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

Percentage Accurate: 85.0% → 99.8%

Time: 16.0s

Alternatives: 11

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: 85.0% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. add-sqr-sqrt 45.7%

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

   2. pow2 45.7%

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

4. Applied egg-rr 45.7%

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

   1. unpow2 45.7%

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

   2. add-sqr-sqrt 85.6%

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

   3. *-commutative 85.6%

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

   4. add-sqr-sqrt 39.7%

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

   5. associate-*r* 39.7%

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

6. Applied egg-rr 39.7%

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

7. Taylor expanded in x around 0 85.6%

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

   1. +-commutative 85.6%

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

   2. *-commutative 85.6%

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

   3. fma-define 85.6%

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

   4. sub-neg 85.6%

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

   5. mul-1-neg 85.6%

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

   6. log1p-define 99.9%

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

   7. mul-1-neg 99.9%

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

   8. *-commutative 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + \left(-0.5 \cdot \left(z \cdot {y}^{2}\right) - z \cdot y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (- (* -0.5 (* z (pow y 2.0))) (* z y))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + ((-0.5 * (z * pow(y, 2.0))) - (z * y))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (((-0.5d0) * (z * (y ** 2.0d0))) - (z * y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + ((-0.5 * (z * Math.pow(y, 2.0))) - (z * y))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + ((-0.5 * (z * math.pow(y, 2.0))) - (z * y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(Float64(-0.5 * Float64(z * (y ^ 2.0))) - Float64(z * y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + ((-0.5 * (z * (y ^ 2.0))) - (z * y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(z * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in y around 0 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \left(z \cdot {y}^{2}\right) - z \cdot y\right)\right) - t \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

   1. associate--l+ 85.6%

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

   2. fma-define 85.6%

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

   3. sub-neg 85.6%

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

   4. log1p-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around 0 99.6%

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

   1. neg-mul-1 99.6%

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

   2. +-commutative 99.6%

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

   3. unsub-neg 99.6%

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

   4. mul-1-neg 99.6%

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

   5. distribute-rgt-neg-in 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -1.22 \cdot 10^{-113} \lor \neg \left(t \leq 4.7 \cdot 10^{-153}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -1.22e-113) (not (<= t 4.7e-153)))
     (- t_1 t)
     (- t_1 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -1.22e-113) || !(t <= 4.7e-153)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((t <= (-1.22d-113)) .or. (.not. (t <= 4.7d-153))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if ((t <= -1.22e-113) || !(t <= 4.7e-153)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -1.22e-113) or not (t <= 4.7e-153):
		tmp = t_1 - t
	else:
		tmp = t_1 - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -1.22e-113) || !(t <= 4.7e-153))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -1.22e-113) || ~((t <= 4.7e-153)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -1.22e-113], N[Not[LessEqual[t, 4.7e-153]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.21999999999999995e-113 or 4.6999999999999999e-153 < t

   1. Initial program 93.5%

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

      1. associate--l+ 93.5%

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

      2. fma-define 93.5%

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

      3. sub-neg 93.5%

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

      4. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 93.4%

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

   if -1.21999999999999995e-113 < t < 4.6999999999999999e-153

   1. Initial program 68.1%

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

      1. associate--l+ 68.1%

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

      2. fma-define 68.1%

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

      3. sub-neg 68.1%

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

      4. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 98.9%

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

      1. neg-mul-1 98.9%

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

      2. +-commutative 98.9%

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

      3. unsub-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-rgt-neg-in 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in t around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. *-commutative 89.5%

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

   10. Simplified 89.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{-113} \lor \neg \left(t \leq 4.7 \cdot 10^{-153}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-67} \lor \neg \left(x \leq 2 \cdot 10^{-92}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.25e-67) (not (<= x 2e-92)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.25e-67) || !(x <= 2e-92)) {
		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 <= -1.25e-67) || !(x <= 2e-92)) {
		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 <= -1.25e-67) or not (x <= 2e-92):
		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 <= -1.25e-67) || !(x <= 2e-92))
		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, -1.25e-67], N[Not[LessEqual[x, 2e-92]], $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 < -1.25e-67 or 1.99999999999999998e-92 < x

   1. Initial program 89.8%

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

      1. associate--l+ 89.8%

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

      2. fma-define 89.8%

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

      3. sub-neg 89.8%

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

      4. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 89.7%

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

   if -1.25e-67 < x < 1.99999999999999998e-92

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0 74.2%

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

      1. sub-neg 74.2%

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

      2. mul-1-neg 74.2%

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

      3. log1p-define 95.7%

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

      4. mul-1-neg 95.7%

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

   5. Simplified 95.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-67} \lor \neg \left(x \leq 2 \cdot 10^{-92}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-66} \lor \neg \left(x \leq 1.6 \cdot 10^{-92}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.4e-66) (not (<= x 1.6e-92)))
   (- (* x (log y)) t)
   (- (- t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.4e-66) || !(x <= 1.6e-92)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -t - (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 ((x <= (-4.4d-66)) .or. (.not. (x <= 1.6d-92))) then
        tmp = (x * log(y)) - t
    else
        tmp = -t - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.4e-66) || !(x <= 1.6e-92)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.4e-66) or not (x <= 1.6e-92):
		tmp = (x * math.log(y)) - t
	else:
		tmp = -t - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.4e-66) || !(x <= 1.6e-92))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(-t) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.4e-66) || ~((x <= 1.6e-92)))
		tmp = (x * log(y)) - t;
	else
		tmp = -t - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.4e-66], N[Not[LessEqual[x, 1.6e-92]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000002e-66 or 1.5999999999999998e-92 < x

   1. Initial program 89.8%

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

      1. associate--l+ 89.8%

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

      2. fma-define 89.8%

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

      3. sub-neg 89.8%

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

      4. log1p-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 89.7%

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

   if -4.4000000000000002e-66 < x < 1.5999999999999998e-92

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0 74.2%

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

      1. sub-neg 74.2%

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

      2. mul-1-neg 74.2%

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

      3. log1p-define 95.7%

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

      4. mul-1-neg 95.7%

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

   5. Simplified 95.7%

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

   6. Taylor expanded in y around 0 95.0%

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

      1. mul-1-neg 95.0%

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

      2. *-commutative 95.0%

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

      3. distribute-rgt-neg-in 95.0%

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

   8. Simplified 95.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-66} \lor \neg \left(x \leq 1.6 \cdot 10^{-92}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 1.1 \cdot 10^{+153}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5e-8) (not (<= x 1.1e+153))) (* x (log y)) (- (- t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 1.1e+153)) {
		tmp = x * log(y);
	} else {
		tmp = -t - (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 ((x <= (-5d-8)) .or. (.not. (x <= 1.1d+153))) then
        tmp = x * log(y)
    else
        tmp = -t - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e-8) || !(x <= 1.1e+153)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5e-8) or not (x <= 1.1e+153):
		tmp = x * math.log(y)
	else:
		tmp = -t - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5e-8) || !(x <= 1.1e+153))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5e-8) || ~((x <= 1.1e+153)))
		tmp = x * log(y);
	else
		tmp = -t - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5e-8], N[Not[LessEqual[x, 1.1e+153]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(z * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8 or 1.1e153 < x

   1. Initial program 95.4%

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

      1. associate--l+ 95.4%

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

      2. fma-define 95.4%

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

      3. sub-neg 95.4%

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

      4. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 99.7%

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

      1. neg-mul-1 99.7%

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

      2. +-commutative 99.7%

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

      3. unsub-neg 99.7%

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

      4. mul-1-neg 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 72.1%

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

   if -4.9999999999999998e-8 < x < 1.1e153

   1. Initial program 80.0%

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

   3. Taylor expanded in x around 0 61.5%

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

      1. sub-neg 61.5%

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

      2. mul-1-neg 61.5%

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

      3. log1p-define 80.8%

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

      4. mul-1-neg 80.8%

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

   5. Simplified 80.8%

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

   6. Taylor expanded in y around 0 80.4%

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

      1. mul-1-neg 80.4%

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

      2. *-commutative 80.4%

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

      3. distribute-rgt-neg-in 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8} \lor \neg \left(x \leq 1.1 \cdot 10^{+153}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in y around 0 99.6%

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

   1. +-commutative 99.6%

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

   2. mul-1-neg 99.6%

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

   3. unsub-neg 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 9: 49.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-113} \lor \neg \left(t \leq 4.5 \cdot 10^{-149}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.8e-113) (not (<= t 4.5e-149))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.8e-113) || !(t <= 4.5e-149)) {
		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 <= (-1.8d-113)) .or. (.not. (t <= 4.5d-149))) 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 <= -1.8e-113) || !(t <= 4.5e-149)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.8e-113) or not (t <= 4.5e-149):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.8e-113) || !(t <= 4.5e-149))
		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 <= -1.8e-113) || ~((t <= 4.5e-149)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.8e-113], N[Not[LessEqual[t, 4.5e-149]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.79999999999999987e-113 or 4.4999999999999998e-149 < t

   1. Initial program 93.5%

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

      1. associate--l+ 93.5%

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

      2. fma-define 93.5%

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

      3. sub-neg 93.5%

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

      4. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 63.6%

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

      1. neg-mul-1 63.6%

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

   7. Simplified 63.6%

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

   if -1.79999999999999987e-113 < t < 4.4999999999999998e-149

   1. Initial program 68.5%

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

   3. Taylor expanded in x around 0 13.5%

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

      1. sub-neg 13.5%

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

      2. mul-1-neg 13.5%

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

      3. log1p-define 43.9%

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

      4. mul-1-neg 43.9%

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

   5. Simplified 43.9%

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

   6. Taylor expanded in y around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. *-commutative 43.1%

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

      3. distribute-rgt-neg-in 43.1%

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

   8. Simplified 43.1%

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

   9. Taylor expanded in z around inf 34.0%

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

       1. associate-*r* 34.0%

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

       2. mul-1-neg 34.0%

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

   11. Simplified 34.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-113} \lor \neg \left(t \leq 4.5 \cdot 10^{-149}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 35.2× 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 85.6%

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

3. Taylor expanded in x around 0 47.8%

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

   1. sub-neg 47.8%

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

   2. mul-1-neg 47.8%

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

   3. log1p-define 61.5%

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

   4. mul-1-neg 61.5%

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

5. Simplified 61.5%

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

6. Taylor expanded in y around 0 61.2%

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

   1. mul-1-neg 61.2%

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

   2. *-commutative 61.2%

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

   3. distribute-rgt-neg-in 61.2%

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

8. Simplified 61.2%

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

9. Final simplification 61.2%

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

Alternative 11: 43.3% 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 85.6%

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

   1. associate--l+ 85.6%

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

   2. fma-define 85.6%

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

   3. sub-neg 85.6%

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

   4. log1p-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in t around inf 47.3%

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

   1. neg-mul-1 47.3%

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

7. Simplified 47.3%

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

8. Final simplification 47.3%

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

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 2024034 
(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))