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

Percentage Accurate: 89.0% → 99.8%

Time: 20.4s

Alternatives: 19

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: 89.0% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. sub-neg 87.8%

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

   2. +-commutative 87.8%

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

   3. associate-+l+ 87.8%

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

   4. fma-define 87.8%

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

   5. sub-neg 87.8%

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

   6. metadata-eval 87.8%

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

   7. sub-neg 87.8%

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

   8. log1p-define 99.8%

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

   9. fma-define 99.8%

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

   10. sub-neg 99.8%

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

   11. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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 87.8%

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

   1. +-commutative 87.8%

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

   2. fma-define 87.8%

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

   3. sub-neg 87.8%

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

   4. metadata-eval 87.8%

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

   5. sub-neg 87.8%

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

   6. log1p-define 99.8%

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

   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 3: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around 0 99.2%

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

   1. associate--l+ 99.2%

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

   2. associate-*r* 99.2%

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

   3. mul-1-neg 99.2%

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

   4. fma-define 99.2%

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

   5. sub-neg 99.2%

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

   6. metadata-eval 99.2%

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

   7. +-commutative 99.2%

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

   8. fma-neg 99.2%

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

   9. sub-neg 99.2%

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

   10. metadata-eval 99.2%

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

   11. +-commutative 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 63.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t - \log y\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) - t\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-163}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-212}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-161}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+142}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t (log y))))
   (if (<= x -2.9e+70)
     t_1
     (if (<= x -6e-124)
       (- (* y (- (* y (* z (+ -0.5 (* y -0.3333333333333333)))) z)) t)
       (if (<= x -4.7e-163)
         t_2
         (if (<= x 2.6e-212)
           (- (- t) (* z y))
           (if (<= x 5.8e-161)
             t_2
             (if (<= x 1.32e+142)
               (- (* y (- (* -0.5 (* z y)) z)) t)
               t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t - log(y);
	double tmp;
	if (x <= -2.9e+70) {
		tmp = t_1;
	} else if (x <= -6e-124) {
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
	} else if (x <= -4.7e-163) {
		tmp = t_2;
	} else if (x <= 2.6e-212) {
		tmp = -t - (z * y);
	} else if (x <= 5.8e-161) {
		tmp = t_2;
	} else if (x <= 1.32e+142) {
		tmp = (y * ((-0.5 * (z * y)) - z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t - log(y)
    if (x <= (-2.9d+70)) then
        tmp = t_1
    else if (x <= (-6d-124)) then
        tmp = (y * ((y * (z * ((-0.5d0) + (y * (-0.3333333333333333d0))))) - z)) - t
    else if (x <= (-4.7d-163)) then
        tmp = t_2
    else if (x <= 2.6d-212) then
        tmp = -t - (z * y)
    else if (x <= 5.8d-161) then
        tmp = t_2
    else if (x <= 1.32d+142) then
        tmp = (y * (((-0.5d0) * (z * y)) - z)) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t - Math.log(y);
	double tmp;
	if (x <= -2.9e+70) {
		tmp = t_1;
	} else if (x <= -6e-124) {
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
	} else if (x <= -4.7e-163) {
		tmp = t_2;
	} else if (x <= 2.6e-212) {
		tmp = -t - (z * y);
	} else if (x <= 5.8e-161) {
		tmp = t_2;
	} else if (x <= 1.32e+142) {
		tmp = (y * ((-0.5 * (z * y)) - z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t - math.log(y)
	tmp = 0
	if x <= -2.9e+70:
		tmp = t_1
	elif x <= -6e-124:
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t
	elif x <= -4.7e-163:
		tmp = t_2
	elif x <= 2.6e-212:
		tmp = -t - (z * y)
	elif x <= 5.8e-161:
		tmp = t_2
	elif x <= 1.32e+142:
		tmp = (y * ((-0.5 * (z * y)) - z)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t - log(y))
	tmp = 0.0
	if (x <= -2.9e+70)
		tmp = t_1;
	elseif (x <= -6e-124)
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(z * Float64(-0.5 + Float64(y * -0.3333333333333333)))) - z)) - t);
	elseif (x <= -4.7e-163)
		tmp = t_2;
	elseif (x <= 2.6e-212)
		tmp = Float64(Float64(-t) - Float64(z * y));
	elseif (x <= 5.8e-161)
		tmp = t_2;
	elseif (x <= 1.32e+142)
		tmp = Float64(Float64(y * Float64(Float64(-0.5 * Float64(z * y)) - z)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t - log(y);
	tmp = 0.0;
	if (x <= -2.9e+70)
		tmp = t_1;
	elseif (x <= -6e-124)
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
	elseif (x <= -4.7e-163)
		tmp = t_2;
	elseif (x <= 2.6e-212)
		tmp = -t - (z * y);
	elseif (x <= 5.8e-161)
		tmp = t_2;
	elseif (x <= 1.32e+142)
		tmp = (y * ((-0.5 * (z * y)) - z)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e+70], t$95$1, If[LessEqual[x, -6e-124], N[(N[(y * N[(N[(y * N[(z * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, -4.7e-163], t$95$2, If[LessEqual[x, 2.6e-212], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-161], t$95$2, If[LessEqual[x, 1.32e+142], N[(N[(y * N[(N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.8999999999999998e70 or 1.3199999999999999e142 < x

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0 93.9%

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

      1. fma-neg 93.9%

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

      2. sub-neg 93.9%

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

      3. metadata-eval 93.9%

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

      4. +-commutative 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around inf 78.3%

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

      1. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   if -2.8999999999999998e70 < x < -6e-124

   1. Initial program 91.1%

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

   3. Taylor expanded in z around inf 63.7%

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

      1. *-commutative 63.7%

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

      2. sub-neg 63.7%

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

      3. log1p-define 72.6%

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

   5. Simplified 72.6%

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

   6. Taylor expanded in y around 0 72.5%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in y around 0 72.5%

      \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)}\right) - t \]
   8. Step-by-step derivation

      1. +-commutative 72.5%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(y \cdot z\right) + -0.5 \cdot z\right)}\right) - t \]

      2. associate-*r* 72.5%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot z} + -0.5 \cdot z\right)\right) - t \]

      3. distribute-rgt-out 72.5%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 \cdot y + -0.5\right)\right)}\right) - t \]

      4. *-commutative 72.5%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.3333333333333333} + -0.5\right)\right)\right) - t \]

   9. Simplified 72.5%

      \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(z \cdot \left(y \cdot -0.3333333333333333 + -0.5\right)\right)}\right) - t \]

   if -6e-124 < x < -4.7e-163 or 2.6e-212 < x < 5.8e-161

   1. Initial program 87.1%

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

   3. Taylor expanded in y around 0 87.1%

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

      1. fma-neg 87.1%

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

      2. sub-neg 87.1%

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

      3. metadata-eval 87.1%

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

      4. +-commutative 87.1%

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

   5. Simplified 87.1%

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

      1. add-cube-cbrt 85.5%

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

      2. pow3 85.6%

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

      3. add-sqr-sqrt 44.3%

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

      4. sqrt-unprod 80.5%

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

      5. sqr-neg 80.5%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 75.4%

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

   7. Applied egg-rr 75.4%

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

   8. Taylor expanded in x around 0 76.6%

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

      1. neg-mul-1 76.6%

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

      2. unsub-neg 76.6%

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

   10. Simplified 76.6%

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

   if -4.7e-163 < x < 2.6e-212

   1. Initial program 80.2%

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

   3. Taylor expanded in z around inf 46.8%

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

      1. *-commutative 46.8%

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

      2. sub-neg 46.8%

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

      3. log1p-define 66.2%

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

   5. Simplified 66.2%

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

   6. Taylor expanded in y around 0 66.2%

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

      1. associate-*r* 66.2%

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

      2. mul-1-neg 66.2%

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

   8. Simplified 66.2%

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

   if 5.8e-161 < x < 1.3199999999999999e142

   1. Initial program 83.0%

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

   3. Taylor expanded in z around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. sub-neg 53.9%

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

      3. log1p-define 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in y around 0 68.7%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + -0.5 \cdot \left(y \cdot z\right)\right)} - t \]
   7. Step-by-step derivation

      1. +-commutative 68.7%

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

      2. mul-1-neg 68.7%

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

      3. unsub-neg 68.7%

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

      4. *-commutative 68.7%

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

   8. Simplified 68.7%

      \[\leadsto \color{blue}{y \cdot \left(\left(y \cdot z\right) \cdot -0.5 - z\right)} - t \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) - t\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-163}:\\ \;\;\;\;t - \log y\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-212}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-161}:\\ \;\;\;\;t - \log y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{+142}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;-1 + x \leq -0.999999999998:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;-1 + x \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= (+ -1.0 x) -5e+70)
     t_1
     (if (<= (+ -1.0 x) -0.999999999998)
       (- (- t) (log y))
       (if (<= (+ -1.0 x) 2e+142) (- (- t) (* z y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((-1.0 + x) <= -5e+70) {
		tmp = t_1;
	} else if ((-1.0 + x) <= -0.999999999998) {
		tmp = -t - log(y);
	} else if ((-1.0 + x) <= 2e+142) {
		tmp = -t - (z * 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 (((-1.0d0) + x) <= (-5d+70)) then
        tmp = t_1
    else if (((-1.0d0) + x) <= (-0.999999999998d0)) then
        tmp = -t - log(y)
    else if (((-1.0d0) + x) <= 2d+142) then
        tmp = -t - (z * 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 ((-1.0 + x) <= -5e+70) {
		tmp = t_1;
	} else if ((-1.0 + x) <= -0.999999999998) {
		tmp = -t - Math.log(y);
	} else if ((-1.0 + x) <= 2e+142) {
		tmp = -t - (z * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (-1.0 + x) <= -5e+70:
		tmp = t_1
	elif (-1.0 + x) <= -0.999999999998:
		tmp = -t - math.log(y)
	elif (-1.0 + x) <= 2e+142:
		tmp = -t - (z * y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (Float64(-1.0 + x) <= -5e+70)
		tmp = t_1;
	elseif (Float64(-1.0 + x) <= -0.999999999998)
		tmp = Float64(Float64(-t) - log(y));
	elseif (Float64(-1.0 + x) <= 2e+142)
		tmp = Float64(Float64(-t) - Float64(z * 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 ((-1.0 + x) <= -5e+70)
		tmp = t_1;
	elseif ((-1.0 + x) <= -0.999999999998)
		tmp = -t - log(y);
	elseif ((-1.0 + x) <= 2e+142)
		tmp = -t - (z * 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[N[(-1.0 + x), $MachinePrecision], -5e+70], t$95$1, If[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.999999999998], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 2e+142], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -5.0000000000000002e70 or 2.0000000000000001e142 < (-.f64 x 1)

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0 93.9%

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

      1. fma-neg 93.9%

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

      2. sub-neg 93.9%

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

      3. metadata-eval 93.9%

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

      4. +-commutative 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around inf 78.3%

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

      1. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   if -5.0000000000000002e70 < (-.f64 x 1) < -0.99999999999800004

   1. Initial program 84.8%

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

   3. Taylor expanded in y around 0 83.8%

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

      1. fma-neg 83.8%

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

      2. sub-neg 83.8%

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

      3. metadata-eval 83.8%

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

      4. +-commutative 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in x around 0 80.9%

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

      1. mul-1-neg 80.9%

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

   8. Simplified 80.9%

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

   if -0.99999999999800004 < (-.f64 x 1) < 2.0000000000000001e142

   1. Initial program 81.8%

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

   3. Taylor expanded in z around inf 56.1%

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

      1. *-commutative 56.1%

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

      2. sub-neg 56.1%

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

      3. log1p-define 74.1%

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

   5. Simplified 74.1%

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

   6. Taylor expanded in y around 0 74.1%

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

      1. associate-*r* 74.1%

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

      2. mul-1-neg 74.1%

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

   8. Simplified 74.1%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -5 \cdot 10^{+70}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;-1 + x \leq -0.999999999998:\\ \;\;\;\;\left(-t\right) - \log y\\ \mathbf{elif}\;-1 + x \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000001:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ -1.0 x) -1.00000001)
   (- (* (log y) (+ -1.0 x)) t)
   (if (<= (+ -1.0 x) 5e+29)
     (- (- (* y (- 1.0 z)) (log y)) t)
     (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -1.00000001) {
		tmp = (log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 5e+29) {
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	} else {
		tmp = (x * log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((-1.0d0) + x) <= (-1.00000001d0)) then
        tmp = (log(y) * ((-1.0d0) + x)) - t
    else if (((-1.0d0) + x) <= 5d+29) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - t
    else
        tmp = (x * log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((-1.0 + x) <= -1.00000001) {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	} else if ((-1.0 + x) <= 5e+29) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = (x * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (-1.0 + x) <= -1.00000001:
		tmp = (math.log(y) * (-1.0 + x)) - t
	elif (-1.0 + x) <= 5e+29:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = (x * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(-1.0 + x) <= -1.00000001)
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	elseif (Float64(-1.0 + x) <= 5e+29)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((-1.0 + x) <= -1.00000001)
		tmp = (log(y) * (-1.0 + x)) - t;
	elseif ((-1.0 + x) <= 5e+29)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = (x * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(-1.0 + x), $MachinePrecision], -1.00000001], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[N[(-1.0 + x), $MachinePrecision], 5e+29], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x 1) < -1.0000000099999999

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0 93.4%

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

   if -1.0000000099999999 < (-.f64 x 1) < 5.0000000000000001e29

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-*r* 99.6%

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

      3. mul-1-neg 99.6%

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

      4. fma-define 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. fma-neg 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. +-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 98.1%

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

      1. +-commutative 98.1%

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

      2. mul-1-neg 98.1%

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

      3. unsub-neg 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. +-commutative 98.1%

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

      7. neg-mul-1 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

      9. distribute-neg-in 98.1%

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

      10. metadata-eval 98.1%

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

      11. unsub-neg 98.1%

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

   8. Simplified 98.1%

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

   if 5.0000000000000001e29 < (-.f64 x 1)

   1. Initial program 93.0%

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

   3. Taylor expanded in x around inf 93.0%

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

      1. associate--l+ 93.0%

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

      2. +-commutative 93.0%

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

      3. associate--l+ 93.0%

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

      4. associate-/l* 93.0%

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

      5. sub-neg 93.0%

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

      6. log1p-define 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. div-sub 99.6%

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

      12. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf 91.3%

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

      1. neg-mul-1 91.3%

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

      2. distribute-neg-frac2 91.3%

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

   8. Simplified 91.3%

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

   9. Taylor expanded in x around 0 91.3%

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

       1. neg-mul-1 91.3%

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

       2. +-commutative 91.3%

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

       3. unsub-neg 91.3%

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

       4. *-commutative 91.3%

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

   11. Simplified 91.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -1.00000001:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \mathbf{elif}\;-1 + x \leq 5 \cdot 10^{+29}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(-1 + x\right)\\ \mathbf{if}\;t \leq -0.00038:\\ \;\;\;\;t \cdot \left(-1 + \log y \cdot \frac{-1 + x}{t}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(1 - z\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) (+ -1.0 x))))
   (if (<= t -0.00038)
     (* t (+ -1.0 (* (log y) (/ (+ -1.0 x) t))))
     (if (<= t 3.4e-9) (+ (* y (- 1.0 z)) t_1) (- t_1 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * (-1.0 + x);
	double tmp;
	if (t <= -0.00038) {
		tmp = t * (-1.0 + (log(y) * ((-1.0 + x) / t)));
	} else if (t <= 3.4e-9) {
		tmp = (y * (1.0 - z)) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * ((-1.0d0) + x)
    if (t <= (-0.00038d0)) then
        tmp = t * ((-1.0d0) + (log(y) * (((-1.0d0) + x) / t)))
    else if (t <= 3.4d-9) then
        tmp = (y * (1.0d0 - z)) + t_1
    else
        tmp = t_1 - t
    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 <= -0.00038) {
		tmp = t * (-1.0 + (Math.log(y) * ((-1.0 + x) / t)));
	} else if (t <= 3.4e-9) {
		tmp = (y * (1.0 - z)) + t_1;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * (-1.0 + x)
	tmp = 0
	if t <= -0.00038:
		tmp = t * (-1.0 + (math.log(y) * ((-1.0 + x) / t)))
	elif t <= 3.4e-9:
		tmp = (y * (1.0 - z)) + t_1
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * Float64(-1.0 + x))
	tmp = 0.0
	if (t <= -0.00038)
		tmp = Float64(t * Float64(-1.0 + Float64(log(y) * Float64(Float64(-1.0 + x) / t))));
	elseif (t <= 3.4e-9)
		tmp = Float64(Float64(y * Float64(1.0 - z)) + t_1);
	else
		tmp = Float64(t_1 - t);
	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 <= -0.00038)
		tmp = t * (-1.0 + (log(y) * ((-1.0 + x) / t)));
	elseif (t <= 3.4e-9)
		tmp = (y * (1.0 - z)) + t_1;
	else
		tmp = t_1 - t;
	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[LessEqual[t, -0.00038], N[(t * N[(-1.0 + N[(N[Log[y], $MachinePrecision] * N[(N[(-1.0 + x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-9], N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.8000000000000002e-4

   1. Initial program 97.2%

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

   3. Taylor expanded in y around 0 96.3%

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

      1. fma-neg 96.3%

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

      2. sub-neg 96.3%

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

      3. metadata-eval 96.3%

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

      4. +-commutative 96.3%

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

   5. Simplified 96.3%

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

   6. Taylor expanded in t around inf 96.3%

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

      1. sub-neg 96.3%

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

      2. associate-/l* 96.3%

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

      3. sub-neg 96.3%

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

      4. metadata-eval 96.3%

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

      5. metadata-eval 96.3%

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

   8. Simplified 96.3%

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

   if -3.8000000000000002e-4 < t < 3.3999999999999998e-9

   1. Initial program 80.2%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. associate-*r* 99.1%

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

      3. mul-1-neg 99.1%

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

      4. fma-define 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. fma-neg 99.1%

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

      9. sub-neg 99.1%

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

      10. metadata-eval 99.1%

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

      11. +-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in t around 0 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. +-commutative 98.8%

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

      4. neg-mul-1 98.8%

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

      5. distribute-rgt-neg-in 98.8%

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

      6. distribute-neg-in 98.8%

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

      7. metadata-eval 98.8%

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

      8. unsub-neg 98.8%

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

      9. sub-neg 98.8%

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

      10. metadata-eval 98.8%

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

   8. Simplified 98.8%

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

   if 3.3999999999999998e-9 < t

   1. Initial program 91.5%

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

   3. Taylor expanded in y around 0 90.8%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00038:\\ \;\;\;\;t \cdot \left(-1 + \log y \cdot \frac{-1 + x}{t}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(1 - z\right) + \log y \cdot \left(-1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+15} \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (+ -1.0 x) -4e+15) (not (<= (+ -1.0 x) -0.5)))
   (- (* x (log y)) t)
   (- (- t) (log y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -4e+15) || !((-1.0 + x) <= -0.5)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - log(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((-1.0d0) + x) <= (-4d+15)) .or. (.not. (((-1.0d0) + x) <= (-0.5d0)))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - log(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((-1.0 + x) <= -4e+15) || !((-1.0 + x) <= -0.5)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - Math.log(y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((-1.0 + x) <= -4e+15) or not ((-1.0 + x) <= -0.5):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - math.log(y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(-1.0 + x) <= -4e+15) || !(Float64(-1.0 + x) <= -0.5))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - log(y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((-1.0 + x) <= -4e+15) || ~(((-1.0 + x) <= -0.5)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - log(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(-1.0 + x), $MachinePrecision], -4e+15], N[Not[LessEqual[N[(-1.0 + x), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[Log[y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x 1) < -4e15 or -0.5 < (-.f64 x 1)

   1. Initial program 92.1%

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

   3. Taylor expanded in x around inf 92.1%

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

      1. associate--l+ 92.1%

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

      2. +-commutative 92.1%

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

      3. associate--l+ 92.1%

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

      4. associate-/l* 92.1%

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

      5. sub-neg 92.1%

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

      6. log1p-define 99.6%

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

      7. sub-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. +-commutative 99.6%

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

      10. associate-*r/ 99.6%

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

      11. div-sub 99.6%

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

      12. mul-1-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in t around inf 90.8%

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

      1. neg-mul-1 90.8%

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

      2. distribute-neg-frac2 90.8%

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

   8. Simplified 90.8%

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

   9. Taylor expanded in x around 0 90.8%

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

       1. neg-mul-1 90.8%

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

       2. +-commutative 90.8%

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

       3. unsub-neg 90.8%

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

       4. *-commutative 90.8%

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

   11. Simplified 90.8%

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

   if -4e15 < (-.f64 x 1) < -0.5

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0 83.4%

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

      1. fma-neg 83.4%

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

      2. sub-neg 83.4%

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

      3. metadata-eval 83.4%

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

      4. +-commutative 83.4%

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

   5. Simplified 83.4%

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

   6. Taylor expanded in x around 0 81.9%

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

      1. mul-1-neg 81.9%

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

   8. Simplified 81.9%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-1 + x \leq -4 \cdot 10^{+15} \lor \neg \left(-1 + x \leq -0.5\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+171} \lor \neg \left(z \leq 1.45 \cdot 10^{+133}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.4e+171) (not (<= z 1.45e+133)))
   (- (* z (log1p (- y))) t)
   (- (* (log y) (+ -1.0 x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.4e+171) || !(z <= 1.45e+133)) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.4e+171) || !(z <= 1.45e+133)) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (Math.log(y) * (-1.0 + x)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.4e+171) or not (z <= 1.45e+133):
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (math.log(y) * (-1.0 + x)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.4e+171) || !(z <= 1.45e+133))
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(log(y) * Float64(-1.0 + x)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.4e+171], N[Not[LessEqual[z, 1.45e+133]], $MachinePrecision]], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.3999999999999996e171 or 1.4500000000000001e133 < z

   1. Initial program 53.7%

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

   3. Taylor expanded in z around inf 38.8%

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

      1. *-commutative 38.8%

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

      2. sub-neg 38.8%

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

      3. log1p-define 82.2%

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

   5. Simplified 82.2%

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

   if -5.3999999999999996e171 < z < 1.4500000000000001e133

   1. Initial program 98.0%

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

   3. Taylor expanded in y around 0 98.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+171} \lor \neg \left(z \leq 1.45 \cdot 10^{+133}\right):\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(-1 + x\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+68} \lor \neg \left(x \leq 1.95 \cdot 10^{+142}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.06e+68) (not (<= x 1.95e+142)))
   (* x (log y))
   (- (* y (- (* y (* z (+ -0.5 (* y -0.3333333333333333)))) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.06e+68) || !(x <= 1.95e+142)) {
		tmp = x * log(y);
	} else {
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.06d+68)) .or. (.not. (x <= 1.95d+142))) then
        tmp = x * log(y)
    else
        tmp = (y * ((y * (z * ((-0.5d0) + (y * (-0.3333333333333333d0))))) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.06e+68) || !(x <= 1.95e+142)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.06e+68) or not (x <= 1.95e+142):
		tmp = x * math.log(y)
	else:
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.06e+68) || !(x <= 1.95e+142))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(z * Float64(-0.5 + Float64(y * -0.3333333333333333)))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.06e+68) || ~((x <= 1.95e+142)))
		tmp = x * log(y);
	else
		tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.06e+68], N[Not[LessEqual[x, 1.95e+142]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * N[(z * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.06e68 or 1.95e142 < x

   1. Initial program 95.0%

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

   3. Taylor expanded in y around 0 93.9%

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

      1. fma-neg 93.9%

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

      2. sub-neg 93.9%

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

      3. metadata-eval 93.9%

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

      4. +-commutative 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in x around inf 78.3%

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

      1. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   if -1.06e68 < x < 1.95e142

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf 48.9%

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

      1. *-commutative 48.9%

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

      2. sub-neg 48.9%

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

      3. log1p-define 63.4%

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

   5. Simplified 63.4%

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

   6. Taylor expanded in y around 0 63.4%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in y around 0 63.4%

      \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)}\right) - t \]
   8. Step-by-step derivation

      1. +-commutative 63.4%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(y \cdot z\right) + -0.5 \cdot z\right)}\right) - t \]

      2. associate-*r* 63.4%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot z} + -0.5 \cdot z\right)\right) - t \]

      3. distribute-rgt-out 63.4%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 \cdot y + -0.5\right)\right)}\right) - t \]

      4. *-commutative 63.4%

         \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.3333333333333333} + -0.5\right)\right)\right) - t \]

   9. Simplified 63.4%

      \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(z \cdot \left(y \cdot -0.3333333333333333 + -0.5\right)\right)}\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06 \cdot 10^{+68} \lor \neg \left(x \leq 1.95 \cdot 10^{+142}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 87.8%

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

3. Taylor expanded in y around 0 99.2%

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

   1. +-commutative 99.2%

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

   2. sub-neg 99.2%

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

   3. metadata-eval 99.2%

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

   4. fma-define 99.2%

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

   5. mul-1-neg 99.2%

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

   6. fma-neg 99.2%

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

   7. +-commutative 99.2%

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

   8. sub-neg 99.2%

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

   9. metadata-eval 99.2%

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

   10. +-commutative 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 12: 45.9% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
  t))

C

double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - 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) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
}

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.9%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 48.9%

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

8. Final simplification 48.9%

   \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t \]
9. Add Preprocessing

Alternative 13: 42.5% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00105 \lor \neg \left(t \leq 1550000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -0.00105) (not (<= t 1550000.0))) (- t) (* y (- 1.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00105) || !(t <= 1550000.0)) {
		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 <= (-0.00105d0)) .or. (.not. (t <= 1550000.0d0))) 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 <= -0.00105) || !(t <= 1550000.0)) {
		tmp = -t;
	} else {
		tmp = y * (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -0.00105) or not (t <= 1550000.0):
		tmp = -t
	else:
		tmp = y * (1.0 - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -0.00105) || !(t <= 1550000.0))
		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 <= -0.00105) || ~((t <= 1550000.0)))
		tmp = -t;
	else
		tmp = y * (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -0.00105], N[Not[LessEqual[t, 1550000.0]], $MachinePrecision]], (-t), N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00104999999999999994 or 1.55e6 < t

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf 67.3%

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

      1. mul-1-neg 67.3%

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

   5. Simplified 67.3%

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

   if -0.00104999999999999994 < t < 1.55e6

   1. Initial program 80.7%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. associate-*r* 99.1%

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

      3. mul-1-neg 99.1%

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

      4. fma-define 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. fma-neg 99.1%

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

      9. sub-neg 99.1%

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

      10. metadata-eval 99.1%

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

      11. +-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in y around inf 20.7%

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

      1. sub-neg 20.7%

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

      2. metadata-eval 20.7%

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

      3. +-commutative 20.7%

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

      4. neg-mul-1 20.7%

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

      5. distribute-rgt-neg-in 20.7%

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

      6. distribute-neg-in 20.7%

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

      7. metadata-eval 20.7%

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

      8. unsub-neg 20.7%

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

   8. Simplified 20.7%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00105 \lor \neg \left(t \leq 1550000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* y (- (* y (* z (+ -0.5 (* y -0.3333333333333333)))) z)) t))

C

double code(double x, double y, double z, double t) {
	return (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - 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 * ((y * (z * ((-0.5d0) + (y * (-0.3333333333333333d0))))) - z)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
}

Python

def code(x, y, z, t):
	return (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(Float64(y * Float64(z * Float64(-0.5 + Float64(y * -0.3333333333333333)))) - z)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * ((y * (z * (-0.5 + (y * -0.3333333333333333)))) - z)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(N[(y * N[(z * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.9%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in y around 0 48.9%

   \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)}\right) - t \]
8. Step-by-step derivation

   1. +-commutative 48.9%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(y \cdot z\right) + -0.5 \cdot z\right)}\right) - t \]

   2. associate-*r* 48.9%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot z} + -0.5 \cdot z\right)\right) - t \]

   3. distribute-rgt-out 48.9%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 \cdot y + -0.5\right)\right)}\right) - t \]

   4. *-commutative 48.9%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.3333333333333333} + -0.5\right)\right)\right) - t \]

9. Simplified 48.9%

   \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(z \cdot \left(y \cdot -0.3333333333333333 + -0.5\right)\right)}\right) - t \]

10. Final simplification 48.9%

    \[\leadsto y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) - t \]
11. Add Preprocessing

Alternative 15: 42.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00105 \lor \neg \left(t \leq 6700000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -0.00105) (not (<= t 6700000.0))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00105) || !(t <= 6700000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-0.00105d0)) .or. (.not. (t <= 6700000.0d0))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -0.00105) || !(t <= 6700000.0)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -0.00105) or not (t <= 6700000.0):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -0.00105) || !(t <= 6700000.0))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -0.00105) || ~((t <= 6700000.0)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -0.00105], N[Not[LessEqual[t, 6700000.0]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00104999999999999994 or 6.7e6 < t

   1. Initial program 94.1%

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

   3. Taylor expanded in t around inf 67.3%

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

      1. mul-1-neg 67.3%

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

   5. Simplified 67.3%

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

   if -0.00104999999999999994 < t < 6.7e6

   1. Initial program 80.7%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. associate-*r* 99.1%

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

      3. mul-1-neg 99.1%

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

      4. fma-define 99.1%

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

      5. sub-neg 99.1%

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

      6. metadata-eval 99.1%

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

      7. +-commutative 99.1%

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

      8. fma-neg 99.1%

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

      9. sub-neg 99.1%

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

      10. metadata-eval 99.1%

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

      11. +-commutative 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in z around inf 20.2%

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

      1. associate-*r* 20.2%

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

      2. mul-1-neg 20.2%

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

   8. Simplified 20.2%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00105 \lor \neg \left(t \leq 6700000\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.8% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t) {
	return (y * ((-0.5 * (z * y)) - z)) - t;
}

Python

def code(x, y, z, t):
	return (y * ((-0.5 * (z * y)) - z)) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.9%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + -0.5 \cdot \left(y \cdot z\right)\right)} - t \]
7. Step-by-step derivation

   1. +-commutative 48.9%

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

   2. mul-1-neg 48.9%

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

   3. unsub-neg 48.9%

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

   4. *-commutative 48.9%

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

8. Simplified 48.9%

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

9. Final simplification 48.9%

   \[\leadsto y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t \]
10. Add Preprocessing

Alternative 17: 45.5% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in z around inf 38.1%

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

   1. *-commutative 38.1%

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

   2. sub-neg 38.1%

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

   3. log1p-define 49.1%

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

5. Simplified 49.1%

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

6. Taylor expanded in y around 0 48.5%

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

   1. associate-*r* 48.5%

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

   2. mul-1-neg 48.5%

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

8. Simplified 48.5%

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

9. Final simplification 48.5%

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

Alternative 18: 34.9% 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 87.8%

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

3. Taylor expanded in t around inf 37.1%

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

   1. mul-1-neg 37.1%

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

5. Simplified 37.1%

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

6. Final simplification 37.1%

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

Alternative 19: 2.3% accurate, 215.0× 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 t
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in y around 0 86.9%

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

   1. fma-neg 86.9%

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

   2. sub-neg 86.9%

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

   3. metadata-eval 86.9%

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

   4. +-commutative 86.9%

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

5. Simplified 86.9%

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

   1. add-cube-cbrt 85.4%

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

   2. pow3 85.4%

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

   3. add-sqr-sqrt 43.0%

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

   4. sqrt-unprod 51.9%

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

   5. sqr-neg 51.9%

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

   6. sqrt-unprod 24.0%

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

   7. add-sqr-sqrt 48.7%

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

7. Applied egg-rr 48.7%

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

8. Taylor expanded in t around inf 2.1%

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

9. Final simplification 2.1%

   \[\leadsto t \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))