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

Percentage Accurate: 85.4% → 99.8%

Time: 13.9s

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: 85.4% 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 84.2%

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

   1. +-commutative 84.2%

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

   2. fma-def 84.2%

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

   3. sub-neg 84.2%

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

   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: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((-0.5 * (z * (y * y))) - (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[(z * y), $MachinePrecision]), $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. fma-def 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. distribute-lft-neg-out 99.3%

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

   3. unsub-neg 99.3%

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

   4. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. fma-def 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. distribute-lft-neg-out 99.3%

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

   3. unsub-neg 99.3%

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

   4. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. associate-+r- 99.3%

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

   2. fma-def 99.3%

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

   3. associate-*l* 99.3%

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

8. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, \log y, \left(y \cdot \left(y \cdot z\right)\right) \cdot -0.5\right) - y \cdot z\right)} - t \]
9. Step-by-step derivation

   1. fma-udef 99.3%

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

10. Applied egg-rr 99.3%

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

11. Final simplification 99.3%

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

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. fma-def 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

   1. fma-udef 99.3%

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

   2. distribute-lft-neg-out 99.3%

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

   3. unsub-neg 99.3%

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

   4. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

7. Taylor expanded in y around inf 99.3%

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

   1. neg-mul-1 99.3%

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

   2. associate-+r+ 99.3%

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

   3. sub-neg 99.3%

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

   4. unpow2 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

   7. mul-1-neg 99.3%

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

   8. unsub-neg 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 5: 90.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-105} \lor \neg \left(x \leq 3.4 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.35e-105) (not (<= x 3.4e-53)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.35e-105) || !(x <= 3.4e-53)) {
		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 <= -1.35e-105) || !(x <= 3.4e-53)) {
		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 <= -1.35e-105) or not (x <= 3.4e-53):
		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 <= -1.35e-105) || !(x <= 3.4e-53))
		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, -1.35e-105], N[Not[LessEqual[x, 3.4e-53]], $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 < -1.34999999999999996e-105 or 3.4e-53 < x

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. fma-def 88.6%

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

      3. sub-neg 88.6%

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

      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 88.2%

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

   if -1.34999999999999996e-105 < x < 3.4e-53

   1. Initial program 77.4%

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

   2. Taylor expanded in x around 0 74.0%

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

      1. sub-neg 74.0%

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

      2. mul-1-neg 74.0%

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

      3. log1p-def 95.4%

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

      4. mul-1-neg 95.4%

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

   4. Simplified 95.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-105} \lor \neg \left(x \leq 3.4 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 6: 90.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-107} \lor \neg \left(x \leq 3.4 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(z \cdot -0.5\right) - z \cdot y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e-107) (not (<= x 3.4e-53)))
   (- (* x (log y)) t)
   (- (- (* (* y y) (* z -0.5)) (* z y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e-107) || !(x <= 3.4e-53)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (((y * y) * (z * -0.5)) - (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 ((x <= (-3.1d-107)) .or. (.not. (x <= 3.4d-53))) then
        tmp = (x * log(y)) - t
    else
        tmp = (((y * y) * (z * (-0.5d0))) - (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 ((x <= -3.1e-107) || !(x <= 3.4e-53)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (((y * y) * (z * -0.5)) - (z * y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.1e-107) or not (x <= 3.4e-53):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (((y * y) * (z * -0.5)) - (z * y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e-107) || !(x <= 3.4e-53))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(Float64(y * y) * Float64(z * -0.5)) - Float64(z * y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.1e-107) || ~((x <= 3.4e-53)))
		tmp = (x * log(y)) - t;
	else
		tmp = (((y * y) * (z * -0.5)) - (z * y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e-107], N[Not[LessEqual[x, 3.4e-53]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(y * y), $MachinePrecision] * N[(z * -0.5), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000022e-107 or 3.4e-53 < x

   1. Initial program 88.6%

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

      1. +-commutative 88.6%

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

      2. fma-def 88.6%

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

      3. sub-neg 88.6%

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

      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 88.2%

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

   if -3.10000000000000022e-107 < x < 3.4e-53

   1. Initial program 77.4%

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

   2. Taylor expanded in y around 0 98.8%

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

      1. fma-def 98.8%

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

      2. unpow2 98.8%

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

      3. associate-*r* 98.8%

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

      4. mul-1-neg 98.8%

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

   4. Simplified 98.8%

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

   5. Taylor expanded in x around 0 94.3%

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

      1. *-commutative 94.3%

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

      2. associate-*l* 94.3%

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

      3. fma-def 94.3%

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

      4. mul-1-neg 94.3%

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

      5. fma-neg 94.3%

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

      6. unpow2 94.3%

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

   7. Simplified 94.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-107} \lor \neg \left(x \leq 3.4 \cdot 10^{-53}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot \left(z \cdot -0.5\right) - z \cdot y\right) - t\\ \end{array} \]

Alternative 7: 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 84.2%

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

2. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. log-pow 47.3%

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

   4. mul-1-neg 47.3%

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

   5. unsub-neg 47.3%

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

   6. log-pow 98.9%

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

   7. *-commutative 98.9%

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

4. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 8: 57.1% accurate, 16.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (((y * y) * (z * -0.5)) - (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 = (((y * y) * (z * (-0.5d0))) - (z * y)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. fma-def 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in x around 0 58.4%

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

   1. *-commutative 58.4%

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

   2. associate-*l* 58.4%

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

   3. fma-def 58.4%

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

   4. mul-1-neg 58.4%

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

   5. fma-neg 58.4%

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

   6. unpow2 58.4%

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

7. Simplified 58.4%

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

8. Final simplification 58.4%

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

Alternative 9: 57.1% accurate, 19.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (z * ((y * (y * -0.5)) - 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 * (y * (-0.5d0))) - y)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. fma-def 99.3%

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

   2. unpow2 99.3%

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

   3. associate-*r* 99.3%

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

   4. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in x around 0 58.4%

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

   1. associate-*r* 58.4%

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

   2. associate-*r* 58.4%

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

   3. distribute-rgt-in 58.4%

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

   4. mul-1-neg 58.4%

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

   5. unsub-neg 58.4%

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

   6. *-commutative 58.4%

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

   7. unpow2 58.4%

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

   8. associate-*l* 58.4%

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

7. Simplified 58.4%

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

8. Final simplification 58.4%

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

Alternative 10: 48.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.04 \cdot 10^{-75}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-63}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.04e-75) (- t) (if (<= t 6.4e-63) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.04e-75) {
		tmp = -t;
	} else if (t <= 6.4e-63) {
		tmp = z * -y;
	} 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 <= (-1.04d-75)) then
        tmp = -t
    else if (t <= 6.4d-63) then
        tmp = z * -y
    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 <= -1.04e-75) {
		tmp = -t;
	} else if (t <= 6.4e-63) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.04e-75:
		tmp = -t
	elif t <= 6.4e-63:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.04e-75)
		tmp = Float64(-t);
	elseif (t <= 6.4e-63)
		tmp = Float64(z * Float64(-y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.04e-75)
		tmp = -t;
	elseif (t <= 6.4e-63)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.04e-75], (-t), If[LessEqual[t, 6.4e-63], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.04e-75 or 6.39999999999999978e-63 < t

   1. Initial program 92.1%

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

      1. +-commutative 92.1%

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

      2. fma-def 92.1%

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

      3. sub-neg 92.1%

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

      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 64.1%

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

      1. mul-1-neg 64.1%

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

   6. Simplified 64.1%

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

   if -1.04e-75 < t < 6.39999999999999978e-63

   1. Initial program 71.3%

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

   2. Taylor expanded in y around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. *-commutative 98.0%

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

      3. log-pow 33.6%

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

      4. mul-1-neg 33.6%

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

      5. unsub-neg 33.6%

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

      6. log-pow 98.0%

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

      7. *-commutative 98.0%

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

   4. Simplified 98.0%

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

   5. Taylor expanded in y around inf 35.0%

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

      1. neg-mul-1 35.0%

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

      2. distribute-rgt-neg-in 35.0%

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

   7. Simplified 35.0%

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

   8. Taylor expanded in y around inf 29.0%

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

      1. associate-*r* 29.0%

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

      2. neg-mul-1 29.0%

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

   10. Simplified 29.0%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.04 \cdot 10^{-75}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-63}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 56.8% 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 84.2%

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

2. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. log-pow 47.3%

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

   4. mul-1-neg 47.3%

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

   5. unsub-neg 47.3%

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

   6. log-pow 98.9%

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

   7. *-commutative 98.9%

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

4. Simplified 98.9%

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

5. Taylor expanded in y around inf 58.0%

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

   1. neg-mul-1 58.0%

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

   2. distribute-rgt-neg-in 58.0%

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

7. Simplified 58.0%

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

8. Final simplification 58.0%

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

Alternative 12: 42.4% 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 84.2%

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

   1. +-commutative 84.2%

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

   2. fma-def 84.2%

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

   3. sub-neg 84.2%

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

   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 43.1%

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

   1. mul-1-neg 43.1%

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

6. Simplified 43.1%

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

7. Final simplification 43.1%

   \[\leadsto -t \]

Alternative 13: 2.2% 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 84.2%

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

   1. +-commutative 84.2%

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

   2. fma-def 84.2%

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

   3. sub-neg 84.2%

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

   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 82.9%

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

   1. fma-neg 82.9%

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

6. Simplified 82.9%

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

   1. add-sqr-sqrt 37.9%

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

   2. pow2 37.9%

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

   3. fma-udef 37.9%

      \[\leadsto {\left(\sqrt{\color{blue}{\log y \cdot x + \left(-t\right)}}\right)}^{2} \]

   4. *-commutative 37.9%

      \[\leadsto {\left(\sqrt{\color{blue}{x \cdot \log y} + \left(-t\right)}\right)}^{2} \]

   5. fma-def 37.9%

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

   6. add-sqr-sqrt 28.4%

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

   7. sqrt-unprod 28.1%

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

   8. sqr-neg 28.1%

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

   9. sqrt-unprod 9.8%

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

   10. add-sqr-sqrt 16.9%

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

8. Applied egg-rr 16.9%

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

9. Taylor expanded in x around 0 2.3%

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

10. Final simplification 2.3%

    \[\leadsto t \]

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 2023240 
(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))