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

Percentage Accurate: 99.9% → 99.9%

Time: 10.8s

Alternatives: 10

Speedup: 1.0×

Specification

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

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

FPCore

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

C

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

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) - (z + (y + (log((1.0d0 / y)) * x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around inf 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+64} \lor \neg \left(x \leq 2.6 \cdot 10^{+71}\right) \land \left(x \leq 1.4 \cdot 10^{+138} \lor \neg \left(x \leq 7.8 \cdot 10^{+201}\right)\right):\\ \;\;\;\;\log t + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.85e+64)
         (and (not (<= x 2.6e+71)) (or (<= x 1.4e+138) (not (<= x 7.8e+201)))))
   (+ (log t) (* x (log y)))
   (- (- (log t) z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.85e+64) || (!(x <= 2.6e+71) && ((x <= 1.4e+138) || !(x <= 7.8e+201)))) {
		tmp = log(t) + (x * log(y));
	} else {
		tmp = (log(t) - z) - 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 ((x <= (-1.85d+64)) .or. (.not. (x <= 2.6d+71)) .and. (x <= 1.4d+138) .or. (.not. (x <= 7.8d+201))) then
        tmp = log(t) + (x * log(y))
    else
        tmp = (log(t) - z) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.85e+64) || (!(x <= 2.6e+71) && ((x <= 1.4e+138) || !(x <= 7.8e+201)))) {
		tmp = Math.log(t) + (x * Math.log(y));
	} else {
		tmp = (Math.log(t) - z) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.85e+64) or (not (x <= 2.6e+71) and ((x <= 1.4e+138) or not (x <= 7.8e+201))):
		tmp = math.log(t) + (x * math.log(y))
	else:
		tmp = (math.log(t) - z) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.85e+64) || (!(x <= 2.6e+71) && ((x <= 1.4e+138) || !(x <= 7.8e+201))))
		tmp = Float64(log(t) + Float64(x * log(y)));
	else
		tmp = Float64(Float64(log(t) - z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.85e+64) || (~((x <= 2.6e+71)) && ((x <= 1.4e+138) || ~((x <= 7.8e+201)))))
		tmp = log(t) + (x * log(y));
	else
		tmp = (log(t) - z) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.85e+64], And[N[Not[LessEqual[x, 2.6e+71]], $MachinePrecision], Or[LessEqual[x, 1.4e+138], N[Not[LessEqual[x, 7.8e+201]], $MachinePrecision]]]], N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.84999999999999992e64 or 2.59999999999999991e71 < x < 1.4e138 or 7.8000000000000001e201 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 99.6%

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

   3. Taylor expanded in x around inf 79.1%

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

      1. mul-1-neg 79.1%

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

      2. *-commutative 79.1%

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

      3. log-rec 79.1%

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

      4. distribute-rgt-neg-in 79.1%

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

      5. remove-double-neg 79.1%

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

      6. *-commutative 79.1%

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

   5. Simplified 79.1%

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

   if -1.84999999999999992e64 < x < 2.59999999999999991e71 or 1.4e138 < x < 7.8000000000000001e201

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 92.8%

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

      1. neg-mul-1 92.8%

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

      2. +-commutative 92.8%

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

      3. distribute-neg-in 92.8%

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

      4. sub-neg 92.8%

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

   4. Simplified 92.8%

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

   5. Taylor expanded in z around 0 92.8%

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

      1. +-commutative 92.8%

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

      2. mul-1-neg 92.8%

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

      3. sub-neg 92.8%

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

   7. Simplified 92.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{+64} \lor \neg \left(x \leq 2.6 \cdot 10^{+71}\right) \land \left(x \leq 1.4 \cdot 10^{+138} \lor \neg \left(x \leq 7.8 \cdot 10^{+201}\right)\right):\\ \;\;\;\;\log t + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+62} \lor \neg \left(x \leq 3 \cdot 10^{+70}\right):\\ \;\;\;\;\log t + \left(x \cdot \log y - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.7e+62) (not (<= x 3e+70)))
   (+ (log t) (- (* x (log y)) y))
   (- (- (log t) z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.7e+62) || !(x <= 3e+70)) {
		tmp = log(t) + ((x * log(y)) - y);
	} else {
		tmp = (log(t) - z) - 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 ((x <= (-3.7d+62)) .or. (.not. (x <= 3d+70))) then
        tmp = log(t) + ((x * log(y)) - y)
    else
        tmp = (log(t) - z) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.7e+62) || !(x <= 3e+70)) {
		tmp = Math.log(t) + ((x * Math.log(y)) - y);
	} else {
		tmp = (Math.log(t) - z) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.7e+62) or not (x <= 3e+70):
		tmp = math.log(t) + ((x * math.log(y)) - y)
	else:
		tmp = (math.log(t) - z) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.7e+62) || !(x <= 3e+70))
		tmp = Float64(log(t) + Float64(Float64(x * log(y)) - y));
	else
		tmp = Float64(Float64(log(t) - z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.7e+62) || ~((x <= 3e+70)))
		tmp = log(t) + ((x * log(y)) - y);
	else
		tmp = (log(t) - z) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.7e+62], N[Not[LessEqual[x, 3e+70]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.70000000000000014e62 or 2.99999999999999976e70 < x

   1. Initial program 99.6%

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

      1. add-cbrt-cube 20.6%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z\right) + \log t \]

      2. pow3 20.6%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot \log y\right)}^{3}}} - y\right) - z\right) + \log t \]

   3. Applied egg-rr 20.6%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{\left(x \cdot \log y\right)}^{3}}} - y\right) - z\right) + \log t \]

   4. Taylor expanded in z around 0 85.7%

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

   if -3.70000000000000014e62 < x < 2.99999999999999976e70

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.5%

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

      1. neg-mul-1 94.5%

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

      2. +-commutative 94.5%

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

      3. distribute-neg-in 94.5%

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

      4. sub-neg 94.5%

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

   4. Simplified 94.5%

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

   5. Taylor expanded in z around 0 94.5%

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

      1. +-commutative 94.5%

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

      2. mul-1-neg 94.5%

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

      3. sub-neg 94.5%

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

   7. Simplified 94.5%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+62} \lor \neg \left(x \leq 3 \cdot 10^{+70}\right):\\ \;\;\;\;\log t + \left(x \cdot \log y - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]

Alternative 4: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;\log t + \left(t_1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t + \left(t_1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= y 6.8e+59) (+ (log t) (- t_1 z)) (+ (log t) (- t_1 y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (y <= 6.8e+59) {
		tmp = log(t) + (t_1 - z);
	} else {
		tmp = log(t) + (t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (y <= 6.8d+59) then
        tmp = log(t) + (t_1 - z)
    else
        tmp = log(t) + (t_1 - y)
    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 tmp;
	if (y <= 6.8e+59) {
		tmp = Math.log(t) + (t_1 - z);
	} else {
		tmp = Math.log(t) + (t_1 - y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if y <= 6.8e+59:
		tmp = math.log(t) + (t_1 - z)
	else:
		tmp = math.log(t) + (t_1 - y)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 6.8e+59)
		tmp = Float64(log(t) + Float64(t_1 - z));
	else
		tmp = Float64(log(t) + Float64(t_1 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (y <= 6.8e+59)
		tmp = log(t) + (t_1 - z);
	else
		tmp = log(t) + (t_1 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6.8e+59], N[(N[Log[t], $MachinePrecision] + N[(t$95$1 - z), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] + N[(t$95$1 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 6.80000000000000012e59

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 97.6%

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

   if 6.80000000000000012e59 < y

   1. Initial program 99.9%

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

      1. add-cbrt-cube 59.8%

         \[\leadsto \left(\left(\color{blue}{\sqrt[3]{\left(\left(x \cdot \log y\right) \cdot \left(x \cdot \log y\right)\right) \cdot \left(x \cdot \log y\right)}} - y\right) - z\right) + \log t \]

      2. pow3 59.8%

         \[\leadsto \left(\left(\sqrt[3]{\color{blue}{{\left(x \cdot \log y\right)}^{3}}} - y\right) - z\right) + \log t \]

   3. Applied egg-rr 59.8%

      \[\leadsto \left(\left(\color{blue}{\sqrt[3]{{\left(x \cdot \log y\right)}^{3}}} - y\right) - z\right) + \log t \]

   4. Taylor expanded in z around 0 82.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;\log t + \left(x \cdot \log y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t + \left(x \cdot \log y - y\right)\\ \end{array} \]

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) + (((x * log(y)) - y) - z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 6: 60.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2060000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+14}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2060000.0) (- z) (if (<= z 2.35e+14) (- (log t) y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2060000.0) {
		tmp = -z;
	} else if (z <= 2.35e+14) {
		tmp = log(t) - y;
	} else {
		tmp = -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 (z <= (-2060000.0d0)) then
        tmp = -z
    else if (z <= 2.35d+14) then
        tmp = log(t) - y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2060000.0) {
		tmp = -z;
	} else if (z <= 2.35e+14) {
		tmp = Math.log(t) - y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2060000.0:
		tmp = -z
	elif z <= 2.35e+14:
		tmp = math.log(t) - y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2060000.0)
		tmp = Float64(-z);
	elseif (z <= 2.35e+14)
		tmp = Float64(log(t) - y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2060000.0)
		tmp = -z;
	elseif (z <= 2.35e+14)
		tmp = log(t) - y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2060000.0], (-z), If[LessEqual[z, 2.35e+14], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -2.06e6 or 2.35e14 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 79.2%

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

      1. neg-mul-1 79.2%

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

      2. +-commutative 79.2%

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

      3. distribute-neg-in 79.2%

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

      4. sub-neg 79.2%

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

   4. Simplified 79.2%

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

   5. Taylor expanded in z around inf 61.9%

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

      1. mul-1-neg 61.9%

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

   7. Simplified 61.9%

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

   if -2.06e6 < z < 2.35e14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 57.1%

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

      1. neg-mul-1 57.1%

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

      2. +-commutative 57.1%

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

      3. distribute-neg-in 57.1%

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

      4. sub-neg 57.1%

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

   4. Simplified 57.1%

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

   5. Taylor expanded in z around 0 55.8%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2060000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+14}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 7: 60.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+59}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 3e+59) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3e+59) {
		tmp = log(t) - z;
	} else {
		tmp = -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 <= 3d+59) then
        tmp = log(t) - z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3e+59) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3e+59:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3e+59)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3e+59)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3e+59], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 3e59

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 60.4%

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

      1. neg-mul-1 60.4%

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

   4. Simplified 60.4%

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

   5. Taylor expanded in z around 0 60.4%

      \[\leadsto \color{blue}{-1 \cdot z + \log t} \]
   6. Step-by-step derivation

      1. mul-1-neg 60.4%

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

      2. +-commutative 60.4%

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

      3. sub-neg 60.4%

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

   7. Simplified 60.4%

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

   if 3e59 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. +-commutative 76.9%

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

      3. distribute-neg-in 76.9%

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

      4. sub-neg 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in y around inf 59.7%

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

      1. neg-mul-1 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+59}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 8: 70.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 68.3%

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

   1. neg-mul-1 68.3%

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

   2. +-commutative 68.3%

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

   3. distribute-neg-in 68.3%

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

   4. sub-neg 68.3%

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

4. Simplified 68.3%

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

5. Taylor expanded in z around 0 68.3%

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

   1. +-commutative 68.3%

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

   2. mul-1-neg 68.3%

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

   3. sub-neg 68.3%

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

7. Simplified 68.3%

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

8. Final simplification 68.3%

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

Alternative 9: 48.7% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+59}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 6.2e+59) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.2e+59) {
		tmp = -z;
	} else {
		tmp = -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 <= 6.2d+59) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.2e+59) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 6.2e+59:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.2e+59)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.2e+59)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6.2e+59], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 6.20000000000000029e59

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 62.5%

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

      1. neg-mul-1 62.5%

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

      2. +-commutative 62.5%

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

      3. distribute-neg-in 62.5%

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

      4. sub-neg 62.5%

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

   4. Simplified 62.5%

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

   5. Taylor expanded in z around inf 43.4%

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

      1. mul-1-neg 43.4%

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

   7. Simplified 43.4%

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

   if 6.20000000000000029e59 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. +-commutative 76.9%

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

      3. distribute-neg-in 76.9%

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

      4. sub-neg 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in y around inf 59.7%

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

      1. neg-mul-1 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+59}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 10: 30.0% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 68.3%

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

   1. neg-mul-1 68.3%

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

   2. +-commutative 68.3%

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

   3. distribute-neg-in 68.3%

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

   4. sub-neg 68.3%

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

4. Simplified 68.3%

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

5. Taylor expanded in y around inf 26.9%

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

   1. neg-mul-1 26.9%

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

7. Simplified 26.9%

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

8. Final simplification 26.9%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023194 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))