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

Percentage Accurate: 99.9% → 99.9%

Time: 9.2s

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 99.9%

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

   1. +-commutative 99.9%

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

   2. associate--l- 99.9%

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

   3. associate-+r- 99.9%

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

   4. +-commutative 99.9%

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

   5. associate--l+ 99.9%

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

   6. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+14} \lor \neg \left(x \leq 7.6 \cdot 10^{+56}\right):\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.6e+14) (not (<= x 7.6e+56)))
   (- (+ (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 <= -8.6e+14) || !(x <= 7.6e+56)) {
		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 <= (-8.6d+14)) .or. (.not. (x <= 7.6d+56))) 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 <= -8.6e+14) || !(x <= 7.6e+56)) {
		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 <= -8.6e+14) or not (x <= 7.6e+56):
		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 <= -8.6e+14) || !(x <= 7.6e+56))
		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 <= -8.6e+14) || ~((x <= 7.6e+56)))
		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, -8.6e+14], N[Not[LessEqual[x, 7.6e+56]], $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 < -8.6e14 or 7.59999999999999991e56 < 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 85.2%

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

   if -8.6e14 < x < 7.59999999999999991e56

   1. Initial program 100.0%

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

   2. 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 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{+14} \lor \neg \left(x \leq 7.6 \cdot 10^{+56}\right):\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 3: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t + x \cdot \log y\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{+14}:\\ \;\;\;\;t_1 - y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (log t) (* x (log y)))))
   (if (<= x -3.9e+14)
     (- t_1 y)
     (if (<= x 4.2e+61) (- (log t) (+ y z)) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) + (x * log(y));
	double tmp;
	if (x <= -3.9e+14) {
		tmp = t_1 - y;
	} else if (x <= 4.2e+61) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) + (x * log(y))
    if (x <= (-3.9d+14)) then
        tmp = t_1 - y
    else if (x <= 4.2d+61) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) + (x * Math.log(y));
	double tmp;
	if (x <= -3.9e+14) {
		tmp = t_1 - y;
	} else if (x <= 4.2e+61) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) + (x * math.log(y))
	tmp = 0
	if x <= -3.9e+14:
		tmp = t_1 - y
	elif x <= 4.2e+61:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) + Float64(x * log(y)))
	tmp = 0.0
	if (x <= -3.9e+14)
		tmp = Float64(t_1 - y);
	elseif (x <= 4.2e+61)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(t_1 - z);
	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 <= -3.9e+14)
		tmp = t_1 - y;
	elseif (x <= 4.2e+61)
		tmp = log(t) - (y + z);
	else
		tmp = t_1 - z;
	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, -3.9e+14], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 4.2e+61], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9e14

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 86.9%

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

   if -3.9e14 < x < 4.2000000000000002e61

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

   if 4.2000000000000002e61 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 85.5%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{+14}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = log(t) - (z + (y + (x * log((1.0d0 / y)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 6: 60.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - y\\ t_2 := x \cdot \log y\\ t_3 := \log t - z\\ \mathbf{if}\;x \leq -1.72 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-85}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-263}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) y)) (t_2 (* x (log y))) (t_3 (- (log t) z)))
   (if (<= x -1.72e+16)
     t_2
     (if (<= x -2.7e-85)
       t_3
       (if (<= x -5.7e-111)
         t_1
         (if (<= x -5.4e-263) t_3 (if (<= x 5.5e+66) 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 t_3 = log(t) - z;
	double tmp;
	if (x <= -1.72e+16) {
		tmp = t_2;
	} else if (x <= -2.7e-85) {
		tmp = t_3;
	} else if (x <= -5.7e-111) {
		tmp = t_1;
	} else if (x <= -5.4e-263) {
		tmp = t_3;
	} else if (x <= 5.5e+66) {
		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) :: t_3
    real(8) :: tmp
    t_1 = log(t) - y
    t_2 = x * log(y)
    t_3 = log(t) - z
    if (x <= (-1.72d+16)) then
        tmp = t_2
    else if (x <= (-2.7d-85)) then
        tmp = t_3
    else if (x <= (-5.7d-111)) then
        tmp = t_1
    else if (x <= (-5.4d-263)) then
        tmp = t_3
    else if (x <= 5.5d+66) 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 t_3 = Math.log(t) - z;
	double tmp;
	if (x <= -1.72e+16) {
		tmp = t_2;
	} else if (x <= -2.7e-85) {
		tmp = t_3;
	} else if (x <= -5.7e-111) {
		tmp = t_1;
	} else if (x <= -5.4e-263) {
		tmp = t_3;
	} else if (x <= 5.5e+66) {
		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)
	t_3 = math.log(t) - z
	tmp = 0
	if x <= -1.72e+16:
		tmp = t_2
	elif x <= -2.7e-85:
		tmp = t_3
	elif x <= -5.7e-111:
		tmp = t_1
	elif x <= -5.4e-263:
		tmp = t_3
	elif x <= 5.5e+66:
		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))
	t_3 = Float64(log(t) - z)
	tmp = 0.0
	if (x <= -1.72e+16)
		tmp = t_2;
	elseif (x <= -2.7e-85)
		tmp = t_3;
	elseif (x <= -5.7e-111)
		tmp = t_1;
	elseif (x <= -5.4e-263)
		tmp = t_3;
	elseif (x <= 5.5e+66)
		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);
	t_3 = log(t) - z;
	tmp = 0.0;
	if (x <= -1.72e+16)
		tmp = t_2;
	elseif (x <= -2.7e-85)
		tmp = t_3;
	elseif (x <= -5.7e-111)
		tmp = t_1;
	elseif (x <= -5.4e-263)
		tmp = t_3;
	elseif (x <= 5.5e+66)
		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]}, Block[{t$95$3 = N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -1.72e+16], t$95$2, If[LessEqual[x, -2.7e-85], t$95$3, If[LessEqual[x, -5.7e-111], t$95$1, If[LessEqual[x, -5.4e-263], t$95$3, If[LessEqual[x, 5.5e+66], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.72e16 or 5.5e66 < 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 85.1%

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

   3. Taylor expanded in y around 0 66.9%

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

   4. Taylor expanded in x around inf 66.9%

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

   if -1.72e16 < x < -2.7000000000000001e-85 or -5.7e-111 < x < -5.40000000000000007e-263

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 81.1%

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

      1. associate--l+ 81.1%

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

      2. fma-def 81.1%

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

   4. Simplified 81.1%

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

   5. Taylor expanded in x around 0 81.0%

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

   if -2.7000000000000001e-85 < x < -5.7e-111 or -5.40000000000000007e-263 < x < 5.5e66

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 66.3%

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

   3. Taylor expanded in x around 0 65.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-85}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq -5.7 \cdot 10^{-111}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq -5.4 \cdot 10^{-263}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+66}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 7: 47.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -52000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-177}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-226}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -52000000000000.0)
     t_1
     (if (<= x -4e-177)
       (- z)
       (if (<= x 1.06e-226) (log t) (if (<= x 3.9e+86) (- z) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -52000000000000.0) {
		tmp = t_1;
	} else if (x <= -4e-177) {
		tmp = -z;
	} else if (x <= 1.06e-226) {
		tmp = log(t);
	} else if (x <= 3.9e+86) {
		tmp = -z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-52000000000000.0d0)) then
        tmp = t_1
    else if (x <= (-4d-177)) then
        tmp = -z
    else if (x <= 1.06d-226) then
        tmp = log(t)
    else if (x <= 3.9d+86) then
        tmp = -z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -52000000000000.0) {
		tmp = t_1;
	} else if (x <= -4e-177) {
		tmp = -z;
	} else if (x <= 1.06e-226) {
		tmp = Math.log(t);
	} else if (x <= 3.9e+86) {
		tmp = -z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -52000000000000.0:
		tmp = t_1
	elif x <= -4e-177:
		tmp = -z
	elif x <= 1.06e-226:
		tmp = math.log(t)
	elif x <= 3.9e+86:
		tmp = -z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -52000000000000.0)
		tmp = t_1;
	elseif (x <= -4e-177)
		tmp = Float64(-z);
	elseif (x <= 1.06e-226)
		tmp = log(t);
	elseif (x <= 3.9e+86)
		tmp = Float64(-z);
	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 <= -52000000000000.0)
		tmp = t_1;
	elseif (x <= -4e-177)
		tmp = -z;
	elseif (x <= 1.06e-226)
		tmp = log(t);
	elseif (x <= 3.9e+86)
		tmp = -z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -52000000000000.0], t$95$1, If[LessEqual[x, -4e-177], (-z), If[LessEqual[x, 1.06e-226], N[Log[t], $MachinePrecision], If[LessEqual[x, 3.9e+86], (-z), t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2e13 or 3.9000000000000002e86 < 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 86.3%

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

   3. Taylor expanded in y around 0 67.5%

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

   4. Taylor expanded in x around inf 67.5%

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

   if -5.2e13 < x < -3.99999999999999981e-177 or 1.0599999999999999e-226 < x < 3.9000000000000002e86

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 60.8%

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

      1. associate--l+ 60.8%

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

      2. fma-def 60.8%

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

   4. Simplified 60.8%

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

   5. Taylor expanded in z around inf 44.1%

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

      1. neg-mul-1 44.1%

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

   7. Simplified 44.1%

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

   if -3.99999999999999981e-177 < x < 1.0599999999999999e-226

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 72.2%

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

   3. Taylor expanded in x around 0 72.2%

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

   4. Taylor expanded in y around 0 47.5%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -52000000000000:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-177}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{-226}:\\ \;\;\;\;\log t\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 8: 59.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log t - y\\ \mathbf{if}\;x \leq -900000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-166}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+60}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (log t) y)))
   (if (<= x -900000000000.0)
     t_1
     (if (<= x -2.2e-121)
       t_2
       (if (<= x -1.5e-166) (- z) (if (<= x 2.5e+60) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = log(t) - y;
	double tmp;
	if (x <= -900000000000.0) {
		tmp = t_1;
	} else if (x <= -2.2e-121) {
		tmp = t_2;
	} else if (x <= -1.5e-166) {
		tmp = -z;
	} else if (x <= 2.5e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = log(t) - y
    if (x <= (-900000000000.0d0)) then
        tmp = t_1
    else if (x <= (-2.2d-121)) then
        tmp = t_2
    else if (x <= (-1.5d-166)) then
        tmp = -z
    else if (x <= 2.5d+60) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = Math.log(t) - y;
	double tmp;
	if (x <= -900000000000.0) {
		tmp = t_1;
	} else if (x <= -2.2e-121) {
		tmp = t_2;
	} else if (x <= -1.5e-166) {
		tmp = -z;
	} else if (x <= 2.5e+60) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = math.log(t) - y
	tmp = 0
	if x <= -900000000000.0:
		tmp = t_1
	elif x <= -2.2e-121:
		tmp = t_2
	elif x <= -1.5e-166:
		tmp = -z
	elif x <= 2.5e+60:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(log(t) - y)
	tmp = 0.0
	if (x <= -900000000000.0)
		tmp = t_1;
	elseif (x <= -2.2e-121)
		tmp = t_2;
	elseif (x <= -1.5e-166)
		tmp = Float64(-z);
	elseif (x <= 2.5e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = log(t) - y;
	tmp = 0.0;
	if (x <= -900000000000.0)
		tmp = t_1;
	elseif (x <= -2.2e-121)
		tmp = t_2;
	elseif (x <= -1.5e-166)
		tmp = -z;
	elseif (x <= 2.5e+60)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[x, -900000000000.0], t$95$1, If[LessEqual[x, -2.2e-121], t$95$2, If[LessEqual[x, -1.5e-166], (-z), If[LessEqual[x, 2.5e+60], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e11 or 2.49999999999999987e60 < 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 84.4%

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

   3. Taylor expanded in y around 0 66.3%

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

   4. Taylor expanded in x around inf 66.3%

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

   if -9e11 < x < -2.20000000000000021e-121 or -1.5000000000000001e-166 < x < 2.49999999999999987e60

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 65.1%

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

   3. Taylor expanded in x around 0 64.4%

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

   if -2.20000000000000021e-121 < x < -1.5000000000000001e-166

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 91.0%

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

      1. associate--l+ 91.0%

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

      2. fma-def 91.0%

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

   4. Simplified 91.0%

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

   5. Taylor expanded in z around inf 82.1%

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

      1. neg-mul-1 82.1%

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

   7. Simplified 82.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -900000000000:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-121}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-166}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+60}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]

Alternative 9: 84.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+165}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.4e+163)
   (* x (log y))
   (if (<= x 1.75e+165) (- (log t) (+ y z)) (* x (- (log (/ 1.0 y)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e+163) {
		tmp = x * log(y);
	} else if (x <= 1.75e+165) {
		tmp = log(t) - (y + z);
	} else {
		tmp = x * -log((1.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.4d+163)) then
        tmp = x * log(y)
    else if (x <= 1.75d+165) then
        tmp = log(t) - (y + z)
    else
        tmp = x * -log((1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.4e+163) {
		tmp = x * Math.log(y);
	} else if (x <= 1.75e+165) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = x * -Math.log((1.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.4e+163:
		tmp = x * math.log(y)
	elif x <= 1.75e+165:
		tmp = math.log(t) - (y + z)
	else:
		tmp = x * -math.log((1.0 / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.4e+163)
		tmp = Float64(x * log(y));
	elseif (x <= 1.75e+165)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(x * Float64(-log(Float64(1.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.4e+163)
		tmp = x * log(y);
	elseif (x <= 1.75e+165)
		tmp = log(t) - (y + z);
	else
		tmp = x * -log((1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.4e+163], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e+165], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(x * (-N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3999999999999999e163

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0 92.4%

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

   3. Taylor expanded in y around 0 79.0%

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

   4. Taylor expanded in x around inf 79.0%

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

   if -2.3999999999999999e163 < x < 1.74999999999999998e165

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.9%

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

   if 1.74999999999999998e165 < 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 90.2%

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

   3. Taylor expanded in y around 0 79.2%

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

   4. Taylor expanded in y around inf 79.3%

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

   5. Taylor expanded in x around inf 79.3%

      \[\leadsto \color{blue}{-1 \cdot \left(\log \left(\frac{1}{y}\right) \cdot x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+163}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{+165}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-\log \left(\frac{1}{y}\right)\right)\\ \end{array} \]

Alternative 10: 84.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+163} \lor \neg \left(x \leq 4.1 \cdot 10^{+164}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.15e+163) (not (<= x 4.1e+164)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.15e+163) || !(x <= 4.1e+164)) {
		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 <= (-2.15d+163)) .or. (.not. (x <= 4.1d+164))) 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 <= -2.15e+163) || !(x <= 4.1e+164)) {
		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 <= -2.15e+163) or not (x <= 4.1e+164):
		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 <= -2.15e+163) || !(x <= 4.1e+164))
		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 <= -2.15e+163) || ~((x <= 4.1e+164)))
		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, -2.15e+163], N[Not[LessEqual[x, 4.1e+164]], $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 < -2.1500000000000001e163 or 4.10000000000000016e164 < 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 91.5%

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

   3. Taylor expanded in y around 0 79.1%

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

   4. Taylor expanded in x around inf 79.1%

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

   if -2.1500000000000001e163 < x < 4.10000000000000016e164

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 89.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+163} \lor \neg \left(x \leq 4.1 \cdot 10^{+164}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 11: 41.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1800000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-36}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1800000000000.0) (- z) (if (<= z 3.7e-36) (log t) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1800000000000.0) {
		tmp = -z;
	} else if (z <= 3.7e-36) {
		tmp = log(t);
	} 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 <= (-1800000000000.0d0)) then
        tmp = -z
    else if (z <= 3.7d-36) then
        tmp = log(t)
    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 <= -1800000000000.0) {
		tmp = -z;
	} else if (z <= 3.7e-36) {
		tmp = Math.log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1800000000000.0:
		tmp = -z
	elif z <= 3.7e-36:
		tmp = math.log(t)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1800000000000.0)
		tmp = Float64(-z);
	elseif (z <= 3.7e-36)
		tmp = log(t);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1800000000000.0)
		tmp = -z;
	elseif (z <= 3.7e-36)
		tmp = log(t);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1800000000000.0], (-z), If[LessEqual[z, 3.7e-36], N[Log[t], $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e12 or 3.70000000000000002e-36 < 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 74.4%

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

      1. associate--l+ 74.4%

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

      2. fma-def 74.4%

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

   4. Simplified 74.4%

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

   5. Taylor expanded in z around inf 52.6%

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

      1. neg-mul-1 52.6%

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

   7. Simplified 52.6%

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

   if -1.8e12 < z < 3.70000000000000002e-36

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 99.8%

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

   3. Taylor expanded in x around 0 62.8%

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

   4. Taylor expanded in y around 0 30.0%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1800000000000:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-36}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 12: 30.6% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around 0 70.9%

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

   1. associate--l+ 70.9%

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

   2. fma-def 70.9%

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

4. Simplified 70.9%

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

5. Taylor expanded in z around inf 29.8%

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

   1. neg-mul-1 29.8%

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

7. Simplified 29.8%

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

8. Final simplification 29.8%

   \[\leadsto -z \]

Reproduce

herbie shell --seed 2023181 
(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)))