Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 2.0s

Alternatives: 9

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 89.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq -500000:\\ \;\;\;\;\left(\log y - y\right) - z\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+143)
     t_2
     (if (<= t_2 -500000.0)
       (- (- (log y) y) z)
       (if (<= t_2 4e-16) (- (log t) z) (- t_1 z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+143) {
		tmp = t_2;
	} else if (t_2 <= -500000.0) {
		tmp = (log(y) - y) - z;
	} else if (t_2 <= 4e-16) {
		tmp = log(t) - z;
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-5d+143)) then
        tmp = t_2
    else if (t_2 <= (-500000.0d0)) then
        tmp = (log(y) - y) - z
    else if (t_2 <= 4d-16) then
        tmp = log(t) - z
    else
        tmp = t_1 - z
    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 - y;
	double tmp;
	if (t_2 <= -5e+143) {
		tmp = t_2;
	} else if (t_2 <= -500000.0) {
		tmp = (Math.log(y) - y) - z;
	} else if (t_2 <= 4e-16) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -5e+143:
		tmp = t_2
	elif t_2 <= -500000.0:
		tmp = (math.log(y) - y) - z
	elif t_2 <= 4e-16:
		tmp = math.log(t) - z
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -5e+143)
		tmp = t_2;
	elseif (t_2 <= -500000.0)
		tmp = Float64(Float64(log(y) - y) - z);
	elseif (t_2 <= 4e-16)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -5e+143)
		tmp = t_2;
	elseif (t_2 <= -500000.0)
		tmp = (log(y) - y) - z;
	elseif (t_2 <= 4e-16)
		tmp = log(t) - z;
	else
		tmp = t_1 - z;
	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 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+143], t$95$2, If[LessEqual[t$95$2, -500000.0], N[(N[(N[Log[y], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$2, 4e-16], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5.00000000000000012e143

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.1%

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

   if -5.00000000000000012e143 < (-.f64 (*.f64 x (log.f64 y)) y) < -5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.9%

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

   if -5e5 < (-.f64 (*.f64 x (log.f64 y)) y) < 3.9999999999999999e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.8%

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

   if 3.9999999999999999e-16 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.5%

      \[\leadsto x \cdot \log y - z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq -500000:\\ \;\;\;\;\left(\log y - y\right) - z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)))
   (if (<= t_1 -5e+143)
     t_1
     (if (<= t_1 -500000.0)
       (- (- (log y) y) z)
       (if (<= t_1 4e-16) (- (log t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double tmp;
	if (t_1 <= -5e+143) {
		tmp = t_1;
	} else if (t_1 <= -500000.0) {
		tmp = (log(y) - y) - z;
	} else if (t_1 <= 4e-16) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = (x * log(y)) - y
    if (t_1 <= (-5d+143)) then
        tmp = t_1
    else if (t_1 <= (-500000.0d0)) then
        tmp = (log(y) - y) - z
    else if (t_1 <= 4d-16) then
        tmp = log(t) - z
    else
        tmp = t_1
    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)) - y;
	double tmp;
	if (t_1 <= -5e+143) {
		tmp = t_1;
	} else if (t_1 <= -500000.0) {
		tmp = (Math.log(y) - y) - z;
	} else if (t_1 <= 4e-16) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	tmp = 0
	if t_1 <= -5e+143:
		tmp = t_1
	elif t_1 <= -500000.0:
		tmp = (math.log(y) - y) - z
	elif t_1 <= 4e-16:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	tmp = 0.0
	if (t_1 <= -5e+143)
		tmp = t_1;
	elseif (t_1 <= -500000.0)
		tmp = Float64(Float64(log(y) - y) - z);
	elseif (t_1 <= 4e-16)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	tmp = 0.0;
	if (t_1 <= -5e+143)
		tmp = t_1;
	elseif (t_1 <= -500000.0)
		tmp = (log(y) - y) - z;
	elseif (t_1 <= 4e-16)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+143], t$95$1, If[LessEqual[t$95$1, -500000.0], N[(N[(N[Log[y], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$1, 4e-16], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5.00000000000000012e143 or 3.9999999999999999e-16 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.9%

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

   if -5.00000000000000012e143 < (-.f64 (*.f64 x (log.f64 y)) y) < -5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 85.9%

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

   if -5e5 < (-.f64 (*.f64 x (log.f64 y)) y) < 3.9999999999999999e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\log t - z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ t_2 := t\_1 - z\\ \mathbf{if}\;t\_1 \leq -500000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)) (t_2 (- t_1 z)))
   (if (<= t_1 -500000.0) t_2 (if (<= t_1 4e-16) (- (log t) z) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double t_2 = t_1 - z;
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = t_2;
	} else if (t_1 <= 4e-16) {
		tmp = log(t) - 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)) - y
    t_2 = t_1 - z
    if (t_1 <= (-500000.0d0)) then
        tmp = t_2
    else if (t_1 <= 4d-16) then
        tmp = log(t) - 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)) - y;
	double t_2 = t_1 - z;
	double tmp;
	if (t_1 <= -500000.0) {
		tmp = t_2;
	} else if (t_1 <= 4e-16) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	t_2 = t_1 - z
	tmp = 0
	if t_1 <= -500000.0:
		tmp = t_2
	elif t_1 <= 4e-16:
		tmp = math.log(t) - z
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	t_2 = Float64(t_1 - z)
	tmp = 0.0
	if (t_1 <= -500000.0)
		tmp = t_2;
	elseif (t_1 <= 4e-16)
		tmp = Float64(log(t) - z);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	t_2 = t_1 - z;
	tmp = 0.0;
	if (t_1 <= -500000.0)
		tmp = t_2;
	elseif (t_1 <= 4e-16)
		tmp = log(t) - z;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - z), $MachinePrecision]}, If[LessEqual[t$95$1, -500000.0], t$95$2, If[LessEqual[t$95$1, 4e-16], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e5 or 3.9999999999999999e-16 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.4%

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

   if -5e5 < (-.f64 (*.f64 x (log.f64 y)) y) < 3.9999999999999999e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\log t - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;\left(\log y - y\right) - z\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -500000.0)
     (- (- (log y) y) z)
     (if (<= t_2 4e-16) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = (log(y) - y) - z;
	} else if (t_2 <= 4e-16) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	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 - y
    if (t_2 <= (-500000.0d0)) then
        tmp = (log(y) - y) - z
    else if (t_2 <= 4d-16) then
        tmp = log(t) - z
    else
        tmp = t_1
    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 - y;
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = (Math.log(y) - y) - z;
	} else if (t_2 <= 4e-16) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -500000.0:
		tmp = (math.log(y) - y) - z
	elif t_2 <= 4e-16:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = Float64(Float64(log(y) - y) - z);
	elseif (t_2 <= 4e-16)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -500000.0)
		tmp = (log(y) - y) - z;
	elseif (t_2 <= 4e-16)
		tmp = log(t) - z;
	else
		tmp = t_1;
	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 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], N[(N[(N[Log[y], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[t$95$2, 4e-16], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -5e5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.4%

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

   if -5e5 < (-.f64 (*.f64 x (log.f64 y)) y) < 3.9999999999999999e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 46.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 99.8%

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

   if 3.9999999999999999e-16 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.1%

      \[\leadsto x \cdot \color{blue}{\log y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 67.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+110}:\\ \;\;\;\;\log y - y\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -1e+110) (- (log y) y) (if (<= t_2 4e-16) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -1e+110) {
		tmp = log(y) - y;
	} else if (t_2 <= 4e-16) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	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 - y
    if (t_2 <= (-1d+110)) then
        tmp = log(y) - y
    else if (t_2 <= 4d-16) then
        tmp = log(t) - z
    else
        tmp = t_1
    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 - y;
	double tmp;
	if (t_2 <= -1e+110) {
		tmp = Math.log(y) - y;
	} else if (t_2 <= 4e-16) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -1e+110:
		tmp = math.log(y) - y
	elif t_2 <= 4e-16:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -1e+110)
		tmp = Float64(log(y) - y);
	elseif (t_2 <= 4e-16)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -1e+110)
		tmp = log(y) - y;
	elseif (t_2 <= 4e-16)
		tmp = log(t) - z;
	else
		tmp = t_1;
	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 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+110], N[(N[Log[y], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[t$95$2, 4e-16], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e110

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.8%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 56.0%

       \[\leadsto \log y - \color{blue}{y} \]

   if -1e110 < (-.f64 (*.f64 x (log.f64 y)) y) < 3.9999999999999999e-16

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 81.9%

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

   if 3.9999999999999999e-16 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.1%

      \[\leadsto x \cdot \color{blue}{\log y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{+99}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.75e+99) (- (log t) z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.75e+99) {
		tmp = log(t) - z;
	} else {
		tmp = log(t) - y;
	}
	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 (y <= 1.75d+99) then
        tmp = log(t) - z
    else
        tmp = log(t) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.75e+99) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.75e+99:
		tmp = math.log(t) - z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.75e+99)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.75e+99)
		tmp = log(t) - z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.75e+99], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7499999999999999e99

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 55.6%

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

   if 1.7499999999999999e99 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in x around 0

      \[\leadsto x \cdot \log y - y \]

   11. Applied rewrites 68.2%

       \[\leadsto \log t - y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 41.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 86.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 59.2%

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

9. Taylor expanded in x around 0

   \[\leadsto x \cdot \log y - y \]

11. Applied rewrites 43.3%

    \[\leadsto \log t - y \]
12. Add Preprocessing

Alternative 9: 14.4% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return log(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 = log(t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Log[t], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 86.3%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 15.5%

   \[\leadsto \log t \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  :pre (TRUE)
  (+ (- (- (* x (log y)) y) z) (log t)))