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

Percentage Accurate: 99.9% → 99.9%

Time: 9.9s

Alternatives: 12

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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $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. +-commutative 99.8%

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

   2. associate--l- 99.8%

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

   3. associate-+r- 99.8%

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

   4. +-commutative 99.8%

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

   5. associate--l+ 99.8%

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

   6. fma-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+32} \lor \neg \left(z \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.45e+32) (not (<= z 1.25e+76)))
   (- (log t) (+ y z))
   (- (+ (log t) (* x (log y))) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.45e+32) || !(z <= 1.25e+76)) {
		tmp = log(t) - (y + z);
	} else {
		tmp = (log(t) + (x * log(y))) - 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 <= (-2.45d+32)) .or. (.not. (z <= 1.25d+76))) then
        tmp = log(t) - (y + z)
    else
        tmp = (log(t) + (x * log(y))) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.45e+32) || !(z <= 1.25e+76)) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = (Math.log(t) + (x * Math.log(y))) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.45e+32) or not (z <= 1.25e+76):
		tmp = math.log(t) - (y + z)
	else:
		tmp = (math.log(t) + (x * math.log(y))) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.45e+32) || !(z <= 1.25e+76))
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(Float64(log(t) + Float64(x * log(y))) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.45e+32) || ~((z <= 1.25e+76)))
		tmp = log(t) - (y + z);
	else
		tmp = (log(t) + (x * log(y))) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.45e+32], N[Not[LessEqual[z, 1.25e+76]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.4500000000000001e32 or 1.24999999999999998e76 < 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 83.5%

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

   if -2.4500000000000001e32 < z < 1.24999999999999998e76

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 98.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+32} \lor \neg \left(z \leq 1.25 \cdot 10^{+76}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 3: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t + x \cdot \log y\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+66}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+40}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (log t) (* x (log y)))))
   (if (<= x -7.8e+66)
     (- t_1 z)
     (if (<= x 2e+40) (- (log t) (+ y 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 (x <= -7.8e+66) {
		tmp = t_1 - z;
	} else if (x <= 2e+40) {
		tmp = log(t) - (y + 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 (x <= (-7.8d+66)) then
        tmp = t_1 - z
    else if (x <= 2d+40) then
        tmp = log(t) - (y + 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 (x <= -7.8e+66) {
		tmp = t_1 - z;
	} else if (x <= 2e+40) {
		tmp = Math.log(t) - (y + 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 x <= -7.8e+66:
		tmp = t_1 - z
	elif x <= 2e+40:
		tmp = math.log(t) - (y + 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 (x <= -7.8e+66)
		tmp = Float64(t_1 - z);
	elseif (x <= 2e+40)
		tmp = Float64(log(t) - Float64(y + 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 (x <= -7.8e+66)
		tmp = t_1 - z;
	elseif (x <= 2e+40)
		tmp = log(t) - (y + 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[LessEqual[x, -7.8e+66], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[x, 2e+40], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.8000000000000007e66

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 89.8%

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

   if -7.8000000000000007e66 < x < 2.00000000000000006e40

   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%

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

   if 2.00000000000000006e40 < 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 88.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+66}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+40}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 4: 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 5: 48.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-306}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-261}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+71}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -8.5e+115)
     t_1
     (if (<= x -1.85e+83)
       (- z)
       (if (<= x -2.4e+62)
         t_1
         (if (<= x -8.5e-306)
           (- y)
           (if (<= x 1.4e-261) (- z) (if (<= x 7.4e+71) (- y) t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -8.5e+115) {
		tmp = t_1;
	} else if (x <= -1.85e+83) {
		tmp = -z;
	} else if (x <= -2.4e+62) {
		tmp = t_1;
	} else if (x <= -8.5e-306) {
		tmp = -y;
	} else if (x <= 1.4e-261) {
		tmp = -z;
	} else if (x <= 7.4e+71) {
		tmp = -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-8.5d+115)) then
        tmp = t_1
    else if (x <= (-1.85d+83)) then
        tmp = -z
    else if (x <= (-2.4d+62)) then
        tmp = t_1
    else if (x <= (-8.5d-306)) then
        tmp = -y
    else if (x <= 1.4d-261) then
        tmp = -z
    else if (x <= 7.4d+71) then
        tmp = -y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -8.5e+115) {
		tmp = t_1;
	} else if (x <= -1.85e+83) {
		tmp = -z;
	} else if (x <= -2.4e+62) {
		tmp = t_1;
	} else if (x <= -8.5e-306) {
		tmp = -y;
	} else if (x <= 1.4e-261) {
		tmp = -z;
	} else if (x <= 7.4e+71) {
		tmp = -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -8.5e+115:
		tmp = t_1
	elif x <= -1.85e+83:
		tmp = -z
	elif x <= -2.4e+62:
		tmp = t_1
	elif x <= -8.5e-306:
		tmp = -y
	elif x <= 1.4e-261:
		tmp = -z
	elif x <= 7.4e+71:
		tmp = -y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -8.5e+115)
		tmp = t_1;
	elseif (x <= -1.85e+83)
		tmp = Float64(-z);
	elseif (x <= -2.4e+62)
		tmp = t_1;
	elseif (x <= -8.5e-306)
		tmp = Float64(-y);
	elseif (x <= 1.4e-261)
		tmp = Float64(-z);
	elseif (x <= 7.4e+71)
		tmp = Float64(-y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -8.5e+115)
		tmp = t_1;
	elseif (x <= -1.85e+83)
		tmp = -z;
	elseif (x <= -2.4e+62)
		tmp = t_1;
	elseif (x <= -8.5e-306)
		tmp = -y;
	elseif (x <= 1.4e-261)
		tmp = -z;
	elseif (x <= 7.4e+71)
		tmp = -y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e+115], t$95$1, If[LessEqual[x, -1.85e+83], (-z), If[LessEqual[x, -2.4e+62], t$95$1, If[LessEqual[x, -8.5e-306], (-y), If[LessEqual[x, 1.4e-261], (-z), If[LessEqual[x, 7.4e+71], (-y), t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000057e115 or -1.8500000000000001e83 < x < -2.4e62 or 7.4e71 < x

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. add-cube-cbrt 98.5%

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

      5. associate-*l* 98.6%

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

      6. fma-neg 98.6%

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

      7. pow2 98.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log y}\right)}^{2}}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right) \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right)} \]

   4. Taylor expanded in x around inf 76.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 76.5%

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

      2. *-lft-identity 76.5%

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

   6. Simplified 76.5%

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

   if -8.50000000000000057e115 < x < -1.8500000000000001e83 or -8.5000000000000002e-306 < x < 1.40000000000000005e-261

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 73.2%

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

      1. neg-mul-1 73.2%

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

   4. Simplified 73.2%

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

   if -2.4e62 < x < -8.5000000000000002e-306 or 1.40000000000000005e-261 < x < 7.4e71

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 45.3%

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

      1. neg-mul-1 45.3%

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

   4. Simplified 45.3%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{-306}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-261}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+71}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 6: 59.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ t_2 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.3 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-262}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)) (t_2 (* x (log y))))
   (if (<= x -1.3e+116)
     t_2
     (if (<= x -1.4e+83)
       (- z)
       (if (<= x -1.6e+62)
         t_2
         (if (<= x -5.3e-306)
           t_1
           (if (<= x 1.5e-262) (- z) (if (<= x 9.5e+70) t_1 t_2))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double t_2 = x * log(y);
	double tmp;
	if (x <= -1.3e+116) {
		tmp = t_2;
	} else if (x <= -1.4e+83) {
		tmp = -z;
	} else if (x <= -1.6e+62) {
		tmp = t_2;
	} else if (x <= -5.3e-306) {
		tmp = t_1;
	} else if (x <= 1.5e-262) {
		tmp = -z;
	} else if (x <= 9.5e+70) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(t) - y
    t_2 = x * log(y)
    if (x <= (-1.3d+116)) then
        tmp = t_2
    else if (x <= (-1.4d+83)) then
        tmp = -z
    else if (x <= (-1.6d+62)) then
        tmp = t_2
    else if (x <= (-5.3d-306)) then
        tmp = t_1
    else if (x <= 1.5d-262) then
        tmp = -z
    else if (x <= 9.5d+70) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) - y;
	double t_2 = x * Math.log(y);
	double tmp;
	if (x <= -1.3e+116) {
		tmp = t_2;
	} else if (x <= -1.4e+83) {
		tmp = -z;
	} else if (x <= -1.6e+62) {
		tmp = t_2;
	} else if (x <= -5.3e-306) {
		tmp = t_1;
	} else if (x <= 1.5e-262) {
		tmp = -z;
	} else if (x <= 9.5e+70) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - y
	t_2 = x * math.log(y)
	tmp = 0
	if x <= -1.3e+116:
		tmp = t_2
	elif x <= -1.4e+83:
		tmp = -z
	elif x <= -1.6e+62:
		tmp = t_2
	elif x <= -5.3e-306:
		tmp = t_1
	elif x <= 1.5e-262:
		tmp = -z
	elif x <= 9.5e+70:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	t_2 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -1.3e+116)
		tmp = t_2;
	elseif (x <= -1.4e+83)
		tmp = Float64(-z);
	elseif (x <= -1.6e+62)
		tmp = t_2;
	elseif (x <= -5.3e-306)
		tmp = t_1;
	elseif (x <= 1.5e-262)
		tmp = Float64(-z);
	elseif (x <= 9.5e+70)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - y;
	t_2 = x * log(y);
	tmp = 0.0;
	if (x <= -1.3e+116)
		tmp = t_2;
	elseif (x <= -1.4e+83)
		tmp = -z;
	elseif (x <= -1.6e+62)
		tmp = t_2;
	elseif (x <= -5.3e-306)
		tmp = t_1;
	elseif (x <= 1.5e-262)
		tmp = -z;
	elseif (x <= 9.5e+70)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.3e+116], t$95$2, If[LessEqual[x, -1.4e+83], (-z), If[LessEqual[x, -1.6e+62], t$95$2, If[LessEqual[x, -5.3e-306], t$95$1, If[LessEqual[x, 1.5e-262], (-z), If[LessEqual[x, 9.5e+70], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999993e116 or -1.4e83 < x < -1.59999999999999992e62 or 9.5000000000000002e70 < x

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. add-cube-cbrt 98.5%

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

      5. associate-*l* 98.6%

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

      6. fma-neg 98.6%

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

      7. pow2 98.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log y}\right)}^{2}}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right) \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right)} \]

   4. Taylor expanded in x around inf 76.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 76.5%

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

      2. *-lft-identity 76.5%

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

   6. Simplified 76.5%

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

   if -1.29999999999999993e116 < x < -1.4e83 or -5.2999999999999998e-306 < x < 1.50000000000000009e-262

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 73.2%

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

      1. neg-mul-1 73.2%

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

   4. Simplified 73.2%

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

   if -1.59999999999999992e62 < x < -5.2999999999999998e-306 or 1.50000000000000009e-262 < x < 9.5000000000000002e70

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 74.7%

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

   3. Taylor expanded in x around 0 71.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+116}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-306}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-262}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+70}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 7: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ t_2 := x \cdot \log y\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+115}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-306}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-258}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)) (t_2 (* x (log y))))
   (if (<= x -8.2e+115)
     t_2
     (if (<= x -1.85e+83)
       (- z)
       (if (<= x -3.7e+62)
         t_2
         (if (<= x -6.5e-306)
           t_1
           (if (<= x 7e-258) (- (log t) z) (if (<= x 1.1e+71) t_1 t_2))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - y;
	double t_2 = x * log(y);
	double tmp;
	if (x <= -8.2e+115) {
		tmp = t_2;
	} else if (x <= -1.85e+83) {
		tmp = -z;
	} else if (x <= -3.7e+62) {
		tmp = t_2;
	} else if (x <= -6.5e-306) {
		tmp = t_1;
	} else if (x <= 7e-258) {
		tmp = log(t) - z;
	} else if (x <= 1.1e+71) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(t) - y
    t_2 = x * log(y)
    if (x <= (-8.2d+115)) then
        tmp = t_2
    else if (x <= (-1.85d+83)) then
        tmp = -z
    else if (x <= (-3.7d+62)) then
        tmp = t_2
    else if (x <= (-6.5d-306)) then
        tmp = t_1
    else if (x <= 7d-258) then
        tmp = log(t) - z
    else if (x <= 1.1d+71) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) - y;
	double t_2 = x * Math.log(y);
	double tmp;
	if (x <= -8.2e+115) {
		tmp = t_2;
	} else if (x <= -1.85e+83) {
		tmp = -z;
	} else if (x <= -3.7e+62) {
		tmp = t_2;
	} else if (x <= -6.5e-306) {
		tmp = t_1;
	} else if (x <= 7e-258) {
		tmp = Math.log(t) - z;
	} else if (x <= 1.1e+71) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - y
	t_2 = x * math.log(y)
	tmp = 0
	if x <= -8.2e+115:
		tmp = t_2
	elif x <= -1.85e+83:
		tmp = -z
	elif x <= -3.7e+62:
		tmp = t_2
	elif x <= -6.5e-306:
		tmp = t_1
	elif x <= 7e-258:
		tmp = math.log(t) - z
	elif x <= 1.1e+71:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - y)
	t_2 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -8.2e+115)
		tmp = t_2;
	elseif (x <= -1.85e+83)
		tmp = Float64(-z);
	elseif (x <= -3.7e+62)
		tmp = t_2;
	elseif (x <= -6.5e-306)
		tmp = t_1;
	elseif (x <= 7e-258)
		tmp = Float64(log(t) - z);
	elseif (x <= 1.1e+71)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - y;
	t_2 = x * log(y);
	tmp = 0.0;
	if (x <= -8.2e+115)
		tmp = t_2;
	elseif (x <= -1.85e+83)
		tmp = -z;
	elseif (x <= -3.7e+62)
		tmp = t_2;
	elseif (x <= -6.5e-306)
		tmp = t_1;
	elseif (x <= 7e-258)
		tmp = log(t) - z;
	elseif (x <= 1.1e+71)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.2e+115], t$95$2, If[LessEqual[x, -1.85e+83], (-z), If[LessEqual[x, -3.7e+62], t$95$2, If[LessEqual[x, -6.5e-306], t$95$1, If[LessEqual[x, 7e-258], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.1e+71], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.19999999999999925e115 or -1.8500000000000001e83 < x < -3.70000000000000014e62 or 1.09999999999999997e71 < x

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. add-cube-cbrt 98.5%

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

      5. associate-*l* 98.6%

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

      6. fma-neg 98.6%

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

      7. pow2 98.6%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log y}\right)}^{2}}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right) \]

   3. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right)} \]

   4. Taylor expanded in x around inf 76.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 76.5%

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

      2. *-lft-identity 76.5%

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

   6. Simplified 76.5%

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

   if -8.19999999999999925e115 < x < -1.8500000000000001e83

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 67.8%

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

      1. neg-mul-1 67.8%

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

   4. Simplified 67.8%

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

   if -3.70000000000000014e62 < x < -6.5000000000000004e-306 or 7.00000000000000003e-258 < x < 1.09999999999999997e71

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 74.7%

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

   3. Taylor expanded in x around 0 71.8%

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

   if -6.5000000000000004e-306 < x < 7.00000000000000003e-258

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate--l- 100.0%

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

      3. associate-+r- 100.0%

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

      4. +-commutative 100.0%

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

      5. associate--l+ 100.0%

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

      6. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 83.4%

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

   5. Taylor expanded in x around 0 83.4%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+115}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+83}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-306}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-258}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+71}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 8: 84.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+125} \lor \neg \left(x \leq 4.2 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e+125) (not (<= x 4.2e+135)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e+125) || !(x <= 4.2e+135)) {
		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 <= (-2.5d+125)) .or. (.not. (x <= 4.2d+135))) 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 <= -2.5e+125) || !(x <= 4.2e+135)) {
		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 <= -2.5e+125) or not (x <= 4.2e+135):
		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 <= -2.5e+125) || !(x <= 4.2e+135))
		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 <= -2.5e+125) || ~((x <= 4.2e+135)))
		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, -2.5e+125], N[Not[LessEqual[x, 4.2e+135]], $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 < -2.49999999999999981e125 or 4.20000000000000019e135 < x

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. associate-+l- 99.6%

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

      3. *-commutative 99.6%

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

      4. add-cube-cbrt 98.4%

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

      5. associate-*l* 98.4%

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

      6. fma-neg 98.4%

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

      7. pow2 98.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log y}\right)}^{2}}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right) \]

   3. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, \sqrt[3]{\log y} \cdot x, -\left(\left(y + z\right) - \log t\right)\right)} \]

   4. Taylor expanded in x around inf 82.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   5. Step-by-step derivation

      1. pow-base-1 82.5%

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

      2. *-lft-identity 82.5%

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

   6. Simplified 82.5%

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

   if -2.49999999999999981e125 < x < 4.20000000000000019e135

   1. Initial program 100.0%

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

   2. 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 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+125} \lor \neg \left(x \leq 4.2 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 9: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+127} \lor \neg \left(x \leq 4.3 \cdot 10^{+36}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -9.2e+127) (not (<= x 4.3e+36)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -9.2e+127) || !(x <= 4.3e+36)) {
		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 <= (-9.2d+127)) .or. (.not. (x <= 4.3d+36))) 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 <= -9.2e+127) || !(x <= 4.3e+36)) {
		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 <= -9.2e+127) or not (x <= 4.3e+36):
		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 <= -9.2e+127) || !(x <= 4.3e+36))
		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 <= -9.2e+127) || ~((x <= 4.3e+36)))
		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, -9.2e+127], N[Not[LessEqual[x, 4.3e+36]], $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 < -9.2000000000000007e127 or 4.30000000000000005e36 < 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 87.1%

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

   3. Taylor expanded in x around inf 87.1%

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

   if -9.2000000000000007e127 < x < 4.30000000000000005e36

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 95.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+127} \lor \neg \left(x \leq 4.3 \cdot 10^{+36}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 10: 46.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-24}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-254}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+74}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.7e+34)
   (- z)
   (if (<= z -9.6e-24)
     (- y)
     (if (<= z -1.52e-254) (log t) (if (<= z 7.8e+74) (- y) (- z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.7e+34) {
		tmp = -z;
	} else if (z <= -9.6e-24) {
		tmp = -y;
	} else if (z <= -1.52e-254) {
		tmp = log(t);
	} else if (z <= 7.8e+74) {
		tmp = -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 <= (-2.7d+34)) then
        tmp = -z
    else if (z <= (-9.6d-24)) then
        tmp = -y
    else if (z <= (-1.52d-254)) then
        tmp = log(t)
    else if (z <= 7.8d+74) then
        tmp = -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 <= -2.7e+34) {
		tmp = -z;
	} else if (z <= -9.6e-24) {
		tmp = -y;
	} else if (z <= -1.52e-254) {
		tmp = Math.log(t);
	} else if (z <= 7.8e+74) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.7e+34:
		tmp = -z
	elif z <= -9.6e-24:
		tmp = -y
	elif z <= -1.52e-254:
		tmp = math.log(t)
	elif z <= 7.8e+74:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.7e+34)
		tmp = Float64(-z);
	elseif (z <= -9.6e-24)
		tmp = Float64(-y);
	elseif (z <= -1.52e-254)
		tmp = log(t);
	elseif (z <= 7.8e+74)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.7e+34)
		tmp = -z;
	elseif (z <= -9.6e-24)
		tmp = -y;
	elseif (z <= -1.52e-254)
		tmp = log(t);
	elseif (z <= 7.8e+74)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.7e+34], (-z), If[LessEqual[z, -9.6e-24], (-y), If[LessEqual[z, -1.52e-254], N[Log[t], $MachinePrecision], If[LessEqual[z, 7.8e+74], (-y), (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.7e34 or 7.80000000000000015e74 < 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 64.7%

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

      1. neg-mul-1 64.7%

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

   4. Simplified 64.7%

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

   if -2.7e34 < z < -9.5999999999999993e-24 or -1.52e-254 < z < 7.80000000000000015e74

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 39.3%

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

      1. neg-mul-1 39.3%

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

   4. Simplified 39.3%

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

   if -9.5999999999999993e-24 < z < -1.52e-254

   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. associate--l- 99.8%

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

      3. associate-+r- 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--l+ 99.8%

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

      6. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 76.2%

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

   5. Taylor expanded in z around 0 76.2%

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

   6. Taylor expanded in x around 0 38.4%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -9.6 \cdot 10^{-24}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq -1.52 \cdot 10^{-254}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;z \leq 7.8 \cdot 10^{+74}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 11: 49.0% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+74}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.8e+34) (- z) (if (<= z 5e+74) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.8e+34) {
		tmp = -z;
	} else if (z <= 5e+74) {
		tmp = -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 <= (-9.8d+34)) then
        tmp = -z
    else if (z <= 5d+74) then
        tmp = -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 <= -9.8e+34) {
		tmp = -z;
	} else if (z <= 5e+74) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9.8e+34:
		tmp = -z
	elif z <= 5e+74:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.8e+34)
		tmp = Float64(-z);
	elseif (z <= 5e+74)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9.8e+34)
		tmp = -z;
	elseif (z <= 5e+74)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.8e+34], (-z), If[LessEqual[z, 5e+74], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -9.8000000000000005e34 or 4.99999999999999963e74 < 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 64.7%

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

      1. neg-mul-1 64.7%

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

   4. Simplified 64.7%

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

   if -9.8000000000000005e34 < z < 4.99999999999999963e74

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 35.8%

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

      1. neg-mul-1 35.8%

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

   4. Simplified 35.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+34}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+74}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 12: 30.3% 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 y around inf 30.1%

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

   1. neg-mul-1 30.1%

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

4. Simplified 30.1%

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

5. Final simplification 30.1%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023238 
(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)))