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

Percentage Accurate: 85.5% → 99.8%

Time: 13.8s

Alternatives: 10

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, 0.7× 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 85.3%

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

   1. +-commutative 85.3%

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

   2. associate--l+ 85.3%

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

   3. fma-define 85.3%

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

   4. sub-neg 85.3%

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

   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: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -6 \cdot 10^{-95} \lor \neg \left(t \leq 2 \cdot 10^{-66}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -6e-95) (not (<= t 2e-66))) (- 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 <= -6e-95) || !(t <= 2e-66)) {
		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 <= (-6d-95)) .or. (.not. (t <= 2d-66))) 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 <= -6e-95) || !(t <= 2e-66)) {
		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 <= -6e-95) or not (t <= 2e-66):
		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 <= -6e-95) || !(t <= 2e-66))
		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 <= -6e-95) || ~((t <= 2e-66)))
		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, -6e-95], N[Not[LessEqual[t, 2e-66]], $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 < -6e-95 or 2e-66 < t

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. associate--l+ 94.1%

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

      3. fma-define 94.1%

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

      4. sub-neg 94.1%

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

      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. Taylor expanded in z around 0 92.5%

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

   if -6e-95 < t < 2e-66

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

      2. associate--l+ 68.9%

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

      3. fma-define 68.9%

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

      4. sub-neg 68.9%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around inf 41.1%

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

      1. associate--l+ 41.1%

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

      2. associate-/l* 41.0%

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

      3. fma-define 41.0%

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

      4. associate-/l* 41.0%

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

      5. fma-neg 41.0%

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

      6. sub-neg 41.0%

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

      7. log1p-define 63.6%

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

      8. metadata-eval 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in y around 0 64.6%

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

   9. Taylor expanded in t around 0 94.4%

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

       1. +-commutative 94.4%

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

       2. mul-1-neg 94.4%

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

       3. unsub-neg 94.4%

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

   11. Simplified 94.4%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-95} \lor \neg \left(t \leq 2 \cdot 10^{-66}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+19} \lor \neg \left(x \leq 0.47\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.7e+19) (not (<= x 0.47)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.7e+19) || !(x <= 0.47)) {
		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 <= -6.7e+19) || !(x <= 0.47)) {
		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 <= -6.7e+19) or not (x <= 0.47):
		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 <= -6.7e+19) || !(x <= 0.47))
		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, -6.7e+19], N[Not[LessEqual[x, 0.47]], $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 < -6.7e19 or 0.46999999999999997 < x

   1. Initial program 95.5%

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

      1. +-commutative 95.5%

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

      2. associate--l+ 95.5%

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

      3. fma-define 95.5%

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

      4. sub-neg 95.5%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 95.5%

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

   if -6.7e19 < x < 0.46999999999999997

   1. Initial program 73.1%

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

      1. +-commutative 73.1%

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

      2. associate--l+ 73.1%

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

      3. fma-define 73.1%

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

      4. sub-neg 73.1%

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

      5. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 62.1%

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

      1. sub-neg 62.1%

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

      2. log1p-define 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+19} \lor \neg \left(x \leq 0.47\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-118} \lor \neg \left(x \leq 0.0136\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.3e-118) (not (<= x 0.0136)))
   (- (* x (log y)) t)
   (- (- t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.3e-118) || !(x <= 0.0136)) {
		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 <= (-3.3d-118)) .or. (.not. (x <= 0.0136d0))) 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 <= -3.3e-118) || !(x <= 0.0136)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.3e-118) or not (x <= 0.0136):
		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 <= -3.3e-118) || !(x <= 0.0136))
		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 <= -3.3e-118) || ~((x <= 0.0136)))
		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, -3.3e-118], N[Not[LessEqual[x, 0.0136]], $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 < -3.3e-118 or 0.0135999999999999992 < x

   1. Initial program 93.3%

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

      1. +-commutative 93.3%

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

      2. associate--l+ 93.3%

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

      3. fma-define 93.3%

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

      4. sub-neg 93.3%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 92.7%

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

   if -3.3e-118 < x < 0.0135999999999999992

   1. Initial program 70.7%

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

      1. add-cube-cbrt 70.5%

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

      2. pow3 70.5%

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

   4. Applied egg-rr 70.5%

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

   5. Taylor expanded in y around 0 98.4%

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

      1. mul-1-neg 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in z around inf 98.5%

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

      1. +-commutative 98.5%

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

      2. mul-1-neg 98.5%

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

      3. unsub-neg 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-/l* 98.6%

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

   10. Simplified 98.6%

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

   11. Taylor expanded in z around inf 89.1%

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

       1. mul-1-neg 89.1%

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

       2. *-commutative 89.1%

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

       3. distribute-rgt-neg-in 89.1%

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

   13. Simplified 89.1%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-118} \lor \neg \left(x \leq 0.0136\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 6: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+69} \lor \neg \left(x \leq 205000000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.8e+69) (not (<= x 205000000000.0)))
   (* x (log y))
   (- (- t) (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.8e+69) || !(x <= 205000000000.0)) {
		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 <= (-5.8d+69)) .or. (.not. (x <= 205000000000.0d0))) 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 <= -5.8e+69) || !(x <= 205000000000.0)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.8e+69) or not (x <= 205000000000.0):
		tmp = x * math.log(y)
	else:
		tmp = -t - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.8e+69) || !(x <= 205000000000.0))
		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 <= -5.8e+69) || ~((x <= 205000000000.0)))
		tmp = x * log(y);
	else
		tmp = -t - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.8e+69], N[Not[LessEqual[x, 205000000000.0]], $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 < -5.7999999999999997e69 or 2.05e11 < x

   1. Initial program 96.2%

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

      1. +-commutative 96.2%

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

      2. associate--l+ 96.2%

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

      3. fma-define 96.2%

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

      4. sub-neg 96.2%

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

      5. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around inf 73.0%

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

      1. associate--l+ 73.0%

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

      2. associate-/l* 72.9%

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

      3. fma-define 72.9%

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

      4. associate-/l* 72.9%

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

      5. fma-neg 72.9%

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

      6. sub-neg 72.9%

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

      7. log1p-define 75.1%

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

      8. metadata-eval 75.1%

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

   7. Simplified 75.1%

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

   8. Taylor expanded in x around inf 80.6%

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

   if -5.7999999999999997e69 < x < 2.05e11

   1. Initial program 75.4%

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

      1. add-cube-cbrt 75.2%

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

      2. pow3 75.2%

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

   4. Applied egg-rr 75.2%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. mul-1-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in z around inf 97.0%

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

      1. +-commutative 97.0%

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

      2. mul-1-neg 97.0%

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

      3. unsub-neg 97.0%

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

      4. *-commutative 97.0%

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

      5. associate-/l* 97.0%

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

   10. Simplified 97.0%

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

   11. Taylor expanded in z around inf 83.7%

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

       1. mul-1-neg 83.7%

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

       2. *-commutative 83.7%

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

       3. distribute-rgt-neg-in 83.7%

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

   13. Simplified 83.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+69} \lor \neg \left(x \leq 205000000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 85.3%

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

   1. +-commutative 85.3%

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

   2. associate--l+ 85.3%

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

   3. fma-define 85.3%

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

   4. sub-neg 85.3%

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

   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. Taylor expanded in y around 0 99.0%

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

   1. +-commutative 99.0%

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

   2. mul-1-neg 99.0%

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

   3. unsub-neg 99.0%

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

7. Simplified 99.0%

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

8. Final simplification 99.0%

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

Alternative 8: 48.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-95} \lor \neg \left(t \leq 1.12 \cdot 10^{-66}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.35e-95) (not (<= t 1.12e-66))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.35e-95) || !(t <= 1.12e-66)) {
		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 <= (-1.35d-95)) .or. (.not. (t <= 1.12d-66))) 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 <= -1.35e-95) || !(t <= 1.12e-66)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.35e-95) or not (t <= 1.12e-66):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.35e-95) || !(t <= 1.12e-66))
		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 <= -1.35e-95) || ~((t <= 1.12e-66)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.35e-95], N[Not[LessEqual[t, 1.12e-66]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e-95 or 1.12000000000000004e-66 < t

   1. Initial program 94.1%

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

      1. +-commutative 94.1%

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

      2. associate--l+ 94.1%

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

      3. fma-define 94.1%

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

      4. sub-neg 94.1%

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

      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. Taylor expanded in t around inf 55.2%

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

      1. neg-mul-1 55.2%

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

   7. Simplified 55.2%

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

   if -1.35e-95 < t < 1.12000000000000004e-66

   1. Initial program 68.9%

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

      1. +-commutative 68.9%

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

      2. associate--l+ 68.9%

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

      3. fma-define 68.9%

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

      4. sub-neg 68.9%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around inf 41.1%

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

      1. associate--l+ 41.1%

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

      2. associate-/l* 41.0%

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

      3. fma-define 41.0%

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

      4. associate-/l* 41.0%

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

      5. fma-neg 41.0%

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

      6. sub-neg 41.0%

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

      7. log1p-define 63.6%

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

      8. metadata-eval 63.6%

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

   7. Simplified 63.6%

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

   8. Taylor expanded in y around 0 64.6%

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

   9. Taylor expanded in y around inf 32.0%

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

       1. associate-*r* 32.0%

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

       2. mul-1-neg 32.0%

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

   11. Simplified 32.0%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-95} \lor \neg \left(t \leq 1.12 \cdot 10^{-66}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.2% 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 85.3%

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

   1. add-cube-cbrt 84.6%

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

   2. pow3 84.6%

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

4. Applied egg-rr 84.6%

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

5. Taylor expanded in y around 0 98.2%

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

   1. mul-1-neg 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in z around inf 85.2%

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

   1. +-commutative 85.2%

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

   2. mul-1-neg 85.2%

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

   3. unsub-neg 85.2%

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

   4. *-commutative 85.2%

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

   5. associate-/l* 85.1%

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

10. Simplified 85.1%

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

11. Taylor expanded in z around inf 53.0%

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

    1. mul-1-neg 53.0%

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

    2. *-commutative 53.0%

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

    3. distribute-rgt-neg-in 53.0%

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

13. Simplified 53.0%

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

14. Final simplification 53.0%

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

Alternative 10: 42.9% 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 85.3%

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

   1. +-commutative 85.3%

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

   2. associate--l+ 85.3%

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

   3. fma-define 85.3%

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

   4. sub-neg 85.3%

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

   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. Taylor expanded in t around inf 38.7%

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

   1. neg-mul-1 38.7%

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

7. Simplified 38.7%

   \[\leadsto \color{blue}{-t} \]
8. 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 2024111 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (- (* (- 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))