Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 85.5% → 99.8%

Time: 14.9s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. +-commutative 84.2%

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

   2. associate--l+ 84.2%

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

   3. +-commutative 84.2%

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

   4. associate-+l- 84.2%

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

   5. fma-neg 84.2%

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

   6. sub0-neg 84.2%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - \left(t - z \cdot \log \left(1 - y\right)\right)}\right) \]

   7. associate-+l- 84.2%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\left(0 - t\right) + z \cdot \log \left(1 - y\right)}\right) \]

   8. neg-sub0 84.2%

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

   9. +-commutative 84.2%

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

   10. fma-def 84.2%

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

   11. sub-neg 84.2%

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

   12. log1p-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. *-lft-identity 84.2%

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

   2. *-lft-identity 84.2%

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

   3. fma-def 84.2%

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

   4. sub-neg 84.2%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+141} \lor \neg \left(x \leq -1.38 \cdot 10^{+103}\right) \land \left(x \leq -3.1 \cdot 10^{+34} \lor \neg \left(x \leq 7.5 \cdot 10^{-5}\right) \land \left(x \leq 1.4 \cdot 10^{+31} \lor \neg \left(x \leq 4.6 \cdot 10^{+125}\right)\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.28e+141)
         (and (not (<= x -1.38e+103))
              (or (<= x -3.1e+34)
                  (and (not (<= x 7.5e-5))
                       (or (<= x 1.4e+31) (not (<= x 4.6e+125)))))))
   (* x (log y))
   (- (- t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.28e+141) || (!(x <= -1.38e+103) && ((x <= -3.1e+34) || (!(x <= 7.5e-5) && ((x <= 1.4e+31) || !(x <= 4.6e+125)))))) {
		tmp = x * log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.28d+141)) .or. (.not. (x <= (-1.38d+103))) .and. (x <= (-3.1d+34)) .or. (.not. (x <= 7.5d-5)) .and. (x <= 1.4d+31) .or. (.not. (x <= 4.6d+125))) then
        tmp = x * log(y)
    else
        tmp = -t - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.28e+141) || (!(x <= -1.38e+103) && ((x <= -3.1e+34) || (!(x <= 7.5e-5) && ((x <= 1.4e+31) || !(x <= 4.6e+125)))))) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.28e+141) or (not (x <= -1.38e+103) and ((x <= -3.1e+34) or (not (x <= 7.5e-5) and ((x <= 1.4e+31) or not (x <= 4.6e+125))))):
		tmp = x * math.log(y)
	else:
		tmp = -t - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.28e+141) || (!(x <= -1.38e+103) && ((x <= -3.1e+34) || (!(x <= 7.5e-5) && ((x <= 1.4e+31) || !(x <= 4.6e+125))))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.28e+141) || (~((x <= -1.38e+103)) && ((x <= -3.1e+34) || (~((x <= 7.5e-5)) && ((x <= 1.4e+31) || ~((x <= 4.6e+125)))))))
		tmp = x * log(y);
	else
		tmp = -t - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.28e+141], And[N[Not[LessEqual[x, -1.38e+103]], $MachinePrecision], Or[LessEqual[x, -3.1e+34], And[N[Not[LessEqual[x, 7.5e-5]], $MachinePrecision], Or[LessEqual[x, 1.4e+31], N[Not[LessEqual[x, 4.6e+125]], $MachinePrecision]]]]]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.28000000000000004e141 or -1.37999999999999993e103 < x < -3.09999999999999977e34 or 7.49999999999999934e-5 < x < 1.40000000000000008e31 or 4.60000000000000026e125 < x

   1. Initial program 96.3%

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

      1. *-lft-identity 96.3%

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

      2. *-lft-identity 96.3%

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

      3. fma-def 96.3%

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

      4. sub-neg 96.3%

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

      5. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 99.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(-1 \cdot \left(y \cdot z\right) + \log y \cdot x\right)\right)} - t \]
   5. Step-by-step derivation

      1. associate-+r+ 99.7%

         \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + -1 \cdot \left(y \cdot z\right)\right) + \log y \cdot x\right)} - t \]

      2. associate-*r* 99.7%

         \[\leadsto \left(\left(\color{blue}{\left(-0.5 \cdot {y}^{2}\right) \cdot z} + -1 \cdot \left(y \cdot z\right)\right) + \log y \cdot x\right) - t \]

      3. associate-*r* 99.7%

         \[\leadsto \left(\left(\left(-0.5 \cdot {y}^{2}\right) \cdot z + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) + \log y \cdot x\right) - t \]

      4. distribute-rgt-out 99.7%

         \[\leadsto \left(\color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)} + \log y \cdot x\right) - t \]

      5. mul-1-neg 99.7%

         \[\leadsto \left(z \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right) + \log y \cdot x\right) - t \]

      6. unsub-neg 99.7%

         \[\leadsto \left(z \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)} + \log y \cdot x\right) - t \]

      7. *-commutative 99.7%

         \[\leadsto \left(z \cdot \left(\color{blue}{{y}^{2} \cdot -0.5} - y\right) + \log y \cdot x\right) - t \]

      8. unpow2 99.7%

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

      9. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 82.8%

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

   if -1.28000000000000004e141 < x < -1.37999999999999993e103 or -3.09999999999999977e34 < x < 7.49999999999999934e-5 or 1.40000000000000008e31 < x < 4.60000000000000026e125

   1. Initial program 78.2%

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

   2. Taylor expanded in x around 0 60.7%

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

      1. sub-neg 60.7%

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

      2. mul-1-neg 60.7%

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

      3. log1p-def 82.5%

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

      4. mul-1-neg 82.5%

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

   4. Simplified 82.5%

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

   5. Taylor expanded in y around 0 81.2%

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

      1. neg-mul-1 81.2%

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

      2. +-commutative 81.2%

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

      3. unsub-neg 81.2%

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

      4. mul-1-neg 81.2%

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

      5. distribute-rgt-neg-in 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+141} \lor \neg \left(x \leq -1.38 \cdot 10^{+103}\right) \land \left(x \leq -3.1 \cdot 10^{+34} \lor \neg \left(x \leq 7.5 \cdot 10^{-5}\right) \land \left(x \leq 1.4 \cdot 10^{+31} \lor \neg \left(x \leq 4.6 \cdot 10^{+125}\right)\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 4: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-t\right) - y \cdot z\\ \mathbf{if}\;x \leq -1.28 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+31} \lor \neg \left(x \leq 4.6 \cdot 10^{+125}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- t) (* y z))))
   (if (<= x -1.28e+141)
     t_1
     (if (<= x -5.5e+101)
       t_2
       (if (<= x -1.8e+36)
         t_1
         (if (<= x 4e-5)
           (- (fma y z t))
           (if (or (<= x 1.3e+31) (not (<= x 4.6e+125))) t_1 t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = -t - (y * z);
	double tmp;
	if (x <= -1.28e+141) {
		tmp = t_1;
	} else if (x <= -5.5e+101) {
		tmp = t_2;
	} else if (x <= -1.8e+36) {
		tmp = t_1;
	} else if (x <= 4e-5) {
		tmp = -fma(y, z, t);
	} else if ((x <= 1.3e+31) || !(x <= 4.6e+125)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(-t) - Float64(y * z))
	tmp = 0.0
	if (x <= -1.28e+141)
		tmp = t_1;
	elseif (x <= -5.5e+101)
		tmp = t_2;
	elseif (x <= -1.8e+36)
		tmp = t_1;
	elseif (x <= 4e-5)
		tmp = Float64(-fma(y, z, t));
	elseif ((x <= 1.3e+31) || !(x <= 4.6e+125))
		tmp = t_1;
	else
		tmp = t_2;
	end
	return 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[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.28e+141], t$95$1, If[LessEqual[x, -5.5e+101], t$95$2, If[LessEqual[x, -1.8e+36], t$95$1, If[LessEqual[x, 4e-5], (-N[(y * z + t), $MachinePrecision]), If[Or[LessEqual[x, 1.3e+31], N[Not[LessEqual[x, 4.6e+125]], $MachinePrecision]], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.28000000000000004e141 or -5.50000000000000018e101 < x < -1.7999999999999999e36 or 4.00000000000000033e-5 < x < 1.3e31 or 4.60000000000000026e125 < x

   1. Initial program 96.3%

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

      1. *-lft-identity 96.3%

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

      2. *-lft-identity 96.3%

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

      3. fma-def 96.3%

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

      4. sub-neg 96.3%

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

      5. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 99.7%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(-1 \cdot \left(y \cdot z\right) + \log y \cdot x\right)\right)} - t \]
   5. Step-by-step derivation

      1. associate-+r+ 99.7%

         \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + -1 \cdot \left(y \cdot z\right)\right) + \log y \cdot x\right)} - t \]

      2. associate-*r* 99.7%

         \[\leadsto \left(\left(\color{blue}{\left(-0.5 \cdot {y}^{2}\right) \cdot z} + -1 \cdot \left(y \cdot z\right)\right) + \log y \cdot x\right) - t \]

      3. associate-*r* 99.7%

         \[\leadsto \left(\left(\left(-0.5 \cdot {y}^{2}\right) \cdot z + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) + \log y \cdot x\right) - t \]

      4. distribute-rgt-out 99.7%

         \[\leadsto \left(\color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)} + \log y \cdot x\right) - t \]

      5. mul-1-neg 99.7%

         \[\leadsto \left(z \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right) + \log y \cdot x\right) - t \]

      6. unsub-neg 99.7%

         \[\leadsto \left(z \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)} + \log y \cdot x\right) - t \]

      7. *-commutative 99.7%

         \[\leadsto \left(z \cdot \left(\color{blue}{{y}^{2} \cdot -0.5} - y\right) + \log y \cdot x\right) - t \]

      8. unpow2 99.7%

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

      9. associate-*l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around inf 82.8%

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

   if -1.28000000000000004e141 < x < -5.50000000000000018e101 or 1.3e31 < x < 4.60000000000000026e125

   1. Initial program 86.8%

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

   2. Taylor expanded in x around 0 61.7%

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

      1. sub-neg 61.7%

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

      2. mul-1-neg 61.7%

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

      3. log1p-def 75.0%

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

      4. mul-1-neg 75.0%

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

   4. Simplified 75.0%

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

   5. Taylor expanded in y around 0 73.3%

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

      1. neg-mul-1 73.3%

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

      2. +-commutative 73.3%

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

      3. unsub-neg 73.3%

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

      4. mul-1-neg 73.3%

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

      5. distribute-rgt-neg-in 73.3%

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

   7. Simplified 73.3%

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

   if -1.7999999999999999e36 < x < 4.00000000000000033e-5

   1. Initial program 76.3%

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

   2. Taylor expanded in x around 0 60.5%

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

      1. sub-neg 60.5%

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

      2. mul-1-neg 60.5%

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

      3. log1p-def 84.1%

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

      4. mul-1-neg 84.1%

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

   4. Simplified 84.1%

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

   5. Taylor expanded in y around 0 83.0%

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

      1. neg-mul-1 83.0%

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

      2. +-commutative 83.0%

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

      3. unsub-neg 83.0%

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

      4. mul-1-neg 83.0%

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

      5. distribute-rgt-neg-in 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in y around 0 83.0%

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

      1. distribute-lft-out 83.0%

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

      2. +-commutative 83.0%

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

      3. fma-def 83.0%

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

      4. neg-mul-1 83.0%

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

   10. Simplified 83.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.28 \cdot 10^{+141}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+101}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-5}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+31} \lor \neg \left(x \leq 4.6 \cdot 10^{+125}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. *-lft-identity 84.2%

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

   2. *-lft-identity 84.2%

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

   3. fma-def 84.2%

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

   4. sub-neg 84.2%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in y around 0 99.3%

   \[\leadsto \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(-1 \cdot \left(y \cdot z\right) + \log y \cdot x\right)\right)} - t \]
5. Step-by-step derivation

   1. associate-+r+ 99.3%

      \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + -1 \cdot \left(y \cdot z\right)\right) + \log y \cdot x\right)} - t \]

   2. associate-*r* 99.3%

      \[\leadsto \left(\left(\color{blue}{\left(-0.5 \cdot {y}^{2}\right) \cdot z} + -1 \cdot \left(y \cdot z\right)\right) + \log y \cdot x\right) - t \]

   3. associate-*r* 99.3%

      \[\leadsto \left(\left(\left(-0.5 \cdot {y}^{2}\right) \cdot z + \color{blue}{\left(-1 \cdot y\right) \cdot z}\right) + \log y \cdot x\right) - t \]

   4. distribute-rgt-out 99.3%

      \[\leadsto \left(\color{blue}{z \cdot \left(-0.5 \cdot {y}^{2} + -1 \cdot y\right)} + \log y \cdot x\right) - t \]

   5. mul-1-neg 99.3%

      \[\leadsto \left(z \cdot \left(-0.5 \cdot {y}^{2} + \color{blue}{\left(-y\right)}\right) + \log y \cdot x\right) - t \]

   6. unsub-neg 99.3%

      \[\leadsto \left(z \cdot \color{blue}{\left(-0.5 \cdot {y}^{2} - y\right)} + \log y \cdot x\right) - t \]

   7. *-commutative 99.3%

      \[\leadsto \left(z \cdot \left(\color{blue}{{y}^{2} \cdot -0.5} - y\right) + \log y \cdot x\right) - t \]

   8. unpow2 99.3%

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

   9. associate-*l* 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

   \[\leadsto \left(z \cdot \left(y \cdot \left(y \cdot -0.5\right) - y\right) + x \cdot \log y\right) - t \]

Alternative 6: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-76} \lor \neg \left(x \leq 1.3 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.5e-76) (not (<= x 1.3e-43)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.5e-76) || !(x <= 1.3e-43)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.5e-76) || !(x <= 1.3e-43)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.5e-76) or not (x <= 1.3e-43):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.5e-76) || !(x <= 1.3e-43))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.5e-76], N[Not[LessEqual[x, 1.3e-43]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999997e-76 or 1.3e-43 < x

   1. Initial program 93.8%

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

   2. Taylor expanded in y around 0 92.4%

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

   if -3.49999999999999997e-76 < x < 1.3e-43

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0 62.9%

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

      1. sub-neg 62.9%

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

      2. mul-1-neg 62.9%

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

      3. log1p-def 92.5%

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

      4. mul-1-neg 92.5%

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

   4. Simplified 92.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-76} \lor \neg \left(x \leq 1.3 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 7: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(-t\right) - x \cdot \log \left(\frac{1}{y}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1e-76)
   (- (- t) (* x (log (/ 1.0 y))))
   (if (<= x 1.2e-43) (- (* z (log1p (- y))) t) (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1e-76) {
		tmp = -t - (x * log((1.0 / y)));
	} else if (x <= 1.2e-43) {
		tmp = (z * log1p(-y)) - t;
	} else {
		tmp = (x * log(y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1e-76) {
		tmp = -t - (x * Math.log((1.0 / y)));
	} else if (x <= 1.2e-43) {
		tmp = (z * Math.log1p(-y)) - t;
	} else {
		tmp = (x * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1e-76:
		tmp = -t - (x * math.log((1.0 / y)))
	elif x <= 1.2e-43:
		tmp = (z * math.log1p(-y)) - t
	else:
		tmp = (x * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1e-76)
		tmp = Float64(Float64(-t) - Float64(x * log(Float64(1.0 / y))));
	elseif (x <= 1.2e-43)
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1e-76], N[((-t) - N[(x * N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-43], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999927e-77

   1. Initial program 94.2%

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

   2. Taylor expanded in y around inf 94.2%

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

   3. Taylor expanded in z around 0 91.4%

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

   if -9.99999999999999927e-77 < x < 1.2000000000000001e-43

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0 62.9%

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

      1. sub-neg 62.9%

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

      2. mul-1-neg 62.9%

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

      3. log1p-def 92.5%

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

      4. mul-1-neg 92.5%

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

   4. Simplified 92.5%

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

   if 1.2000000000000001e-43 < x

   1. Initial program 93.4%

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

   2. Taylor expanded in y around 0 93.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-76}:\\ \;\;\;\;\left(-t\right) - x \cdot \log \left(\frac{1}{y}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]

Alternative 8: 89.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-77} \lor \neg \left(x \leq 1.2 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.5e-77) (not (<= x 1.2e-43)))
   (- (* x (log y)) t)
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.5e-77) || !(x <= 1.2e-43)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.5e-77) || !(x <= 1.2e-43))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.5e-77], N[Not[LessEqual[x, 1.2e-43]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -6.4999999999999999e-77 or 1.2000000000000001e-43 < x

   1. Initial program 93.8%

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

   2. Taylor expanded in y around 0 92.4%

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

   if -6.4999999999999999e-77 < x < 1.2000000000000001e-43

   1. Initial program 70.4%

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

   2. Taylor expanded in x around 0 62.9%

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

      1. sub-neg 62.9%

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

      2. mul-1-neg 62.9%

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

      3. log1p-def 92.5%

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

      4. mul-1-neg 92.5%

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

   4. Simplified 92.5%

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

   5. Taylor expanded in y around 0 91.7%

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

      1. neg-mul-1 91.7%

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

      2. +-commutative 91.7%

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

      3. unsub-neg 91.7%

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

      4. mul-1-neg 91.7%

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

      5. distribute-rgt-neg-in 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in y around 0 91.7%

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

      1. distribute-lft-out 91.7%

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

      2. +-commutative 91.7%

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

      3. fma-def 91.7%

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

      4. neg-mul-1 91.7%

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

   10. Simplified 91.7%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-77} \lor \neg \left(x \leq 1.2 \cdot 10^{-43}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 9: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in y around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. *-commutative 99.0%

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

   3. mul-1-neg 99.0%

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

   4. unsub-neg 99.0%

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

   5. *-commutative 99.0%

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

4. Simplified 99.0%

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

5. Final simplification 99.0%

   \[\leadsto \left(x \cdot \log y - y \cdot z\right) - t \]

Alternative 10: 48.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-20}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-34}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8e-20) (- t) (if (<= t 1.26e-34) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8e-20) {
		tmp = -t;
	} else if (t <= 1.26e-34) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8d-20)) then
        tmp = -t
    else if (t <= 1.26d-34) then
        tmp = y * -z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8e-20) {
		tmp = -t;
	} else if (t <= 1.26e-34) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -8e-20:
		tmp = -t
	elif t <= 1.26e-34:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8e-20)
		tmp = Float64(-t);
	elseif (t <= 1.26e-34)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8e-20)
		tmp = -t;
	elseif (t <= 1.26e-34)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8e-20], (-t), If[LessEqual[t, 1.26e-34], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -7.99999999999999956e-20 or 1.26000000000000009e-34 < t

   1. Initial program 94.9%

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

   2. Taylor expanded in t around inf 72.5%

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

      1. mul-1-neg 72.5%

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

   4. Simplified 72.5%

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

   if -7.99999999999999956e-20 < t < 1.26000000000000009e-34

   1. Initial program 72.5%

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

   2. Taylor expanded in x around 0 15.9%

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

      1. sub-neg 15.9%

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

      2. mul-1-neg 15.9%

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

      3. log1p-def 43.3%

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

      4. mul-1-neg 43.3%

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

   4. Simplified 43.3%

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

   5. Taylor expanded in y around 0 41.6%

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

      1. neg-mul-1 41.6%

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

      2. +-commutative 41.6%

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

      3. unsub-neg 41.6%

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

      4. mul-1-neg 41.6%

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

      5. distribute-rgt-neg-in 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in y around inf 30.8%

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

      1. mul-1-neg 30.8%

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

      2. distribute-rgt-neg-in 30.8%

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

   10. Simplified 30.8%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-20}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.26 \cdot 10^{-34}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 57.1% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in x around 0 45.7%

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

   1. sub-neg 45.7%

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

   2. mul-1-neg 45.7%

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

   3. log1p-def 61.4%

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

   4. mul-1-neg 61.4%

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

4. Simplified 61.4%

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

5. Taylor expanded in y around 0 60.6%

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

   1. neg-mul-1 60.6%

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

   2. +-commutative 60.6%

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

   3. unsub-neg 60.6%

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

   4. mul-1-neg 60.6%

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

   5. distribute-rgt-neg-in 60.6%

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

7. Simplified 60.6%

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

8. Final simplification 60.6%

   \[\leadsto \left(-t\right) - y \cdot z \]

Alternative 12: 43.0% accurate, 105.5× 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 84.2%

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

2. Taylor expanded in t around inf 44.3%

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

   1. mul-1-neg 44.3%

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

4. Simplified 44.3%

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

5. Final simplification 44.3%

   \[\leadsto -t \]

Alternative 13: 2.3% accurate, 211.0× 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 84.2%

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

   1. sub-neg 84.2%

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

   2. associate-+r+ 84.2%

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

   3. sub-neg 84.2%

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

   4. log1p-udef 99.8%

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

   5. fma-udef 99.9%

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

   6. *-commutative 99.9%

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

   7. add-sqr-sqrt 51.4%

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

   8. associate-*r* 51.4%

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

   9. fma-def 51.4%

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

   10. add-sqr-sqrt 0.0%

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

   11. sqrt-unprod 42.3%

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

   12. sqr-neg 42.3%

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

   13. sqrt-unprod 42.2%

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

   14. add-sqr-sqrt 42.2%

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

   15. add-sqr-sqrt 20.7%

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

   16. sqrt-unprod 23.9%

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

   17. sqr-neg 23.9%

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

   18. sqrt-unprod 10.6%

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

   19. add-sqr-sqrt 19.7%

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

3. Applied egg-rr 19.7%

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

4. Taylor expanded in t around inf 2.2%

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

5. Final simplification 2.2%

   \[\leadsto t \]

Developer target: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023200 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))