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

Percentage Accurate: 85.4% → 99.8%

Time: 14.1s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

   1. +-commutative 88.2%

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

   2. associate--l+ 88.2%

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

   3. fma-def 88.2%

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

   4. sub-neg 88.2%

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

   5. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot \log y + -0.5 \cdot \left(z \cdot {y}^{2}\right)\right) - z \cdot y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (- (+ (* x (log y)) (* -0.5 (* z (pow y 2.0)))) (* z y)) t))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) + (-0.5 * (z * pow(y, 2.0)))) - (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)) + ((-0.5d0) * (z * (y ** 2.0d0)))) - (z * y)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * Math.log(y)) + (-0.5 * (z * Math.pow(y, 2.0)))) - (z * y)) - t;
}

Python

def code(x, y, z, t):
	return (((x * math.log(y)) + (-0.5 * (z * math.pow(y, 2.0)))) - (z * y)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) + Float64(-0.5 * Float64(z * (y ^ 2.0)))) - Float64(z * y)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) + (-0.5 * (z * (y ^ 2.0)))) - (z * y)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(z * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in y around 0 99.4%

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

3. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

   1. +-commutative 99.3%

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

   2. *-commutative 99.3%

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

   3. fma-def 99.3%

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

6. Applied egg-rr 99.3%

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

7. Final simplification 99.3%

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

Alternative 4: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+26} \lor \neg \left(x \leq 6 \cdot 10^{-13} \lor \neg \left(x \leq 180000000\right) \land x \leq 1.22 \cdot 10^{+116}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.2e+26)
         (not
          (or (<= x 6e-13) (and (not (<= x 180000000.0)) (<= x 1.22e+116)))))
   (* x (log y))
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+26) || !((x <= 6e-13) || (!(x <= 180000000.0) && (x <= 1.22e+116)))) {
		tmp = x * log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-3.2d+26)) .or. (.not. (x <= 6d-13) .or. (.not. (x <= 180000000.0d0)) .and. (x <= 1.22d+116))) then
        tmp = x * log(y)
    else
        tmp = (y * -z) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.2e+26) || !((x <= 6e-13) || (!(x <= 180000000.0) && (x <= 1.22e+116)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.2e+26) or not ((x <= 6e-13) or (not (x <= 180000000.0) and (x <= 1.22e+116))):
		tmp = x * math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.2e+26) || !((x <= 6e-13) || (!(x <= 180000000.0) && (x <= 1.22e+116))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.2e+26) || ~(((x <= 6e-13) || (~((x <= 180000000.0)) && (x <= 1.22e+116)))))
		tmp = x * log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.2e+26], N[Not[Or[LessEqual[x, 6e-13], And[N[Not[LessEqual[x, 180000000.0]], $MachinePrecision], LessEqual[x, 1.22e+116]]]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.20000000000000029e26 or 5.99999999999999968e-13 < x < 1.8e8 or 1.21999999999999993e116 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 99.6%

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

   3. Taylor expanded in x around inf 86.3%

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

   if -3.20000000000000029e26 < x < 5.99999999999999968e-13 or 1.8e8 < x < 1.21999999999999993e116

   1. Initial program 81.2%

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

   2. Taylor expanded in y around 0 99.1%

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

      1. mul-1-neg 99.1%

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

   4. Simplified 99.1%

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

   5. Taylor expanded in x around 0 84.4%

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

      1. associate-*r* 84.4%

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

      2. neg-mul-1 84.4%

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

      3. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+26} \lor \neg \left(x \leq 6 \cdot 10^{-13} \lor \neg \left(x \leq 180000000\right) \land x \leq 1.22 \cdot 10^{+116}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 5: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;x \leq 110000000000 \lor \neg \left(x \leq 2.9 \cdot 10^{+114}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.8e+25)
     t_1
     (if (<= x 1.8e-11)
       (- (fma y z t))
       (if (or (<= x 110000000000.0) (not (<= x 2.9e+114)))
         t_1
         (- (* y (- z)) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.8e+25) {
		tmp = t_1;
	} else if (x <= 1.8e-11) {
		tmp = -fma(y, z, t);
	} else if ((x <= 110000000000.0) || !(x <= 2.9e+114)) {
		tmp = t_1;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3.8e+25)
		tmp = t_1;
	elseif (x <= 1.8e-11)
		tmp = Float64(-fma(y, z, t));
	elseif ((x <= 110000000000.0) || !(x <= 2.9e+114))
		tmp = t_1;
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+25], t$95$1, If[LessEqual[x, 1.8e-11], (-N[(y * z + t), $MachinePrecision]), If[Or[LessEqual[x, 110000000000.0], N[Not[LessEqual[x, 2.9e+114]], $MachinePrecision]], t$95$1, N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e25 or 1.79999999999999992e-11 < x < 1.1e11 or 2.9e114 < x

   1. Initial program 98.8%

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

   2. Taylor expanded in y around 0 99.6%

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

   3. Taylor expanded in x around inf 86.3%

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

   if -3.8e25 < x < 1.79999999999999992e-11

   1. Initial program 79.5%

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

   2. Taylor expanded in y around 0 98.9%

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

      1. mul-1-neg 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in x around 0 86.3%

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

      1. sub-neg 86.3%

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

      2. mul-1-neg 86.3%

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

      3. distribute-neg-out 86.3%

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

      4. fma-udef 86.3%

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

   7. Simplified 86.3%

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

   if 1.1e11 < x < 2.9e114

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. mul-1-neg 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in x around 0 71.7%

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

      1. associate-*r* 71.7%

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

      2. neg-mul-1 71.7%

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

      3. *-commutative 71.7%

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

   7. Simplified 71.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \mathbf{elif}\;x \leq 110000000000 \lor \neg \left(x \leq 2.9 \cdot 10^{+114}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 6: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-68} \lor \neg \left(x \leq 7.5 \cdot 10^{-116}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.42e-68) (not (<= x 7.5e-116)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.42e-68) || !(x <= 7.5e-116)) {
		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 <= -1.42e-68) || !(x <= 7.5e-116)) {
		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 <= -1.42e-68) or not (x <= 7.5e-116):
		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 <= -1.42e-68) || !(x <= 7.5e-116))
		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, -1.42e-68], N[Not[LessEqual[x, 7.5e-116]], $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 < -1.42e-68 or 7.5000000000000004e-116 < x

   1. Initial program 94.3%

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

   2. Taylor expanded in y around 0 94.3%

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

   if -1.42e-68 < x < 7.5000000000000004e-116

   1. Initial program 76.8%

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

   2. Taylor expanded in x around 0 71.7%

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

      1. sub-neg 71.7%

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

      2. mul-1-neg 71.7%

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

      3. log1p-def 94.8%

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

      4. mul-1-neg 94.8%

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

   4. Simplified 94.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-68} \lor \neg \left(x \leq 7.5 \cdot 10^{-116}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 7: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-70} \lor \neg \left(x \leq 6.8 \cdot 10^{-116}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -9.2e-70) (not (<= x 6.8e-116)))
   (- (* x (log y)) t)
   (- (fma y z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -9.2e-70) || !(x <= 6.8e-116)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = -fma(y, z, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -9.2e-70) || !(x <= 6.8e-116))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(-fma(y, z, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -9.2e-70], N[Not[LessEqual[x, 6.8e-116]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(y * z + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -9.20000000000000002e-70 or 6.79999999999999985e-116 < x

   1. Initial program 94.3%

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

   2. Taylor expanded in y around 0 94.3%

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

   if -9.20000000000000002e-70 < x < 6.79999999999999985e-116

   1. Initial program 76.8%

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

   2. Taylor expanded in y around 0 98.5%

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

      1. mul-1-neg 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in x around 0 93.4%

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

      1. sub-neg 93.4%

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

      2. mul-1-neg 93.4%

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

      3. distribute-neg-out 93.4%

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

      4. fma-udef 93.4%

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

   7. Simplified 93.4%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-70} \lor \neg \left(x \leq 6.8 \cdot 10^{-116}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(y, z, t\right)\\ \end{array} \]

Alternative 8: 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 88.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 9: 48.0% accurate, 25.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-173} \lor \neg \left(t \leq 6.5 \cdot 10^{-81}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.6e-173) (not (<= t 6.5e-81))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.6e-173) || !(t <= 6.5e-81)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	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 <= (-1.6d-173)) .or. (.not. (t <= 6.5d-81))) then
        tmp = -t
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.6e-173) || !(t <= 6.5e-81)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.6e-173) or not (t <= 6.5e-81):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.6e-173) || !(t <= 6.5e-81))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.6e-173) || ~((t <= 6.5e-81)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.6e-173], N[Not[LessEqual[t, 6.5e-81]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6e-173 or 6.5000000000000002e-81 < t

   1. Initial program 93.4%

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

   2. Taylor expanded in t around inf 58.1%

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

      1. neg-mul-1 58.1%

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

   4. Simplified 58.1%

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

   if -1.6e-173 < t < 6.5000000000000002e-81

   1. Initial program 73.8%

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

   2. Taylor expanded in y around 0 97.7%

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

      1. mul-1-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in y around inf 29.2%

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

      1. associate-*r* 29.2%

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

      2. neg-mul-1 29.2%

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

      3. *-commutative 29.2%

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

   7. Simplified 29.2%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-173} \lor \neg \left(t \leq 6.5 \cdot 10^{-81}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]

Alternative 10: 56.8% accurate, 35.2× speedup

Math

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

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(y * Float64(-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 88.2%

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

2. Taylor expanded in y around 0 99.3%

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

   1. mul-1-neg 99.3%

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

4. Simplified 99.3%

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

5. Taylor expanded in x around 0 56.4%

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

   1. associate-*r* 56.4%

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

   2. neg-mul-1 56.4%

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

   3. *-commutative 56.4%

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

7. Simplified 56.4%

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

8. Final simplification 56.4%

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

Alternative 11: 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 88.2%

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

2. Taylor expanded in t around inf 44.8%

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

   1. neg-mul-1 44.8%

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

4. Simplified 44.8%

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

5. Final simplification 44.8%

   \[\leadsto -t \]

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 2023320 
(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))