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

Percentage Accurate: 99.9% → 99.9%

Time: 11.3s

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 99.8%

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

Alternative 2: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -40000000000 \lor \neg \left(t\_2 \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;t\_1 - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (or (<= t_2 -40000000000.0) (not (<= t_2 2e-7)))
     (- t_1 (+ y z))
     (- (log t) (+ y 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 <= -40000000000.0) || !(t_2 <= 2e-7)) {
		tmp = t_1 - (y + z);
	} 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if ((t_2 <= (-40000000000.0d0)) .or. (.not. (t_2 <= 2d-7))) then
        tmp = t_1 - (y + z)
    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 t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if ((t_2 <= -40000000000.0) || !(t_2 <= 2e-7)) {
		tmp = t_1 - (y + z);
	} else {
		tmp = Math.log(t) - (y + 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 <= -40000000000.0) or not (t_2 <= 2e-7):
		tmp = t_1 - (y + z)
	else:
		tmp = math.log(t) - (y + 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 <= -40000000000.0) || !(t_2 <= 2e-7))
		tmp = Float64(t_1 - Float64(y + z));
	else
		tmp = Float64(log(t) - Float64(y + 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 <= -40000000000.0) || ~((t_2 <= 2e-7)))
		tmp = t_1 - (y + z);
	else
		tmp = log(t) - (y + 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[Or[LessEqual[t$95$2, -40000000000.0], N[Not[LessEqual[t$95$2, 2e-7]], $MachinePrecision]], N[(t$95$1 - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -4e10 or 1.9999999999999999e-7 < (-.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. Step-by-step derivation

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.0%

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

   if -4e10 < (-.f64 (*.f64 x (log.f64 y)) y) < 1.9999999999999999e-7

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -40000000000 \lor \neg \left(x \cdot \log y - y \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 0.1:\\ \;\;\;\;\left(t\_1 + \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1 - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= y 0.1) (- (+ t_1 (log t)) z) (- t_1 (+ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (y <= 0.1) {
		tmp = (t_1 + log(t)) - z;
	} else {
		tmp = t_1 - (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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (y <= 0.1d0) then
        tmp = (t_1 + log(t)) - z
    else
        tmp = t_1 - (y + 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 tmp;
	if (y <= 0.1) {
		tmp = (t_1 + Math.log(t)) - z;
	} else {
		tmp = t_1 - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if y <= 0.1:
		tmp = (t_1 + math.log(t)) - z
	else:
		tmp = t_1 - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 0.1)
		tmp = Float64(Float64(t_1 + log(t)) - z);
	else
		tmp = Float64(t_1 - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (y <= 0.1)
		tmp = (t_1 + log(t)) - z;
	else
		tmp = t_1 - (y + z);
	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, 0.1], N[(N[(t$95$1 + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision], N[(t$95$1 - N[(y + z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.10000000000000001

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 99.3%

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

   if 0.10000000000000001 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.1:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+132} \lor \neg \left(x \leq -2.05 \cdot 10^{+86} \lor \neg \left(x \leq -1.9 \cdot 10^{+17}\right) \land x \leq 102000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.5e+132)
         (not
          (or (<= x -2.05e+86)
              (and (not (<= x -1.9e+17)) (<= x 102000000.0)))))
   (* x (log y))
   (- (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.5e+132) || !((x <= -2.05e+86) || (!(x <= -1.9e+17) && (x <= 102000000.0)))) {
		tmp = x * log(y);
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-6.5d+132)) .or. (.not. (x <= (-2.05d+86)) .or. (.not. (x <= (-1.9d+17))) .and. (x <= 102000000.0d0))) then
        tmp = x * log(y)
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.5e+132) || !((x <= -2.05e+86) || (!(x <= -1.9e+17) && (x <= 102000000.0)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.5e+132) or not ((x <= -2.05e+86) or (not (x <= -1.9e+17) and (x <= 102000000.0))):
		tmp = x * math.log(y)
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.5e+132) || !((x <= -2.05e+86) || (!(x <= -1.9e+17) && (x <= 102000000.0))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -6.5e+132) || ~(((x <= -2.05e+86) || (~((x <= -1.9e+17)) && (x <= 102000000.0)))))
		tmp = x * log(y);
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.5e+132], N[Not[Or[LessEqual[x, -2.05e+86], And[N[Not[LessEqual[x, -1.9e+17]], $MachinePrecision], LessEqual[x, 102000000.0]]]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4999999999999994e132 or -2.05e86 < x < -1.9e17 or 1.02e8 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around -inf 76.1%

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

      1. associate-*r* 76.1%

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

      2. mul-1-neg 76.1%

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

      3. mul-1-neg 76.1%

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

      4. associate--l+ 76.1%

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

   7. Simplified 76.1%

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

   8. Taylor expanded in x around inf 73.5%

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

   if -6.4999999999999994e132 < x < -2.05e86 or -1.9e17 < x < 1.02e8

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around -inf 85.9%

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

      1. associate-*r* 85.9%

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

      2. mul-1-neg 85.9%

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

      3. mul-1-neg 85.9%

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

      4. associate--l+ 85.9%

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

   7. Simplified 85.9%

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

   8. Taylor expanded in y around inf 67.2%

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

      1. associate-*r/ 67.2%

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

      2. mul-1-neg 67.2%

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

   10. Simplified 67.2%

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

   11. Taylor expanded in z around 0 80.6%

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

       1. mul-1-neg 80.6%

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

       2. unsub-neg 80.6%

          \[\leadsto \color{blue}{-1 \cdot y - z} \]

       3. mul-1-neg 80.6%

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

   13. Simplified 80.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+132} \lor \neg \left(x \leq -2.05 \cdot 10^{+86} \lor \neg \left(x \leq -1.9 \cdot 10^{+17}\right) \land x \leq 102000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -130 \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -130.0) (not (<= x 40000000.0)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -130.0) || !(x <= 40000000.0)) {
		tmp = (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 <= (-130.0d0)) .or. (.not. (x <= 40000000.0d0))) then
        tmp = (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 <= -130.0) || !(x <= 40000000.0)) {
		tmp = (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 <= -130.0) or not (x <= 40000000.0):
		tmp = (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 <= -130.0) || !(x <= 40000000.0))
		tmp = Float64(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 <= -130.0) || ~((x <= 40000000.0)))
		tmp = (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, -130.0], N[Not[LessEqual[x, 40000000.0]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -130 or 4e7 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 82.4%

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

   if -130 < x < 4e7

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.7%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -130 \lor \neg \left(x \leq 40000000\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+132} \lor \neg \left(x \leq 102000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.2e+132) (not (<= x 102000000.0)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.2e+132) || !(x <= 102000000.0)) {
		tmp = 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 <= (-4.2d+132)) .or. (.not. (x <= 102000000.0d0))) then
        tmp = 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 <= -4.2e+132) || !(x <= 102000000.0)) {
		tmp = x * Math.log(y);
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.2e+132) or not (x <= 102000000.0):
		tmp = 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 <= -4.2e+132) || !(x <= 102000000.0))
		tmp = 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 <= -4.2e+132) || ~((x <= 102000000.0)))
		tmp = x * log(y);
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.2e+132], N[Not[LessEqual[x, 102000000.0]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.19999999999999987e132 or 1.02e8 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around -inf 78.3%

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

      1. associate-*r* 78.3%

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

      2. mul-1-neg 78.3%

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

      3. mul-1-neg 78.3%

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

      4. associate--l+ 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in x around inf 74.3%

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

   if -4.19999999999999987e132 < x < 1.02e8

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 91.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+132} \lor \neg \left(x \leq 102000000\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.0% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+60} \lor \neg \left(z \leq 1.9 \cdot 10^{+71}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.05e+60) (not (<= z 1.9e+71))) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.05e+60) || !(z <= 1.9e+71)) {
		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 ((z <= (-3.05d+60)) .or. (.not. (z <= 1.9d+71))) 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 ((z <= -3.05e+60) || !(z <= 1.9e+71)) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.05e+60) or not (z <= 1.9e+71):
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.05e+60) || !(z <= 1.9e+71))
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.05e+60) || ~((z <= 1.9e+71)))
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.05e+60], N[Not[LessEqual[z, 1.9e+71]], $MachinePrecision]], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -3.05e60 or 1.9e71 < z

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 72.5%

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

      1. mul-1-neg 72.5%

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

   7. Simplified 72.5%

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

   if -3.05e60 < z < 1.9e71

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 40.5%

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

      1. mul-1-neg 40.5%

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

   7. Simplified 40.5%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.05 \cdot 10^{+60} \lor \neg \left(z \leq 1.9 \cdot 10^{+71}\right):\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.4% accurate, 52.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. associate--l- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in z around -inf 82.0%

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

   1. associate-*r* 82.0%

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

   2. mul-1-neg 82.0%

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

   3. mul-1-neg 82.0%

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

   4. associate--l+ 82.0%

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

7. Simplified 82.0%

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

8. Taylor expanded in y around inf 49.4%

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

   1. associate-*r/ 49.4%

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

   2. mul-1-neg 49.4%

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

10. Simplified 49.4%

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

11. Taylor expanded in z around 0 58.5%

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

    1. mul-1-neg 58.5%

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

    2. unsub-neg 58.5%

       \[\leadsto \color{blue}{-1 \cdot y - z} \]

    3. mul-1-neg 58.5%

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

13. Simplified 58.5%

    \[\leadsto \color{blue}{\left(-y\right) - z} \]
14. Add Preprocessing

Alternative 9: 29.4% 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. Step-by-step derivation

   1. associate-+l- 99.8%

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

   2. associate--l- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around inf 30.4%

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

   1. mul-1-neg 30.4%

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

7. Simplified 30.4%

   \[\leadsto \color{blue}{-y} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(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)))