%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9200000 \lor \neg \left(t \leq 3350000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.0% accurate, 26.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[((-z) - -1.0), $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 98.8%

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

   1. +-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. unsub-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. +-commutative 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in x around 0 60.9%

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

   1. mul-1-neg 60.9%

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

8. Simplified 60.9%

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

9. Taylor expanded in y around -inf 42.8%

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

10. Final simplification 42.8%

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

Alternative 19: 45.8% 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 98.8%

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

   1. +-commutative 98.8%

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

   2. sub-neg 98.8%

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

   3. metadata-eval 98.8%

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

   4. *-commutative 98.8%

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

   5. mul-1-neg 98.8%

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

   6. unsub-neg 98.8%

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

   7. *-commutative 98.8%

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

   8. +-commutative 98.8%

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

   9. sub-neg 98.8%

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

   10. metadata-eval 98.8%

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

   11. +-commutative 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in z around inf 42.6%

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

   1. associate-*r* 42.6%

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

   2. mul-1-neg 42.6%

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

8. Simplified 42.6%

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

9. Final simplification 42.6%

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

Alternative 20: 35.9% accurate, 107.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 31.3%

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

   1. neg-mul-1 31.3%

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

7. Simplified 31.3%

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

8. Final simplification 31.3%

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

Reproduce

herbie shell --seed 2024018 
(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))