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

Percentage Accurate: 84.8% → 99.8%

Time: 15.1s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% 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 85.6%

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

   1. +-commutative 85.6%

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

   2. associate--l+ 85.6%

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

   3. fma-define 85.6%

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

   4. sub-neg 85.6%

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

   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: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-65} \lor \neg \left(t \leq 1.8 \cdot 10^{-146}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -1.4e-65) (not (<= t 1.8e-146))) (- t_1 t) (- t_1 (* z y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -1.4e-65) || !(t <= 1.8e-146)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((t <= (-1.4d-65)) .or. (.not. (t <= 1.8d-146))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if ((t <= -1.4e-65) || !(t <= 1.8e-146)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -1.4e-65) or not (t <= 1.8e-146):
		tmp = t_1 - t
	else:
		tmp = t_1 - (z * y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -1.4e-65) || !(t <= 1.8e-146))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -1.4e-65) || ~((t <= 1.8e-146)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -1.4e-65], N[Not[LessEqual[t, 1.8e-146]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(z * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.4e-65 or 1.79999999999999989e-146 < t

   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 99.9%

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

   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 z around inf 68.7%

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

      1. associate--l+ 68.7%

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

      2. div-sub 69.0%

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

      3. sub-neg 69.0%

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

      4. log1p-define 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in z around 0 92.0%

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

   if -1.4e-65 < t < 1.79999999999999989e-146

   1. Initial program 69.1%

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

   3. Taylor expanded in y around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. mul-1-neg 97.6%

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

      3. unsub-neg 97.6%

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

   5. Simplified 97.6%

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

      1. add-cube-cbrt 96.5%

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

      2. pow3 96.5%

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

   7. Applied egg-rr 96.5%

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

   8. Taylor expanded in t around 0 93.0%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-65} \lor \neg \left(t \leq 1.8 \cdot 10^{-146}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-103} \lor \neg \left(x \leq 1.3 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.5e-103) (not (<= x 1.3e-56)))
   (- (* x (log y)) t)
   (- (* y (* z (+ (* y (- (* y -0.3333333333333333) 0.5)) -1.0))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.5e-103) || !(x <= 1.3e-56)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-8.5d-103)) .or. (.not. (x <= 1.3d-56))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * (z * ((y * ((y * (-0.3333333333333333d0)) - 0.5d0)) + (-1.0d0)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.5e-103) || !(x <= 1.3e-56)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8.5e-103) or not (x <= 1.3e-56):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.5e-103) || !(x <= 1.3e-56))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5)) + -1.0))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8.5e-103) || ~((x <= 1.3e-56)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8.5e-103], N[Not[LessEqual[x, 1.3e-56]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(z * N[(N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.50000000000000032e-103 or 1.29999999999999998e-56 < x

   1. Initial program 92.6%

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

      1. +-commutative 92.6%

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

      2. associate--l+ 92.6%

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

      3. fma-define 92.6%

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

      4. sub-neg 92.6%

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

      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 inf 67.9%

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

      1. associate--l+ 67.9%

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

      2. div-sub 68.2%

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

      3. sub-neg 68.2%

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

      4. log1p-define 75.4%

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

   7. Simplified 75.4%

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

   8. Taylor expanded in z around 0 90.3%

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

   if -8.50000000000000032e-103 < x < 1.29999999999999998e-56

   1. Initial program 70.7%

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

   3. Taylor expanded in x around 0 64.8%

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

      1. sub-neg 64.8%

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

      2. log1p-define 94.1%

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

   5. Simplified 94.1%

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

   6. Taylor expanded in y around 0 94.1%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in z around 0 94.1%

      \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

   8. Taylor expanded in y around 0 94.1%

      \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(\color{blue}{-0.3333333333333333 \cdot y} - 0.5\right) - 1\right)\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-103} \lor \neg \left(x \leq 1.3 \cdot 10^{-56}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+84} \lor \neg \left(x \leq 6.5 \cdot 10^{+124}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.5e+84) (not (<= x 6.5e+124)))
   (* x (log y))
   (-
    (* y (- (* y (* z (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5))) z))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e+84) || !(x <= 6.5e+124)) {
		tmp = x * log(y);
	} else {
		tmp = (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.5d+84)) .or. (.not. (x <= 6.5d+124))) then
        tmp = x * log(y)
    else
        tmp = (y * ((y * (z * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.5e+84) || !(x <= 6.5e+124)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.5e+84) or not (x <= 6.5e+124):
		tmp = x * math.log(y)
	else:
		tmp = (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.5e+84) || !(x <= 6.5e+124))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(z * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.5e+84) || ~((x <= 6.5e+124)))
		tmp = x * log(y);
	else
		tmp = (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.5e+84], N[Not[LessEqual[x, 6.5e+124]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * N[(z * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999997e84 or 6.50000000000000008e124 < x

   1. Initial program 97.5%

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

      1. +-commutative 97.5%

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

      2. associate--l+ 97.5%

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

      3. fma-define 97.5%

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

      4. sub-neg 97.5%

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

      5. log1p-define 99.6%

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

   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 z around inf 61.6%

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

      1. associate--l+ 61.6%

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

      2. div-sub 62.1%

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

      3. sub-neg 62.1%

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

      4. log1p-define 64.2%

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

   7. Simplified 64.2%

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

   8. Taylor expanded in x around inf 77.0%

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

   if -4.4999999999999997e84 < x < 6.50000000000000008e124

   1. Initial program 78.5%

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

   3. Taylor expanded in x around 0 58.3%

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

      1. sub-neg 58.3%

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

      2. log1p-define 79.6%

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

   5. Simplified 79.6%

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

   6. Taylor expanded in y around 0 79.4%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

   7. Taylor expanded in z around 0 79.4%

      \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right)\right)}\right) - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+84} \lor \neg \left(x \leq 6.5 \cdot 10^{+124}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in y around 0 98.6%

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

   1. +-commutative 98.6%

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

   2. mul-1-neg 98.6%

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

   3. unsub-neg 98.6%

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 6: 58.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right) - z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (* y (- (* y (* z (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5))) z))
  t))

C

double code(double x, double y, double z, double t) {
	return (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * ((y * (z * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))) - z)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t;
}

Python

def code(x, y, z, t):
	return (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(Float64(y * Float64(z * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * ((y * (z * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))) - z)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(N[(y * N[(z * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 44.5%

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

   1. sub-neg 44.5%

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

   2. log1p-define 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in y around 0 58.2%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 58.2%

   \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right)\right)}\right) - t \]

8. Final simplification 58.2%

   \[\leadsto y \cdot \left(y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right) - z\right) - t \]
9. Add Preprocessing

Alternative 7: 58.7% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (* y (* z (+ -1.0 (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)))))
  t))

C

double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (z * ((-1.0d0) + (y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
}

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5))))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * (z * (-1.0 + (y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5))))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(z * N[(-1.0 + N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 44.5%

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

   1. sub-neg 44.5%

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

   2. log1p-define 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in y around 0 58.2%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 58.2%

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

8. Final simplification 58.2%

   \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right)\right)\right) - t \]
9. Add Preprocessing

Alternative 8: 58.7% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (y * ((y * (z * ((y * -0.3333333333333333) + -0.5))) - z)) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 44.5%

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

   1. sub-neg 44.5%

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

   2. log1p-define 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in y around 0 58.2%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in y around 0 58.0%

   \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)}\right) - t \]
8. Step-by-step derivation

   1. +-commutative 58.0%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(-0.3333333333333333 \cdot \left(y \cdot z\right) + -0.5 \cdot z\right)}\right) - t \]

   2. associate-*r* 58.0%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(\color{blue}{\left(-0.3333333333333333 \cdot y\right) \cdot z} + -0.5 \cdot z\right)\right) - t \]

   3. distribute-rgt-out 58.0%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \color{blue}{\left(z \cdot \left(-0.3333333333333333 \cdot y + -0.5\right)\right)}\right) - t \]

   4. *-commutative 58.0%

      \[\leadsto y \cdot \left(-1 \cdot z + y \cdot \left(z \cdot \left(\color{blue}{y \cdot -0.3333333333333333} + -0.5\right)\right)\right) - t \]

9. Simplified 58.0%

   \[\leadsto y \cdot \left(-1 \cdot z + \color{blue}{y \cdot \left(z \cdot \left(y \cdot -0.3333333333333333 + -0.5\right)\right)}\right) - t \]

10. Final simplification 58.0%

    \[\leadsto y \cdot \left(y \cdot \left(z \cdot \left(y \cdot -0.3333333333333333 + -0.5\right)\right) - z\right) - t \]
11. Add Preprocessing

Alternative 9: 58.6% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 44.5%

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

   1. sub-neg 44.5%

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

   2. log1p-define 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in y around 0 58.2%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + y \cdot \left(-0.3333333333333333 \cdot z + -0.25 \cdot \left(y \cdot z\right)\right)\right)\right)} - t \]

7. Taylor expanded in z around 0 58.2%

   \[\leadsto \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

8. Taylor expanded in y around 0 58.0%

   \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(\color{blue}{-0.3333333333333333 \cdot y} - 0.5\right) - 1\right)\right) - t \]

9. Final simplification 58.0%

   \[\leadsto y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right) - t \]
10. Add Preprocessing

Alternative 10: 49.8% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-55} \lor \neg \left(t \leq 6.4 \cdot 10^{-145}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.2e-55) (not (<= t 6.4e-145))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.2e-55) || !(t <= 6.4e-145)) {
		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 <= (-3.2d-55)) .or. (.not. (t <= 6.4d-145))) 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 <= -3.2e-55) || !(t <= 6.4e-145)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.2e-55) or not (t <= 6.4e-145):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.2e-55) || !(t <= 6.4e-145))
		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 <= -3.2e-55) || ~((t <= 6.4e-145)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.2e-55], N[Not[LessEqual[t, 6.4e-145]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.2000000000000001e-55 or 6.40000000000000017e-145 < t

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. associate--l+ 93.2%

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

      3. fma-define 93.2%

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

      4. sub-neg 93.2%

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

      5. log1p-define 99.9%

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

   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 z around inf 68.3%

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

      1. associate--l+ 68.3%

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

      2. div-sub 68.6%

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

      3. sub-neg 68.6%

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

      4. log1p-define 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in t around inf 59.6%

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

      1. neg-mul-1 59.6%

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

   10. Simplified 59.6%

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

   if -3.2000000000000001e-55 < t < 6.40000000000000017e-145

   1. Initial program 69.9%

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

   3. Taylor expanded in y around 0 97.7%

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

      1. +-commutative 97.7%

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

      2. mul-1-neg 97.7%

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

      3. unsub-neg 97.7%

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

   5. Simplified 97.7%

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

      1. add-cube-cbrt 96.5%

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

      2. pow3 96.5%

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

   7. Applied egg-rr 96.5%

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

   8. Taylor expanded in y around inf 33.5%

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

      1. mul-1-neg 33.5%

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

      2. distribute-rgt-neg-in 33.5%

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

   10. Simplified 33.5%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-55} \lor \neg \left(t \leq 6.4 \cdot 10^{-145}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.5% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (y * (z * (-1.0 + (y * -0.5)))) - t

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 44.5%

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

   1. sub-neg 44.5%

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

   2. log1p-define 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in y around 0 57.8%

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

   1. +-commutative 57.8%

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

   2. associate-*r* 57.8%

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

   3. distribute-rgt-out 57.8%

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

   4. *-commutative 57.8%

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

8. Simplified 57.8%

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

9. Final simplification 57.8%

   \[\leadsto y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t \]
10. Add Preprocessing

Alternative 12: 58.2% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.6%

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

3. Taylor expanded in x around 0 44.5%

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

   1. sub-neg 44.5%

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

   2. log1p-define 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in y around 0 57.1%

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

   1. mul-1-neg 57.1%

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

   2. *-commutative 57.1%

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

   3. distribute-rgt-neg-in 57.1%

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

8. Simplified 57.1%

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

Alternative 13: 43.1% 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 85.6%

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

   1. +-commutative 85.6%

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

   2. associate--l+ 85.6%

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

   3. fma-define 85.6%

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

   4. sub-neg 85.6%

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

   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 inf 65.8%

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

   1. associate--l+ 65.8%

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

   2. div-sub 66.1%

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

   3. sub-neg 66.1%

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

   4. log1p-define 80.2%

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

7. Simplified 80.2%

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

8. Taylor expanded in t around inf 42.8%

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

   1. neg-mul-1 42.8%

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

10. Simplified 42.8%

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

Developer target: 99.5% 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 2024085 
(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))