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

Percentage Accurate: 84.7% → 99.8%

Time: 13.7s

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: 84.7% 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

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

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

   1. +-commutative 83.0%

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

   2. associate--l+ 83.0%

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

   3. fma-define 83.0%

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

   4. sub-neg 83.0%

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

   5. log1p-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

3. Taylor expanded in y around 0 99.3%

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

   1. associate--l+ 99.3%

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

   2. +-commutative 99.3%

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

   3. mul-1-neg 99.3%

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

   4. unsub-neg 99.3%

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

5. Simplified 99.3%

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

   1. associate--l- 99.3%

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

   2. *-commutative 99.3%

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

   3. fmm-def 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 3: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-46} \lor \neg \left(t \leq 3.9 \cdot 10^{-104}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -3.5e-46) (not (<= t 3.9e-104))) (- 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 <= -3.5e-46) || !(t <= 3.9e-104)) {
		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 <= (-3.5d-46)) .or. (.not. (t <= 3.9d-104))) 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 <= -3.5e-46) || !(t <= 3.9e-104)) {
		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 <= -3.5e-46) or not (t <= 3.9e-104):
		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 <= -3.5e-46) || !(t <= 3.9e-104))
		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 <= -3.5e-46) || ~((t <= 3.9e-104)))
		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, -3.5e-46], N[Not[LessEqual[t, 3.9e-104]], $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 < -3.5000000000000002e-46 or 3.9000000000000002e-104 < t

   1. Initial program 92.1%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. fma-define 99.7%

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

      2. fma-define 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

      3. associate-*r* 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

      4. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

   6. Taylor expanded in y around 0 91.6%

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

   if -3.5000000000000002e-46 < t < 3.9000000000000002e-104

   1. Initial program 71.6%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. associate--l- 99.1%

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

      2. *-commutative 99.1%

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

      3. fmm-def 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in t around 0 90.9%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-46} \lor \neg \left(t \leq 3.9 \cdot 10^{-104}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-103} \lor \neg \left(x \leq 9 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.3e-103) (not (<= x 9e-91)))
   (- (* x (log y)) t)
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.3e-103) || !(x <= 9e-91)) {
		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 <= -3.3e-103) || !(x <= 9e-91))
		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, -3.3e-103], N[Not[LessEqual[x, 9e-91]], $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 < -3.2999999999999999e-103 or 8.99999999999999952e-91 < x

   1. Initial program 90.0%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. fma-define 99.8%

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

      2. fma-define 99.8%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

      3. associate-*r* 99.8%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

      4. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

   6. Taylor expanded in y around 0 90.0%

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

   if -3.2999999999999999e-103 < x < 8.99999999999999952e-91

   1. Initial program 69.1%

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

   3. Taylor expanded in y around 0 98.5%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. mul-1-neg 98.5%

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

      4. unsub-neg 98.5%

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

   5. Simplified 98.5%

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

      1. associate--l- 98.5%

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

      2. *-commutative 98.5%

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

      3. fmm-def 98.5%

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

   7. Applied egg-rr 98.5%

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

   8. Taylor expanded in x around 0 88.2%

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

      1. mul-1-neg 88.2%

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

      2. +-commutative 88.2%

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

      3. fma-define 88.2%

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

   10. Simplified 88.2%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-103} \lor \neg \left(x \leq 9 \cdot 10^{-91}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+26} \lor \neg \left(x \leq 5 \cdot 10^{+72}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2e+26) (not (<= x 5e+72))) (* x (log y)) (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2e+26) || !(x <= 5e+72)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2e+26) || !(x <= 5e+72))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2e+26], N[Not[LessEqual[x, 5e+72]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e26 or 4.99999999999999992e72 < x

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. fma-define 99.7%

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

      2. fma-define 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

      3. associate-*r* 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

      4. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

   6. Taylor expanded in x around inf 76.6%

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

   if -2.0000000000000001e26 < x < 4.99999999999999992e72

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. associate--l- 99.1%

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

      2. *-commutative 99.1%

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

      3. fmm-def 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in x around 0 78.3%

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

      1. mul-1-neg 78.3%

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

      2. +-commutative 78.3%

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

      3. fma-define 78.3%

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

   10. Simplified 78.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+26} \lor \neg \left(x \leq 5 \cdot 10^{+72}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+25} \lor \neg \left(x \leq 1.1 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8e+25) (not (<= x 1.1e+73))) (* x (log y)) (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8e+25) || !(x <= 1.1e+73)) {
		tmp = x * log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-8d+25)) .or. (.not. (x <= 1.1d+73))) then
        tmp = x * log(y)
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8e+25) || !(x <= 1.1e+73)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8e+25) or not (x <= 1.1e+73):
		tmp = x * math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8e+25) || !(x <= 1.1e+73))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8e+25) || ~((x <= 1.1e+73)))
		tmp = x * log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8e+25], N[Not[LessEqual[x, 1.1e+73]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000072e25 or 1.1e73 < x

   1. Initial program 93.3%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. fma-define 99.7%

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

      2. fma-define 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

      3. associate-*r* 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

      4. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

   6. Taylor expanded in x around inf 76.6%

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

   if -8.00000000000000072e25 < x < 1.1e73

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around 0 78.3%

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

      1. neg-mul-1 78.3%

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

      2. distribute-neg-in 78.3%

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

      3. unsub-neg 78.3%

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

   8. Simplified 78.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+25} \lor \neg \left(x \leq 1.1 \cdot 10^{+73}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

3. Taylor expanded in y around 0 99.3%

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

   1. associate--l+ 99.3%

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

   2. +-commutative 99.3%

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

   3. mul-1-neg 99.3%

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

   4. unsub-neg 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 8: 49.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-46} \lor \neg \left(t \leq 2.05 \cdot 10^{-103}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.95e-46) (not (<= t 2.05e-103))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.95e-46) || !(t <= 2.05e-103)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.95d-46)) .or. (.not. (t <= 2.05d-103))) then
        tmp = -t
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.95e-46) || !(t <= 2.05e-103)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.95e-46) or not (t <= 2.05e-103):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.95e-46) || !(t <= 2.05e-103))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.95e-46) || ~((t <= 2.05e-103)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.95e-46], N[Not[LessEqual[t, 2.05e-103]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.95e-46 or 2.04999999999999998e-103 < t

   1. Initial program 92.1%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. fma-define 99.7%

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

      2. fma-define 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

      3. associate-*r* 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

      4. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

   6. Taylor expanded in t around inf 59.6%

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

      1. neg-mul-1 59.6%

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

   8. Simplified 59.6%

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

   if -2.95e-46 < t < 2.04999999999999998e-103

   1. Initial program 71.6%

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

   3. Taylor expanded in y around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

      3. mul-1-neg 99.1%

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

      4. unsub-neg 99.1%

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

   5. Simplified 99.1%

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

      1. associate--l- 99.1%

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

      2. *-commutative 99.1%

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

      3. fmm-def 99.1%

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

   7. Applied egg-rr 99.1%

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

   8. Taylor expanded in y around inf 30.4%

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

      1. neg-mul-1 30.4%

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

      2. distribute-rgt-neg-in 30.4%

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

   10. Simplified 30.4%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-46} \lor \neg \left(t \leq 2.05 \cdot 10^{-103}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.0%

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

3. Taylor expanded in y around 0 99.3%

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

   1. associate--l+ 99.3%

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

   2. +-commutative 99.3%

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

   3. mul-1-neg 99.3%

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

   4. unsub-neg 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 54.4%

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

   1. neg-mul-1 54.4%

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

   2. distribute-neg-in 54.4%

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

   3. unsub-neg 54.4%

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

8. Simplified 54.4%

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

9. Final simplification 54.4%

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

Alternative 10: 42.8% 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 83.0%

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

3. Taylor expanded in y around 0 99.5%

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

   1. fma-define 99.5%

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

   2. fma-define 99.5%

      \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

   3. associate-*r* 99.5%

      \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

   4. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

6. Taylor expanded in t around inf 37.6%

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

   1. neg-mul-1 37.6%

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

8. Simplified 37.6%

   \[\leadsto \color{blue}{-t} \]
9. Add Preprocessing

Alternative 11: 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 83.0%

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

3. Taylor expanded in y around 0 99.5%

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

   1. fma-define 99.5%

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

   2. fma-define 99.5%

      \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \color{blue}{\mathsf{fma}\left(-1, z, -0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]

   3. associate-*r* 99.5%

      \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(-0.5 \cdot y\right) \cdot z}\right)\right) - t \]

   4. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \color{blue}{\left(y \cdot -0.5\right)} \cdot z\right)\right) - t \]

5. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, y \cdot \mathsf{fma}\left(-1, z, \left(y \cdot -0.5\right) \cdot z\right)\right) - t} \]

6. Taylor expanded in t around inf 37.6%

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

   1. neg-mul-1 37.6%

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

8. Simplified 37.6%

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

   1. add-sqr-sqrt 19.2%

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

   2. sqrt-unprod 10.2%

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

   3. sqr-neg 10.2%

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

   4. sqrt-unprod 1.0%

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

   5. add-sqr-sqrt 2.3%

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

   6. *-un-lft-identity 2.3%

      \[\leadsto \color{blue}{1 \cdot t} \]

10. Applied egg-rr 2.3%

    \[\leadsto \color{blue}{1 \cdot t} \]

11. Taylor expanded in t around 0 2.3%

    \[\leadsto \color{blue}{t} \]
12. Add Preprocessing

Developer Target 1: 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 2024152 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

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