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

Percentage Accurate: 85.6% → 99.2%

Time: 14.9s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. lower--.f64 N/A

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

5. Applied rewrites 99.7%

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

Alternative 2: 89.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - t\\ t_3 := t\_1 + z \cdot \log \left(1 - y\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-130}:\\ \;\;\;\;-\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y)))
        (t_2 (- t_1 t))
        (t_3 (+ t_1 (* z (log (- 1.0 y))))))
   (if (<= t_3 -2e-72) t_2 (if (<= t_3 1e-130) (- (fma z y 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 = t_1 + (z * log((1.0 - y)));
	double tmp;
	if (t_3 <= -2e-72) {
		tmp = t_2;
	} else if (t_3 <= 1e-130) {
		tmp = -fma(z, y, 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(t_1 + Float64(z * log(Float64(1.0 - y))))
	tmp = 0.0
	if (t_3 <= -2e-72)
		tmp = t_2;
	elseif (t_3 <= 1e-130)
		tmp = Float64(-fma(z, y, t));
	else
		tmp = t_2;
	end
	return 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[(t$95$1 + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-72], t$95$2, If[LessEqual[t$95$3, 1e-130], (-N[(z * y + 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)))) < -1.9999999999999999e-72 or 1.0000000000000001e-130 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 92.3%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 92.3

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

   5. Applied rewrites 92.3%

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

   if -1.9999999999999999e-72 < (+.f64 (*.f64 x (log.f64 y)) (*.f64 z (log.f64 (-.f64 #s(literal 1 binary64) y)))) < 1.0000000000000001e-130

   1. Initial program 72.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.5%

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

Alternative 3: 90.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - t\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;t\_1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 t)))
   (if (<= t -2.8e-100) t_2 (if (<= t 2.9e-60) (- t_1 (* y z)) 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 tmp;
	if (t <= -2.8e-100) {
		tmp = t_2;
	} else if (t <= 2.9e-60) {
		tmp = t_1 - (y * z);
	} 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) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - t
    if (t <= (-2.8d-100)) then
        tmp = t_2
    else if (t <= 2.9d-60) then
        tmp = t_1 - (y * z)
    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 tmp;
	if (t <= -2.8e-100) {
		tmp = t_2;
	} else if (t <= 2.9e-60) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - t
	tmp = 0
	if t <= -2.8e-100:
		tmp = t_2
	elif t <= 2.9e-60:
		tmp = t_1 - (y * z)
	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)
	tmp = 0.0
	if (t <= -2.8e-100)
		tmp = t_2;
	elseif (t <= 2.9e-60)
		tmp = Float64(t_1 - Float64(y * z));
	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;
	tmp = 0.0;
	if (t <= -2.8e-100)
		tmp = t_2;
	elseif (t <= 2.9e-60)
		tmp = t_1 - (y * z);
	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]}, If[LessEqual[t, -2.8e-100], t$95$2, If[LessEqual[t, 2.9e-60], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999995e-100 or 2.8999999999999999e-60 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if -2.79999999999999995e-100 < t < 2.8999999999999999e-60

   1. Initial program 74.2%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 94.6%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+25}:\\ \;\;\;\;-\mathsf{fma}\left(z, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -7.8e+21) t_1 (if (<= x 3.6e+25) (- (fma z y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -7.8e+21) {
		tmp = t_1;
	} else if (x <= 3.6e+25) {
		tmp = -fma(z, y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -7.8e+21)
		tmp = t_1;
	elseif (x <= 3.6e+25)
		tmp = Float64(-fma(z, y, t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e+21], t$95$1, If[LessEqual[x, 3.6e+25], (-N[(z * y + t), $MachinePrecision]), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.8e21 or 3.60000000000000015e25 < x

   1. Initial program 98.0%

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

   3. Taylor expanded in x around inf

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. mul-1-neg N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

      8. mul-1-neg N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower-log.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if -7.8e21 < x < 3.60000000000000015e25

   1. Initial program 75.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.1%

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

Alternative 5: 48.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-101}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5e-101) (- t) (if (<= t 2.9e-60) (* z (- y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5e-101) {
		tmp = -t;
	} else if (t <= 2.9e-60) {
		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 <= (-5d-101)) then
        tmp = -t
    else if (t <= 2.9d-60) 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 <= -5e-101) {
		tmp = -t;
	} else if (t <= 2.9e-60) {
		tmp = z * -y;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5e-101:
		tmp = -t
	elif t <= 2.9e-60:
		tmp = z * -y
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5e-101)
		tmp = Float64(-t);
	elseif (t <= 2.9e-60)
		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 <= -5e-101)
		tmp = -t;
	elseif (t <= 2.9e-60)
		tmp = z * -y;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5e-101], (-t), If[LessEqual[t, 2.9e-60], N[(z * (-y)), $MachinePrecision], (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0000000000000001e-101 or 2.8999999999999999e-60 < t

   1. Initial program 93.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 60.7

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

   5. Applied rewrites 60.7%

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

   if -5.0000000000000001e-101 < t < 2.8999999999999999e-60

   1. Initial program 74.2%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 28.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-101}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-60}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.3% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. lower--.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 54.2%

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

Alternative 7: 43.2% 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 86.9%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 41.8

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

5. Applied rewrites 41.8%

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

Developer Target 1: 99.5% 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 2024219 
(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))