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

Percentage Accurate: 85.8% → 99.8%

Time: 12.2s

Alternatives: 8

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: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-neg N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. sub-neg N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 88.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - t\\ t_3 := \log \left(1 - y\right) \cdot z + t\_1\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-257}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (- t_1 t))
        (t_3 (+ (* (log (- 1.0 y)) z) t_1)))
   (if (<= t_3 -2e-56) t_2 (if (<= t_3 5e-257) (- (* (- y) z) t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - t;
	double t_3 = (log((1.0 - y)) * z) + t_1;
	double tmp;
	if (t_3 <= -2e-56) {
		tmp = t_2;
	} else if (t_3 <= 5e-257) {
		tmp = (-y * z) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - t
    t_3 = (log((1.0d0 - y)) * z) + t_1
    if (t_3 <= (-2d-56)) then
        tmp = t_2
    else if (t_3 <= 5d-257) then
        tmp = (-y * z) - t
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - t;
	double t_3 = (Math.log((1.0 - y)) * z) + t_1;
	double tmp;
	if (t_3 <= -2e-56) {
		tmp = t_2;
	} else if (t_3 <= 5e-257) {
		tmp = (-y * z) - t;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - t
	t_3 = (math.log((1.0 - y)) * z) + t_1
	tmp = 0
	if t_3 <= -2e-56:
		tmp = t_2
	elif t_3 <= 5e-257:
		tmp = (-y * z) - t
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - t)
	t_3 = Float64(Float64(log(Float64(1.0 - y)) * z) + t_1)
	tmp = 0.0
	if (t_3 <= -2e-56)
		tmp = t_2;
	elseif (t_3 <= 5e-257)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - t;
	t_3 = (log((1.0 - y)) * z) + t_1;
	tmp = 0.0;
	if (t_3 <= -2e-56)
		tmp = t_2;
	elseif (t_3 <= 5e-257)
		tmp = (-y * z) - t;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-56], t$95$2, If[LessEqual[t$95$3, 5e-257], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < -2.0000000000000001e-56 or 4.99999999999999989e-257 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. lower-fma.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 92.0

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

   5. Applied rewrites 92.0%

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

   7. Applied rewrites 92.0%

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

   if -2.0000000000000001e-56 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 4.99999999999999989e-257

   1. Initial program 65.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 98.3%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 - y\right) \cdot z + x \cdot \log y \leq -2 \cdot 10^{-56}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;\log \left(1 - y\right) \cdot z + x \cdot \log y \leq 5 \cdot 10^{-257}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. lower-fma.f64 N/A

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

   9. log-rec N/A

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

   10. mul-1-neg N/A

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

   11. mul-1-neg N/A

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

   12. mul-1-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower-log.f64 N/A

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

   15. associate-*r* N/A

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

   16. mul-1-neg N/A

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

   17. lower-*.f64 N/A

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

   18. lower-neg.f64 99.4

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

5. Applied rewrites 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. log-rec N/A

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

   9. lower-fma.f64 N/A

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

   10. log-rec N/A

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

   11. mul-1-neg N/A

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

   12. mul-1-neg N/A

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

   13. mul-1-neg N/A

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

   14. remove-double-neg N/A

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

   15. lower-log.f64 N/A

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

   16. sub-neg N/A

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

   17. mul-1-neg N/A

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

   18. distribute-neg-out N/A

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

   19. lower-neg.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 99.4

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

5. Applied rewrites 99.4%

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

Alternative 5: 57.5% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right) \cdot z - t \]
7. Step-by-step derivation

8. Applied rewrites 53.6%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
9. Add Preprocessing

Alternative 6: 57.4% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \cdot z - t \]
7. Step-by-step derivation

8. Applied rewrites 53.6%

   \[\leadsto \left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y\right) \cdot z - t \]
9. Add Preprocessing

Alternative 7: 57.0% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 53.6

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

5. Applied rewrites 53.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 53.3%

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

Alternative 8: 42.8% accurate, 73.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 41.2

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

5. Applied rewrites 41.2%

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

Developer Target 1: 99.6% accurate, 1.3× 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 2024296 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

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