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

Percentage Accurate: 99.9% → 99.9%

Time: 10.3s

Alternatives: 9

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

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

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. Final simplification 99.9%

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

Alternative 2: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1300000 \lor \neg \left(x \leq 1.95 \cdot 10^{+63}\right):\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1300000.0) (not (<= x 1.95e+63)))
   (- (+ (log t) (* x (log y))) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1300000.0) || !(x <= 1.95e+63)) {
		tmp = (log(t) + (x * log(y))) - y;
	} else {
		tmp = log(t) - (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 ((x <= (-1300000.0d0)) .or. (.not. (x <= 1.95d+63))) then
        tmp = (log(t) + (x * log(y))) - y
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1300000.0) || !(x <= 1.95e+63)) {
		tmp = (Math.log(t) + (x * Math.log(y))) - y;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1300000.0) or not (x <= 1.95e+63):
		tmp = (math.log(t) + (x * math.log(y))) - y
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1300000.0) || !(x <= 1.95e+63))
		tmp = Float64(Float64(log(t) + Float64(x * log(y))) - y);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1300000.0) || ~((x <= 1.95e+63)))
		tmp = (log(t) + (x * log(y))) - y;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1300000.0], N[Not[LessEqual[x, 1.95e+63]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3e6 or 1.95e63 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 83.7%

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

   if -1.3e6 < x < 1.95e63

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1300000 \lor \neg \left(x \leq 1.95 \cdot 10^{+63}\right):\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 3: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t + x \cdot \log y\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{+132} \lor \neg \left(z \leq 1.45 \cdot 10^{+83}\right):\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (log t) (* x (log y)))))
   (if (or (<= z -7.5e+132) (not (<= z 1.45e+83))) (- t_1 z) (- t_1 y))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) + (x * log(y));
	double tmp;
	if ((z <= -7.5e+132) || !(z <= 1.45e+83)) {
		tmp = t_1 - z;
	} else {
		tmp = 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 = log(t) + (x * log(y))
    if ((z <= (-7.5d+132)) .or. (.not. (z <= 1.45d+83))) then
        tmp = t_1 - z
    else
        tmp = 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 = Math.log(t) + (x * Math.log(y));
	double tmp;
	if ((z <= -7.5e+132) || !(z <= 1.45e+83)) {
		tmp = t_1 - z;
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) + (x * math.log(y))
	tmp = 0
	if (z <= -7.5e+132) or not (z <= 1.45e+83):
		tmp = t_1 - z
	else:
		tmp = t_1 - y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) + Float64(x * log(y)))
	tmp = 0.0
	if ((z <= -7.5e+132) || !(z <= 1.45e+83))
		tmp = Float64(t_1 - z);
	else
		tmp = Float64(t_1 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) + (x * log(y));
	tmp = 0.0;
	if ((z <= -7.5e+132) || ~((z <= 1.45e+83)))
		tmp = t_1 - z;
	else
		tmp = t_1 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -7.5e+132], N[Not[LessEqual[z, 1.45e+83]], $MachinePrecision]], N[(t$95$1 - z), $MachinePrecision], N[(t$95$1 - y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000017e132 or 1.45e83 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in y around 0 91.5%

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

   if -7.50000000000000017e132 < z < 1.45e83

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 94.7%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+132} \lor \neg \left(z \leq 1.45 \cdot 10^{+83}\right):\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 4: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+101} \lor \neg \left(x \leq 8 \cdot 10^{+127}\right):\\ \;\;\;\;\log t + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7e+101) (not (<= x 8e+127)))
   (+ (log t) (* x (log y)))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e+101) || !(x <= 8e+127)) {
		tmp = log(t) + (x * log(y));
	} else {
		tmp = log(t) - (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 ((x <= (-7d+101)) .or. (.not. (x <= 8d+127))) then
        tmp = log(t) + (x * log(y))
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e+101) || !(x <= 8e+127)) {
		tmp = Math.log(t) + (x * Math.log(y));
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7e+101) or not (x <= 8e+127):
		tmp = math.log(t) + (x * math.log(y))
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7e+101) || !(x <= 8e+127))
		tmp = Float64(log(t) + Float64(x * log(y)));
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -7e+101) || ~((x <= 8e+127)))
		tmp = log(t) + (x * log(y));
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7e+101], N[Not[LessEqual[x, 8e+127]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.00000000000000046e101 or 7.99999999999999964e127 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in z around 0 87.6%

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

   3. Taylor expanded in y around 0 78.3%

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

   if -7.00000000000000046e101 < x < 7.99999999999999964e127

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 93.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+101} \lor \neg \left(x \leq 8 \cdot 10^{+127}\right):\\ \;\;\;\;\log t + x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 5: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{+133}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+83}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.35e+133) (- z) (if (<= z 4.6e+83) (- (log t) y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.35e+133) {
		tmp = -z;
	} else if (z <= 4.6e+83) {
		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 <= (-3.35d+133)) then
        tmp = -z
    else if (z <= 4.6d+83) 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 <= -3.35e+133) {
		tmp = -z;
	} else if (z <= 4.6e+83) {
		tmp = Math.log(t) - y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.35e+133:
		tmp = -z
	elif z <= 4.6e+83:
		tmp = math.log(t) - y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.35e+133)
		tmp = Float64(-z);
	elseif (z <= 4.6e+83)
		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 <= -3.35e+133)
		tmp = -z;
	elseif (z <= 4.6e+83)
		tmp = log(t) - y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.35e+133], (-z), If[LessEqual[z, 4.6e+83], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -3.35000000000000015e133 or 4.5999999999999999e83 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 73.5%

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

      1. mul-1-neg 73.5%

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

   4. Simplified 73.5%

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

   if -3.35000000000000015e133 < z < 4.5999999999999999e83

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. add-log-exp 20.5%

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

      3. sum-log 20.5%

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

      4. associate--l- 20.5%

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

      5. exp-diff 19.7%

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

      6. *-commutative 19.7%

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

      7. exp-to-pow 19.7%

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

   3. Applied egg-rr 19.7%

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

   4. Taylor expanded in x around 0 19.5%

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

   5. Taylor expanded in z around 0 19.3%

      \[\leadsto \color{blue}{\log \left(\frac{t}{e^{y}}\right)} \]
   6. Step-by-step derivation

      1. log-div 19.3%

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

      2. rem-log-exp 60.3%

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

   7. Simplified 60.3%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.35 \cdot 10^{+133}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+83}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6: 59.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+86}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.2e+86) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.2e+86) {
		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 <= 1.2d+86) 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 <= 1.2e+86) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.2e+86:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.2e+86)
		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 <= 1.2e+86)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.2e+86], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e86

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 93.1%

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

      1. +-commutative 93.1%

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

      2. *-commutative 93.1%

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

      3. log-pow 45.0%

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

      4. log-prod 45.0%

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

   4. Simplified 45.0%

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

   5. Taylor expanded in x around 0 59.2%

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

   if 1.2e86 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.9%

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

      1. neg-mul-1 69.9%

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

   4. Simplified 69.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+86}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 7: 69.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 71.2%

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

3. Final simplification 71.2%

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

Alternative 8: 47.7% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 9.2e+85) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 9.2e+85) {
		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 <= 9.2d+85) 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 <= 9.2e+85) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 9.2e+85:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 9.2e+85)
		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 <= 9.2e+85)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 9.2e+85], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 9.1999999999999996e85

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 43.4%

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

      1. mul-1-neg 43.4%

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

   4. Simplified 43.4%

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

   if 9.1999999999999996e85 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.9%

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

      1. neg-mul-1 69.9%

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

   4. Simplified 69.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 9: 29.7% 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 99.9%

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

2. Taylor expanded in y around inf 32.6%

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

   1. neg-mul-1 32.6%

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

4. Simplified 32.6%

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

5. Final simplification 32.6%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023174 
(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)))