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

Percentage Accurate: 85.6% → 99.8%

Time: 11.2s

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: 85.6% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. *-lft-identity 87.1%

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

   2. *-lft-identity 87.1%

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

   3. fma-def 87.1%

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

   4. sub-neg 87.1%

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

   5. log1p-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-102} \lor \neg \left(t \leq 9.5 \cdot 10^{+23}\right):\\ \;\;\;\;t_1 - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -6.5e-102) (not (<= t 9.5e+23))) (- t_1 t) (- t_1 (* y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -6.5e-102) || !(t <= 9.5e+23)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((t <= (-6.5d-102)) .or. (.not. (t <= 9.5d+23))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (y * z)
    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 <= -6.5e-102) || !(t <= 9.5e+23)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -6.5e-102) or not (t <= 9.5e+23):
		tmp = t_1 - t
	else:
		tmp = t_1 - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -6.5e-102) || !(t <= 9.5e+23))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -6.5e-102) || ~((t <= 9.5e+23)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (y * z);
	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, -6.5e-102], N[Not[LessEqual[t, 9.5e+23]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000003e-102 or 9.50000000000000038e23 < t

   1. Initial program 96.1%

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

      1. *-lft-identity 96.1%

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

      2. *-lft-identity 96.1%

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

      3. fma-def 96.1%

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

      4. sub-neg 96.1%

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

      5. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 96.1%

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

   if -6.5000000000000003e-102 < t < 9.50000000000000038e23

   1. Initial program 74.6%

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

   2. Taylor expanded in y around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. *-commutative 98.4%

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

      3. mul-1-neg 98.4%

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

      4. unsub-neg 98.4%

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

      5. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   5. Taylor expanded in t around 0 92.6%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-102} \lor \neg \left(t \leq 9.5 \cdot 10^{+23}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]

Alternative 3: 90.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-59} \lor \neg \left(x \leq 3.8 \cdot 10^{-86}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.85e-59) (not (<= x 3.8e-86)))
   (- (* x (log y)) t)
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.85e-59) || !(x <= 3.8e-86)) {
		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 <= -1.85e-59) || !(x <= 3.8e-86))
		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, -1.85e-59], N[Not[LessEqual[x, 3.8e-86]], $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 < -1.85e-59 or 3.8e-86 < x

   1. Initial program 94.4%

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

      1. *-lft-identity 94.4%

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

      2. *-lft-identity 94.4%

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

      3. fma-def 94.4%

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

      4. sub-neg 94.4%

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

      5. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 93.1%

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

   if -1.85e-59 < x < 3.8e-86

   1. Initial program 75.1%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. unsub-neg 100.0%

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

      5. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around 0 91.9%

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

      1. fma-def 91.9%

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

      2. neg-mul-1 91.9%

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

   7. Simplified 91.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-59} \lor \neg \left(x \leq 3.8 \cdot 10^{-86}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

2. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. mul-1-neg 99.3%

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

   4. unsub-neg 99.3%

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

   5. *-commutative 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+38} \lor \neg \left(x \leq 6 \cdot 10^{+94}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.6e+38) (not (<= x 6e+94))) (* x (log y)) (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.6e+38) || !(x <= 6e+94)) {
		tmp = x * log(y);
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.6e+38) || !(x <= 6e+94))
		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, -4.6e+38], N[Not[LessEqual[x, 6e+94]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -4.6000000000000002e38 or 6.0000000000000001e94 < x

   1. Initial program 96.8%

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

      1. +-commutative 96.8%

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

      2. associate--l+ 96.8%

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

      3. +-commutative 96.8%

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

      4. associate-+l- 96.8%

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

      5. fma-neg 96.8%

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

      6. sub0-neg 96.8%

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

      7. associate-+l- 96.8%

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

      8. neg-sub0 96.8%

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

      9. +-commutative 96.8%

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

      10. fma-def 96.8%

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

      11. sub-neg 96.8%

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

      12. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. +-commutative 98.6%

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

      3. unsub-neg 98.6%

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

      4. mul-1-neg 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 74.4%

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

   if -4.6000000000000002e38 < x < 6.0000000000000001e94

   1. Initial program 79.1%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 82.3%

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

      1. fma-def 82.3%

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

      2. neg-mul-1 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+38} \lor \neg \left(x \leq 6 \cdot 10^{+94}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 6: 77.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+38} \lor \neg \left(x \leq 1.25 \cdot 10^{+95}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.5e+38) (not (<= x 1.25e+95)))
   (* x (log y))
   (- (- t) (* y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.5e+38) || !(x <= 1.25e+95)) {
		tmp = x * log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-6.5d+38)) .or. (.not. (x <= 1.25d+95))) then
        tmp = x * log(y)
    else
        tmp = -t - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.5e+38) || !(x <= 1.25e+95)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.5e+38) or not (x <= 1.25e+95):
		tmp = x * math.log(y)
	else:
		tmp = -t - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.5e+38) || !(x <= 1.25e+95))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-t) - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -6.5e+38) || ~((x <= 1.25e+95)))
		tmp = x * log(y);
	else
		tmp = -t - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.5e+38], N[Not[LessEqual[x, 1.25e+95]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-t) - N[(y * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5e38 or 1.25000000000000006e95 < x

   1. Initial program 96.8%

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

      1. +-commutative 96.8%

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

      2. associate--l+ 96.8%

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

      3. +-commutative 96.8%

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

      4. associate-+l- 96.8%

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

      5. fma-neg 96.8%

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

      6. sub0-neg 96.8%

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

      7. associate-+l- 96.8%

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

      8. neg-sub0 96.8%

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

      9. +-commutative 96.8%

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

      10. fma-def 96.8%

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

      11. sub-neg 96.8%

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

      12. log1p-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 98.6%

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

      1. mul-1-neg 98.6%

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

      2. +-commutative 98.6%

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

      3. unsub-neg 98.6%

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

      4. mul-1-neg 98.6%

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

      5. distribute-rgt-neg-in 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in x around inf 74.4%

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

   if -6.5e38 < x < 1.25000000000000006e95

   1. Initial program 79.1%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. mul-1-neg 99.9%

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

      4. unsub-neg 99.9%

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

      5. *-commutative 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around inf 82.3%

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

      1. mul-1-neg 82.3%

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

      2. distribute-rgt-neg-in 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+38} \lor \neg \left(x \leq 1.25 \cdot 10^{+95}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) - y \cdot z\\ \end{array} \]

Alternative 7: 56.4% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

2. Taylor expanded in x around 0 45.4%

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

   1. sub-neg 45.4%

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

   2. mul-1-neg 45.4%

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

   3. log1p-def 56.9%

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

   4. mul-1-neg 56.9%

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

4. Simplified 56.9%

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

5. Taylor expanded in y around 0 56.5%

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

   1. +-commutative 56.5%

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

   2. associate-*r* 56.5%

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

   3. associate-*r* 56.5%

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

   4. distribute-rgt-out 56.5%

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

   5. neg-mul-1 56.5%

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

   6. unpow2 56.5%

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

7. Simplified 56.5%

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

8. Final simplification 56.5%

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

Alternative 8: 47.1% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-104}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 39000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.25e-104) (- t) (if (<= t 39000.0) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.25e-104) {
		tmp = -t;
	} else if (t <= 39000.0) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.25d-104)) then
        tmp = -t
    else if (t <= 39000.0d0) then
        tmp = y * -z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.25e-104) {
		tmp = -t;
	} else if (t <= 39000.0) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.25e-104:
		tmp = -t
	elif t <= 39000.0:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.25e-104)
		tmp = Float64(-t);
	elseif (t <= 39000.0)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.25e-104)
		tmp = -t;
	elseif (t <= 39000.0)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.25e-104], (-t), If[LessEqual[t, 39000.0], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2499999999999999e-104 or 39000 < t

   1. Initial program 95.7%

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

      1. *-lft-identity 95.7%

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

      2. *-lft-identity 95.7%

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

      3. fma-def 95.7%

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

      4. sub-neg 95.7%

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

      5. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 68.6%

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

      1. mul-1-neg 68.6%

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

   6. Simplified 68.6%

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

   if -2.2499999999999999e-104 < t < 39000

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

      2. associate--l+ 74.0%

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

      3. +-commutative 74.0%

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

      4. associate-+l- 74.0%

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

      5. fma-neg 74.0%

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

      6. sub0-neg 74.0%

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

      7. associate-+l- 74.0%

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

      8. neg-sub0 74.0%

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

      9. +-commutative 74.0%

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

      10. fma-def 74.0%

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

      11. sub-neg 74.0%

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

      12. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 98.3%

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

      1. mul-1-neg 98.3%

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

      2. +-commutative 98.3%

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

      3. unsub-neg 98.3%

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

      4. mul-1-neg 98.3%

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

      5. distribute-rgt-neg-in 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in y around inf 27.3%

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

      1. associate-*r* 27.3%

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

      2. neg-mul-1 27.3%

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

   9. Simplified 27.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-104}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 39000:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9: 56.1% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

2. Taylor expanded in y around 0 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. mul-1-neg 99.3%

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

   4. unsub-neg 99.3%

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

   5. *-commutative 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in y around inf 56.4%

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

   1. mul-1-neg 56.4%

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

   2. distribute-rgt-neg-in 56.4%

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

7. Simplified 56.4%

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

8. Final simplification 56.4%

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

Alternative 10: 42.1% 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 87.1%

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

   1. *-lft-identity 87.1%

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

   2. *-lft-identity 87.1%

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

   3. fma-def 87.1%

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

   4. sub-neg 87.1%

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

   5. log1p-def 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in t around inf 44.6%

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

   1. mul-1-neg 44.6%

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

6. Simplified 44.6%

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

7. Final simplification 44.6%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023224 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

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

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