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

Percentage Accurate: 85.5% → 99.8%

Time: 14.5s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. +-commutative 84.1%

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

   2. fma-def 84.1%

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

   3. sub-neg 84.1%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(-0.3333333333333333 \cdot \left(z \cdot {y}^{3}\right) - z \cdot y\right)\right) + x \cdot \log y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (+
    (* -0.5 (* z (* y y)))
    (- (* -0.3333333333333333 (* z (pow y 3.0))) (* z y)))
   (* x (log y)))
  t))

C

double code(double x, double y, double z, double t) {
	return (((-0.5 * (z * (y * y))) + ((-0.3333333333333333 * (z * pow(y, 3.0))) - (z * y))) + (x * log(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 = ((((-0.5d0) * (z * (y * y))) + (((-0.3333333333333333d0) * (z * (y ** 3.0d0))) - (z * y))) + (x * log(y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((-0.5 * (z * (y * y))) + ((-0.3333333333333333 * (z * Math.pow(y, 3.0))) - (z * y))) + (x * Math.log(y))) - t;
}

Python

def code(x, y, z, t):
	return (((-0.5 * (z * (y * y))) + ((-0.3333333333333333 * (z * math.pow(y, 3.0))) - (z * y))) + (x * math.log(y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(-0.5 * Float64(z * Float64(y * y))) + Float64(Float64(-0.3333333333333333 * Float64(z * (y ^ 3.0))) - Float64(z * y))) + Float64(x * log(y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((-0.5 * (z * (y * y))) + ((-0.3333333333333333 * (z * (y ^ 3.0))) - (z * y))) + (x * log(y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(-0.5 * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.3333333333333333 * N[(z * N[Power[y, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in y around 0 99.4%

   \[\leadsto \left(x \cdot \log y + \color{blue}{\left(-0.5 \cdot \left({y}^{2} \cdot z\right) + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)}\right) - t \]
3. Step-by-step derivation

   1. expm1-log1p-u 91.7%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({y}^{2} \cdot z\right)\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

   2. expm1-udef 91.7%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({y}^{2} \cdot z\right)} - 1\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

   3. *-commutative 91.7%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{z \cdot {y}^{2}}\right)} - 1\right) + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

   4. unpow2 91.7%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(z \cdot \color{blue}{\left(y \cdot y\right)}\right)} - 1\right) + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

   5. associate-*r* 91.7%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(z \cdot y\right) \cdot y}\right)} - 1\right) + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

4. Applied egg-rr 91.7%

   \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\left(z \cdot y\right) \cdot y\right)} - 1\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]
5. Step-by-step derivation

   1. expm1-def 91.7%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(z \cdot y\right) \cdot y\right)\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

   2. expm1-log1p 99.4%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\left(\left(z \cdot y\right) \cdot y\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

   3. associate-*l* 99.4%

      \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot y\right)\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

6. Simplified 99.4%

   \[\leadsto \left(x \cdot \log y + \left(-0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot y\right)\right)} + \left(-1 \cdot \left(y \cdot z\right) + -0.3333333333333333 \cdot \left({y}^{3} \cdot z\right)\right)\right)\right) - t \]

7. Final simplification 99.4%

   \[\leadsto \left(\left(-0.5 \cdot \left(z \cdot \left(y \cdot y\right)\right) + \left(-0.3333333333333333 \cdot \left(z \cdot {y}^{3}\right) - z \cdot y\right)\right) + x \cdot \log y\right) - t \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in y around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. *-commutative 98.7%

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

   3. log-pow 50.9%

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

   4. mul-1-neg 50.9%

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

   5. unsub-neg 50.9%

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

   6. log-pow 98.7%

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

   7. *-commutative 98.7%

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

4. Simplified 98.7%

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

5. Taylor expanded in y around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. neg-mul-1 98.7%

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

   3. *-commutative 98.7%

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

   4. fma-def 98.7%

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

   5. *-commutative 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 90.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-67} \lor \neg \left(x \leq 7.8 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.6e-67) (not (<= x 7.8e-20)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e-67) || !(x <= 7.8e-20)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.6e-67) || !(x <= 7.8e-20)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.6e-67) or not (x <= 7.8e-20):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.6e-67) || !(x <= 7.8e-20))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.6e-67], N[Not[LessEqual[x, 7.8e-20]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000011e-67 or 7.80000000000000014e-20 < x

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-def 90.4%

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

      3. sub-neg 90.4%

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

      4. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 89.6%

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

   if -1.60000000000000011e-67 < x < 7.80000000000000014e-20

   1. Initial program 77.5%

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

   2. Taylor expanded in x around 0 69.5%

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

      1. sub-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. log1p-def 91.5%

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

      4. mul-1-neg 91.5%

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

   4. Simplified 91.5%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-67} \lor \neg \left(x \leq 7.8 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 5: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-71} \lor \neg \left(x \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.5e-71) (not (<= x 7.5e-20)))
   (- (* x (log y)) t)
   (- (* z (- (* -0.5 (* y y)) y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-71) || !(x <= 7.5e-20)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * ((-0.5 * (y * y)) - y)) - 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 <= (-1.5d-71)) .or. (.not. (x <= 7.5d-20))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * (((-0.5d0) * (y * y)) - y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-71) || !(x <= 7.5e-20)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.5e-71) or not (x <= 7.5e-20):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * ((-0.5 * (y * y)) - y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.5e-71) || !(x <= 7.5e-20))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(Float64(-0.5 * Float64(y * y)) - y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.5e-71) || ~((x <= 7.5e-20)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.5e-71], N[Not[LessEqual[x, 7.5e-20]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5000000000000001e-71 or 7.49999999999999981e-20 < x

   1. Initial program 90.4%

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

      1. +-commutative 90.4%

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

      2. fma-def 90.4%

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

      3. sub-neg 90.4%

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

      4. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 89.6%

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

   if -1.5000000000000001e-71 < x < 7.49999999999999981e-20

   1. Initial program 77.5%

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

   2. Taylor expanded in x around 0 69.5%

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

      1. sub-neg 69.5%

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

      2. mul-1-neg 69.5%

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

      3. log1p-def 91.5%

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

      4. mul-1-neg 91.5%

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

   4. Simplified 91.5%

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

   5. Taylor expanded in y around 0 90.8%

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

      1. +-commutative 90.8%

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

      2. associate-*r* 90.8%

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

      3. associate-*r* 90.8%

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

      4. distribute-rgt-out 90.8%

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

      5. neg-mul-1 90.8%

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

      6. unpow2 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-71} \lor \neg \left(x \leq 7.5 \cdot 10^{-20}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \]

Alternative 6: 99.2% 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 84.1%

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

2. Taylor expanded in y around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. *-commutative 98.7%

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

   3. log-pow 50.9%

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

   4. mul-1-neg 50.9%

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

   5. unsub-neg 50.9%

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

   6. log-pow 98.7%

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

   7. *-commutative 98.7%

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

4. Simplified 98.7%

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

5. Final simplification 98.7%

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

Alternative 7: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in y around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. *-commutative 98.7%

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

   3. log-pow 50.9%

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

   4. mul-1-neg 50.9%

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

   5. unsub-neg 50.9%

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

   6. log-pow 98.7%

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

   7. *-commutative 98.7%

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

4. Simplified 98.7%

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

   1. sub-neg 98.7%

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

   2. sub-neg 98.7%

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

   3. associate-+l+ 98.7%

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

   4. *-commutative 98.7%

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

   5. distribute-rgt-neg-in 98.7%

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

6. Applied egg-rr 98.7%

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

7. Final simplification 98.7%

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

Alternative 8: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.47 \lor \neg \left(x \leq 6.3 \cdot 10^{+171}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.47) (not (<= x 6.3e+171)))
   (* x (log y))
   (- (* z (- (* -0.5 (* y y)) y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.47) || !(x <= 6.3e+171)) {
		tmp = x * log(y);
	} else {
		tmp = (z * ((-0.5 * (y * y)) - y)) - 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 <= (-0.47d0)) .or. (.not. (x <= 6.3d+171))) then
        tmp = x * log(y)
    else
        tmp = (z * (((-0.5d0) * (y * y)) - y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.47) || !(x <= 6.3e+171)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.47) or not (x <= 6.3e+171):
		tmp = x * math.log(y)
	else:
		tmp = (z * ((-0.5 * (y * y)) - y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.47) || !(x <= 6.3e+171))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(z * Float64(Float64(-0.5 * Float64(y * y)) - y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.47) || ~((x <= 6.3e+171)))
		tmp = x * log(y);
	else
		tmp = (z * ((-0.5 * (y * y)) - y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.47], N[Not[LessEqual[x, 6.3e+171]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(-0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.46999999999999997 or 6.3000000000000004e171 < x

   1. Initial program 91.8%

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

      1. +-commutative 91.8%

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

      2. fma-def 91.8%

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

      3. sub-neg 91.8%

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

      4. log1p-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in z around 0 91.2%

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

      1. fma-neg 91.2%

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

   6. Simplified 91.2%

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

   7. Taylor expanded in x around inf 78.6%

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

   if -0.46999999999999997 < x < 6.3000000000000004e171

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 66.2%

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

      1. sub-neg 66.2%

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

      2. mul-1-neg 66.2%

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

      3. log1p-def 86.1%

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

      4. mul-1-neg 86.1%

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

   4. Simplified 86.1%

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

   5. Taylor expanded in y around 0 85.6%

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

      1. +-commutative 85.6%

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

      2. associate-*r* 85.6%

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

      3. associate-*r* 85.6%

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

      4. distribute-rgt-out 85.6%

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

      5. neg-mul-1 85.6%

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

      6. unpow2 85.6%

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

   7. Simplified 85.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.47 \lor \neg \left(x \leq 6.3 \cdot 10^{+171}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot \left(y \cdot y\right) - y\right) - t\\ \end{array} \]

Alternative 9: 57.4% 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 84.1%

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

2. Taylor expanded in x around 0 46.2%

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

   1. sub-neg 46.2%

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

   2. mul-1-neg 46.2%

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

   3. log1p-def 61.4%

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

   4. mul-1-neg 61.4%

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

4. Simplified 61.4%

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

5. Taylor expanded in y around 0 61.1%

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

   1. +-commutative 61.1%

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

   2. associate-*r* 61.1%

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

   3. associate-*r* 61.1%

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

   4. distribute-rgt-out 61.1%

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

   5. neg-mul-1 61.1%

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

   6. unpow2 61.1%

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

7. Simplified 61.1%

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

8. Final simplification 61.1%

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

Alternative 10: 47.3% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-207}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.85e-207) (- t) (if (<= t 2.6e-27) (* y (- z)) (- (* z y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.85e-207) {
		tmp = -t;
	} else if (t <= 2.6e-27) {
		tmp = y * -z;
	} else {
		tmp = (z * y) - 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 <= (-1.85d-207)) then
        tmp = -t
    else if (t <= 2.6d-27) then
        tmp = y * -z
    else
        tmp = (z * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.85e-207) {
		tmp = -t;
	} else if (t <= 2.6e-27) {
		tmp = y * -z;
	} else {
		tmp = (z * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.85e-207:
		tmp = -t
	elif t <= 2.6e-27:
		tmp = y * -z
	else:
		tmp = (z * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.85e-207)
		tmp = Float64(-t);
	elseif (t <= 2.6e-27)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(Float64(z * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.85e-207)
		tmp = -t;
	elseif (t <= 2.6e-27)
		tmp = y * -z;
	else
		tmp = (z * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.85e-207], (-t), If[LessEqual[t, 2.6e-27], N[(y * (-z)), $MachinePrecision], N[(N[(z * y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.84999999999999992e-207

   1. Initial program 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-def 92.0%

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

      3. sub-neg 92.0%

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

      4. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 55.4%

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

      1. mul-1-neg 55.4%

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

   6. Simplified 55.4%

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

   if -1.84999999999999992e-207 < t < 2.60000000000000017e-27

   1. Initial program 68.6%

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

   2. Taylor expanded in y around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-commutative 97.6%

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

      3. log-pow 38.3%

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

      4. mul-1-neg 38.3%

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

      5. unsub-neg 38.3%

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

      6. log-pow 97.6%

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

      7. *-commutative 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in y around 0 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

      3. *-commutative 97.6%

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

      4. fma-def 97.7%

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

      5. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in z around inf 33.6%

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

      1. mul-1-neg 33.6%

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

      2. distribute-rgt-neg-in 33.6%

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

   10. Simplified 33.6%

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

   if 2.60000000000000017e-27 < t

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

      2. add-cube-cbrt 90.8%

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

      3. associate-*r* 90.9%

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

      4. fma-def 90.9%

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

      5. pow2 90.9%

         \[\leadsto \mathsf{fma}\left(\log y \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x}, z \cdot \log \left(1 - y\right)\right) - t \]

      6. sub-neg 90.9%

         \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \log \color{blue}{\left(1 + \left(-y\right)\right)}\right) - t \]

      7. log1p-udef 99.6%

         \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \color{blue}{\mathsf{log1p}\left(-y\right)}\right) - t \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{-y} \cdot \sqrt{-y}}\right)\right) - t \]

      9. sqrt-unprod 90.8%

         \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{\left(-y\right) \cdot \left(-y\right)}}\right)\right) - t \]

      10. sqr-neg 90.8%

          \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \mathsf{log1p}\left(\sqrt{\color{blue}{y \cdot y}}\right)\right) - t \]

      11. sqrt-unprod 90.8%

          \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \mathsf{log1p}\left(\color{blue}{\sqrt{y} \cdot \sqrt{y}}\right)\right) - t \]

      12. add-sqr-sqrt 90.8%

          \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \mathsf{log1p}\left(\color{blue}{y}\right)\right) - t \]

   3. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, z \cdot \mathsf{log1p}\left(y\right)\right)} - t \]

   4. Taylor expanded in y around 0 90.8%

      \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, \color{blue}{y \cdot z}\right) - t \]

   5. Taylor expanded in y around inf 71.1%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-207}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot y - t\\ \end{array} \]

Alternative 11: 47.3% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-204}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.8e-204) (- t) (if (<= t 1.05e-18) (* y (- z)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.8e-204) {
		tmp = -t;
	} else if (t <= 1.05e-18) {
		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 <= (-4.8d-204)) then
        tmp = -t
    else if (t <= 1.05d-18) 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 <= -4.8e-204) {
		tmp = -t;
	} else if (t <= 1.05e-18) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.8e-204:
		tmp = -t
	elif t <= 1.05e-18:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.8e-204)
		tmp = Float64(-t);
	elseif (t <= 1.05e-18)
		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 <= -4.8e-204)
		tmp = -t;
	elseif (t <= 1.05e-18)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.8e-204], (-t), If[LessEqual[t, 1.05e-18], N[(y * (-z)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e-204 or 1.05e-18 < t

   1. Initial program 91.7%

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

      1. +-commutative 91.7%

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

      2. fma-def 91.7%

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

      3. sub-neg 91.7%

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

      4. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 61.5%

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

      1. mul-1-neg 61.5%

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

   6. Simplified 61.5%

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

   if -4.8e-204 < t < 1.05e-18

   1. Initial program 68.6%

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

   2. Taylor expanded in y around 0 97.6%

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

      1. +-commutative 97.6%

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

      2. *-commutative 97.6%

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

      3. log-pow 38.3%

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

      4. mul-1-neg 38.3%

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

      5. unsub-neg 38.3%

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

      6. log-pow 97.6%

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

      7. *-commutative 97.6%

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

   4. Simplified 97.6%

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

   5. Taylor expanded in y around 0 97.6%

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

      1. associate-*r* 97.6%

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

      2. neg-mul-1 97.6%

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

      3. *-commutative 97.6%

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

      4. fma-def 97.7%

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

      5. *-commutative 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in z around inf 33.6%

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

      1. mul-1-neg 33.6%

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

      2. distribute-rgt-neg-in 33.6%

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

   10. Simplified 33.6%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-204}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12: 57.0% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in y around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. *-commutative 98.7%

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

   3. log-pow 50.9%

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

   4. mul-1-neg 50.9%

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

   5. unsub-neg 50.9%

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

   6. log-pow 98.7%

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

   7. *-commutative 98.7%

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

4. Simplified 98.7%

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

5. Taylor expanded in y around inf 60.5%

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

   1. associate-*r* 60.5%

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

   2. neg-mul-1 60.5%

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

   3. *-commutative 60.5%

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

7. Simplified 60.5%

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

8. Final simplification 60.5%

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

Alternative 13: 42.7% 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 84.1%

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

   1. +-commutative 84.1%

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

   2. fma-def 84.1%

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

   3. sub-neg 84.1%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 44.8%

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

   1. mul-1-neg 44.8%

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

6. Simplified 44.8%

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

7. Final simplification 44.8%

   \[\leadsto -t \]

Alternative 14: 2.3% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 84.1%

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

   1. +-commutative 84.1%

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

   2. fma-def 84.1%

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

   3. sub-neg 84.1%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around 0 82.5%

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

   1. fma-neg 82.5%

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

6. Simplified 82.5%

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

   1. add-cube-cbrt 81.0%

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

   2. pow3 80.9%

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

   3. add-sqr-sqrt 40.3%

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

   4. sqrt-unprod 44.7%

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

   5. sqr-neg 44.7%

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

   6. sqrt-unprod 19.8%

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

   7. add-sqr-sqrt 37.2%

      \[\leadsto {\left(\sqrt[3]{\mathsf{fma}\left(\log y, x, \color{blue}{t}\right)}\right)}^{3} \]

8. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\log y, x, t\right)}\right)}^{3}} \]

9. Taylor expanded in x around 0 2.4%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot t} \]
10. Step-by-step derivation

    1. pow-base-1 2.4%

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

    2. *-lft-identity 2.4%

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

11. Simplified 2.4%

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

12. Final simplification 2.4%

    \[\leadsto t \]

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 2023199 
(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))