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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 15

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. sub-neg 99.8%

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

   3. associate--l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\left(\log y \cdot x - y\right) - z\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(-y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.55)
   (- (- (* (log y) x) y) z)
   (if (<= x 1.55) (- (log t) (+ y z)) (fma (log y) x (- (- y) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.55) {
		tmp = ((log(y) * x) - y) - z;
	} else if (x <= 1.55) {
		tmp = log(t) - (y + z);
	} else {
		tmp = fma(log(y), x, (-y - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(Float64(log(y) * x) - y) - z);
	elseif (x <= 1.55)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = fma(log(y), x, Float64(Float64(-y) - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.55], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 1.55], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[((-y) - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   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%

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

      2. sub-neg 99.6%

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

      3. associate--l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around inf 99.6%

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

   5. Taylor expanded in y around inf 98.1%

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

      1. mul-1-neg 98.1%

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

      2. unsub-neg 98.1%

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

      3. mul-1-neg 98.1%

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

      4. log-rec 99.6%

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

   7. Simplified 99.6%

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

   if -1.55000000000000004 < x < 1.55000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.2%

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

   if 1.55000000000000004 < 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%

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

      2. sub-neg 99.6%

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

      3. associate--l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around inf 99.3%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\left(\log y \cdot x - y\right) - z\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(-y\right) - z\right)\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\left(\log t + t_1\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= y 1.9e-6) (- (+ (log t) t_1) z) (- (- t_1 y) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (y <= 1.9e-6) {
		tmp = (log(t) + t_1) - z;
	} else {
		tmp = (t_1 - y) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (y <= 1.9d-6) then
        tmp = (log(t) + t_1) - z
    else
        tmp = (t_1 - y) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (y <= 1.9e-6) {
		tmp = (Math.log(t) + t_1) - z;
	} else {
		tmp = (t_1 - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if y <= 1.9e-6:
		tmp = (math.log(t) + t_1) - z
	else:
		tmp = (t_1 - y) - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (y <= 1.9e-6)
		tmp = Float64(Float64(log(t) + t_1) - z);
	else
		tmp = Float64(Float64(t_1 - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (y <= 1.9e-6)
		tmp = (log(t) + t_1) - z;
	else
		tmp = (t_1 - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, 1.9e-6], N[(N[(N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - z), $MachinePrecision], N[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e-6

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 99.5%

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

   if 1.9e-6 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. associate--l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 99.5%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. unsub-neg 99.5%

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

      3. mul-1-neg 99.5%

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

      4. log-rec 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\left(\log t + \log y \cdot x\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log y \cdot x - y\right) - z\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 69.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-74}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+133}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- (- y) z)))
   (if (<= x -1.16e+218)
     t_1
     (if (<= x -2.4e+177)
       t_2
       (if (<= x -6e+87)
         t_1
         (if (<= x 4.7e-101)
           t_2
           (if (<= x 3.8e-74) (- (log t) y) (if (<= x 8.6e+133) t_2 t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = -y - z;
	double tmp;
	if (x <= -1.16e+218) {
		tmp = t_1;
	} else if (x <= -2.4e+177) {
		tmp = t_2;
	} else if (x <= -6e+87) {
		tmp = t_1;
	} else if (x <= 4.7e-101) {
		tmp = t_2;
	} else if (x <= 3.8e-74) {
		tmp = log(t) - y;
	} else if (x <= 8.6e+133) {
		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 = log(y) * x
    t_2 = -y - z
    if (x <= (-1.16d+218)) then
        tmp = t_1
    else if (x <= (-2.4d+177)) then
        tmp = t_2
    else if (x <= (-6d+87)) then
        tmp = t_1
    else if (x <= 4.7d-101) then
        tmp = t_2
    else if (x <= 3.8d-74) then
        tmp = log(t) - y
    else if (x <= 8.6d+133) 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 = Math.log(y) * x;
	double t_2 = -y - z;
	double tmp;
	if (x <= -1.16e+218) {
		tmp = t_1;
	} else if (x <= -2.4e+177) {
		tmp = t_2;
	} else if (x <= -6e+87) {
		tmp = t_1;
	} else if (x <= 4.7e-101) {
		tmp = t_2;
	} else if (x <= 3.8e-74) {
		tmp = Math.log(t) - y;
	} else if (x <= 8.6e+133) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = -y - z
	tmp = 0
	if x <= -1.16e+218:
		tmp = t_1
	elif x <= -2.4e+177:
		tmp = t_2
	elif x <= -6e+87:
		tmp = t_1
	elif x <= 4.7e-101:
		tmp = t_2
	elif x <= 3.8e-74:
		tmp = math.log(t) - y
	elif x <= 8.6e+133:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (x <= -1.16e+218)
		tmp = t_1;
	elseif (x <= -2.4e+177)
		tmp = t_2;
	elseif (x <= -6e+87)
		tmp = t_1;
	elseif (x <= 4.7e-101)
		tmp = t_2;
	elseif (x <= 3.8e-74)
		tmp = Float64(log(t) - y);
	elseif (x <= 8.6e+133)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = -y - z;
	tmp = 0.0;
	if (x <= -1.16e+218)
		tmp = t_1;
	elseif (x <= -2.4e+177)
		tmp = t_2;
	elseif (x <= -6e+87)
		tmp = t_1;
	elseif (x <= 4.7e-101)
		tmp = t_2;
	elseif (x <= 3.8e-74)
		tmp = log(t) - y;
	elseif (x <= 8.6e+133)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[x, -1.16e+218], t$95$1, If[LessEqual[x, -2.4e+177], t$95$2, If[LessEqual[x, -6e+87], t$95$1, If[LessEqual[x, 4.7e-101], t$95$2, If[LessEqual[x, 3.8e-74], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 8.6e+133], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15999999999999994e218 or -2.4e177 < x < -5.9999999999999998e87 or 8.59999999999999989e133 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. associate--l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around inf 80.3%

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

   if -1.15999999999999994e218 < x < -2.4e177 or -5.9999999999999998e87 < x < 4.6999999999999999e-101 or 3.7999999999999996e-74 < x < 8.59999999999999989e133

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. associate--l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 86.9%

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

   5. Taylor expanded in x around 0 79.1%

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

      1. mul-1-neg 79.1%

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

   7. Simplified 79.1%

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

   if 4.6999999999999999e-101 < x < 3.7999999999999996e-74

   1. Initial program 100.0%

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

   2. Taylor expanded in z around 0 81.4%

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

   3. Taylor expanded in x around 0 81.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+218}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+177}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \leq -6 \cdot 10^{+87}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-101}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-74}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+133}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 6: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-81}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- (- y) z)))
   (if (<= x -1.2e+218)
     t_1
     (if (<= x -2.4e+177)
       t_2
       (if (<= x -7.2e+88)
         t_1
         (if (<= x -2.8e-301)
           t_2
           (if (<= x 4.3e-81) (- (log t) z) (if (<= x 8.5e+137) t_2 t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = -y - z;
	double tmp;
	if (x <= -1.2e+218) {
		tmp = t_1;
	} else if (x <= -2.4e+177) {
		tmp = t_2;
	} else if (x <= -7.2e+88) {
		tmp = t_1;
	} else if (x <= -2.8e-301) {
		tmp = t_2;
	} else if (x <= 4.3e-81) {
		tmp = log(t) - z;
	} else if (x <= 8.5e+137) {
		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 = log(y) * x
    t_2 = -y - z
    if (x <= (-1.2d+218)) then
        tmp = t_1
    else if (x <= (-2.4d+177)) then
        tmp = t_2
    else if (x <= (-7.2d+88)) then
        tmp = t_1
    else if (x <= (-2.8d-301)) then
        tmp = t_2
    else if (x <= 4.3d-81) then
        tmp = log(t) - z
    else if (x <= 8.5d+137) 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 = Math.log(y) * x;
	double t_2 = -y - z;
	double tmp;
	if (x <= -1.2e+218) {
		tmp = t_1;
	} else if (x <= -2.4e+177) {
		tmp = t_2;
	} else if (x <= -7.2e+88) {
		tmp = t_1;
	} else if (x <= -2.8e-301) {
		tmp = t_2;
	} else if (x <= 4.3e-81) {
		tmp = Math.log(t) - z;
	} else if (x <= 8.5e+137) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = -y - z
	tmp = 0
	if x <= -1.2e+218:
		tmp = t_1
	elif x <= -2.4e+177:
		tmp = t_2
	elif x <= -7.2e+88:
		tmp = t_1
	elif x <= -2.8e-301:
		tmp = t_2
	elif x <= 4.3e-81:
		tmp = math.log(t) - z
	elif x <= 8.5e+137:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (x <= -1.2e+218)
		tmp = t_1;
	elseif (x <= -2.4e+177)
		tmp = t_2;
	elseif (x <= -7.2e+88)
		tmp = t_1;
	elseif (x <= -2.8e-301)
		tmp = t_2;
	elseif (x <= 4.3e-81)
		tmp = Float64(log(t) - z);
	elseif (x <= 8.5e+137)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = -y - z;
	tmp = 0.0;
	if (x <= -1.2e+218)
		tmp = t_1;
	elseif (x <= -2.4e+177)
		tmp = t_2;
	elseif (x <= -7.2e+88)
		tmp = t_1;
	elseif (x <= -2.8e-301)
		tmp = t_2;
	elseif (x <= 4.3e-81)
		tmp = log(t) - z;
	elseif (x <= 8.5e+137)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[x, -1.2e+218], t$95$1, If[LessEqual[x, -2.4e+177], t$95$2, If[LessEqual[x, -7.2e+88], t$95$1, If[LessEqual[x, -2.8e-301], t$95$2, If[LessEqual[x, 4.3e-81], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 8.5e+137], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1999999999999999e218 or -2.4e177 < x < -7.2000000000000004e88 or 8.50000000000000028e137 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. associate--l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around inf 80.3%

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

   if -1.1999999999999999e218 < x < -2.4e177 or -7.2000000000000004e88 < x < -2.8000000000000001e-301 or 4.3000000000000003e-81 < x < 8.50000000000000028e137

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. associate--l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 88.6%

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

   5. Taylor expanded in x around 0 79.4%

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

      1. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

   if -2.8000000000000001e-301 < x < 4.3000000000000003e-81

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. associate--l+ 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-def 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(-y\right) - \left(z - \log t\right)\right)} \]
   4. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. *-commutative 100.0%

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

      3. add-cube-cbrt 100.0%

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

      4. unpow2 100.0%

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

      5. associate-*r* 100.0%

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

      6. fma-udef 100.0%

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

      7. add-sqr-sqrt 38.8%

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

      8. pow2 38.8%

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

   5. Applied egg-rr 38.9%

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

   6. Taylor expanded in x around 0 89.0%

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

   7. Taylor expanded in y around 0 89.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+218}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{+177}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{+88}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq -2.8 \cdot 10^{-301}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{-81}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+137}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]

Alternative 7: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+218} \lor \neg \left(x \leq -1.8 \cdot 10^{+175} \lor \neg \left(x \leq -9.6 \cdot 10^{+88}\right) \land x \leq 5.5 \cdot 10^{+138}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.2e+218)
         (not
          (or (<= x -1.8e+175) (and (not (<= x -9.6e+88)) (<= x 5.5e+138)))))
   (* (log y) x)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.2e+218) || !((x <= -1.8e+175) || (!(x <= -9.6e+88) && (x <= 5.5e+138)))) {
		tmp = log(y) * x;
	} 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 <= (-1.2d+218)) .or. (.not. (x <= (-1.8d+175)) .or. (.not. (x <= (-9.6d+88))) .and. (x <= 5.5d+138))) then
        tmp = log(y) * x
    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 <= -1.2e+218) || !((x <= -1.8e+175) || (!(x <= -9.6e+88) && (x <= 5.5e+138)))) {
		tmp = Math.log(y) * x;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.2e+218) or not ((x <= -1.8e+175) or (not (x <= -9.6e+88) and (x <= 5.5e+138))):
		tmp = math.log(y) * x
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.2e+218) || !((x <= -1.8e+175) || (!(x <= -9.6e+88) && (x <= 5.5e+138))))
		tmp = Float64(log(y) * x);
	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 <= -1.2e+218) || ~(((x <= -1.8e+175) || (~((x <= -9.6e+88)) && (x <= 5.5e+138)))))
		tmp = log(y) * x;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.2e+218], N[Not[Or[LessEqual[x, -1.8e+175], And[N[Not[LessEqual[x, -9.6e+88]], $MachinePrecision], LessEqual[x, 5.5e+138]]]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1999999999999999e218 or -1.80000000000000017e175 < x < -9.5999999999999996e88 or 5.4999999999999999e138 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. associate--l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around inf 80.3%

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

   if -1.1999999999999999e218 < x < -1.80000000000000017e175 or -9.5999999999999996e88 < x < 5.4999999999999999e138

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 92.1%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+218} \lor \neg \left(x \leq -1.8 \cdot 10^{+175} \lor \neg \left(x \leq -9.6 \cdot 10^{+88}\right) \land x \leq 5.5 \cdot 10^{+138}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 8: 70.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+218} \lor \neg \left(x \leq -2.4 \cdot 10^{+177}\right) \land \left(x \leq -7.4 \cdot 10^{+87} \lor \neg \left(x \leq 6.3 \cdot 10^{+128}\right)\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.16e+218)
         (and (not (<= x -2.4e+177))
              (or (<= x -7.4e+87) (not (<= x 6.3e+128)))))
   (* (log y) x)
   (- (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.16e+218) || (!(x <= -2.4e+177) && ((x <= -7.4e+87) || !(x <= 6.3e+128)))) {
		tmp = log(y) * x;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.16d+218)) .or. (.not. (x <= (-2.4d+177))) .and. (x <= (-7.4d+87)) .or. (.not. (x <= 6.3d+128))) then
        tmp = log(y) * x
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.16e+218) || (!(x <= -2.4e+177) && ((x <= -7.4e+87) || !(x <= 6.3e+128)))) {
		tmp = Math.log(y) * x;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.16e+218) or (not (x <= -2.4e+177) and ((x <= -7.4e+87) or not (x <= 6.3e+128))):
		tmp = math.log(y) * x
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.16e+218) || (!(x <= -2.4e+177) && ((x <= -7.4e+87) || !(x <= 6.3e+128))))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.16e+218) || (~((x <= -2.4e+177)) && ((x <= -7.4e+87) || ~((x <= 6.3e+128)))))
		tmp = log(y) * x;
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.16e+218], And[N[Not[LessEqual[x, -2.4e+177]], $MachinePrecision], Or[LessEqual[x, -7.4e+87], N[Not[LessEqual[x, 6.3e+128]], $MachinePrecision]]]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15999999999999994e218 or -2.4e177 < x < -7.40000000000000005e87 or 6.2999999999999999e128 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. associate--l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in x around inf 80.3%

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

   if -1.15999999999999994e218 < x < -2.4e177 or -7.40000000000000005e87 < x < 6.2999999999999999e128

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. associate--l+ 99.9%

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

      4. *-commutative 99.9%

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

      5. fma-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in z around inf 84.5%

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

   5. Taylor expanded in x around 0 77.0%

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

      1. mul-1-neg 77.0%

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

   7. Simplified 77.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{+218} \lor \neg \left(x \leq -2.4 \cdot 10^{+177}\right) \land \left(x \leq -7.4 \cdot 10^{+87} \lor \neg \left(x \leq 6.3 \cdot 10^{+128}\right)\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]

Alternative 9: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\left(\log y \cdot x - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.12) (not (<= x 1.05)))
   (- (- (* (log y) x) y) z)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.12) || !(x <= 1.05)) {
		tmp = ((log(y) * x) - y) - z;
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.12d0)) .or. (.not. (x <= 1.05d0))) then
        tmp = ((log(y) * x) - y) - z
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.12) || !(x <= 1.05)) {
		tmp = ((Math.log(y) * x) - y) - z;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.12) or not (x <= 1.05):
		tmp = ((math.log(y) * x) - y) - z
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.12) || !(x <= 1.05))
		tmp = Float64(Float64(Float64(log(y) * x) - y) - z);
	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 <= -1.12) || ~((x <= 1.05)))
		tmp = ((log(y) * x) - y) - z;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.12], N[Not[LessEqual[x, 1.05]], $MachinePrecision]], N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1200000000000001 or 1.05000000000000004 < 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%

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

      2. sub-neg 99.6%

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

      3. associate--l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around inf 99.5%

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

   5. Taylor expanded in y around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. unsub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. log-rec 99.5%

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

   7. Simplified 99.5%

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

   if -1.1200000000000001 < x < 1.05000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.2%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \lor \neg \left(x \leq 1.05\right):\\ \;\;\;\;\left(\log y \cdot x - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 10: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+68} \lor \neg \left(x \leq 2.5 \cdot 10^{+128}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.1e+68) (not (<= x 2.5e+128)))
   (- (* (log y) x) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.1e+68) || !(x <= 2.5e+128)) {
		tmp = (log(y) * x) - 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 <= (-2.1d+68)) .or. (.not. (x <= 2.5d+128))) then
        tmp = (log(y) * x) - 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 <= -2.1e+68) || !(x <= 2.5e+128)) {
		tmp = (Math.log(y) * x) - y;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.1e+68) or not (x <= 2.5e+128):
		tmp = (math.log(y) * x) - y
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.1e+68) || !(x <= 2.5e+128))
		tmp = Float64(Float64(log(y) * x) - 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 <= -2.1e+68) || ~((x <= 2.5e+128)))
		tmp = (log(y) * x) - y;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.1e+68], N[Not[LessEqual[x, 2.5e+128]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.10000000000000001e68 or 2.5e128 < x

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. sub-neg 99.5%

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

      3. associate--l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in z around inf 99.5%

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

   5. Taylor expanded in z around 0 85.7%

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

   if -2.10000000000000001e68 < x < 2.5e128

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 93.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+68} \lor \neg \left(x \leq 2.5 \cdot 10^{+128}\right):\\ \;\;\;\;\log y \cdot x - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 11: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+48}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+127}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -6.2e+48)
     (- t_1 z)
     (if (<= x 4.6e+127) (- (log t) (+ y z)) (- t_1 y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -6.2e+48) {
		tmp = t_1 - z;
	} else if (x <= 4.6e+127) {
		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(y) * x
    if (x <= (-6.2d+48)) then
        tmp = t_1 - z
    else if (x <= 4.6d+127) 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(y) * x;
	double tmp;
	if (x <= -6.2e+48) {
		tmp = t_1 - z;
	} else if (x <= 4.6e+127) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -6.2e+48:
		tmp = t_1 - z
	elif x <= 4.6e+127:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1 - y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -6.2e+48)
		tmp = Float64(t_1 - z);
	elseif (x <= 4.6e+127)
		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(y) * x;
	tmp = 0.0;
	if (x <= -6.2e+48)
		tmp = t_1 - z;
	elseif (x <= 4.6e+127)
		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[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.2e+48], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[x, 4.6e+127], 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 < -6.20000000000000011e48

   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%

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

      2. sub-neg 99.6%

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

      3. associate--l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in z around inf 99.6%

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

   5. Taylor expanded in y around 0 87.1%

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

   if -6.20000000000000011e48 < x < 4.6000000000000003e127

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 94.2%

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

   if 4.6000000000000003e127 < x

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. sub-neg 99.4%

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

      3. associate--l+ 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in z around inf 99.4%

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

   5. Taylor expanded in z around 0 98.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+48}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+127}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - y\\ \end{array} \]

Alternative 12: 44.8% accurate, 41.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+152}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.6e+152) (- y z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+152) {
		tmp = y - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.6d+152) then
        tmp = y - z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.6e+152) {
		tmp = y - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.6e+152:
		tmp = y - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.6e+152)
		tmp = Float64(y - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.6e+152)
		tmp = y - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.6e+152], N[(y - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.6000000000000001e152

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. sub-neg 99.8%

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

      3. associate--l+ 99.8%

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

      4. *-commutative 99.8%

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

      5. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(-y\right) - \left(z - \log t\right)\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.8%

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

      2. *-commutative 99.8%

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

      3. add-cube-cbrt 99.4%

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

      4. unpow2 99.4%

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

      5. associate-*r* 99.4%

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

      6. fma-udef 99.4%

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

      7. add-sqr-sqrt 48.7%

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

      8. pow2 48.7%

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

   5. Applied egg-rr 48.5%

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

   6. Taylor expanded in x around 0 59.1%

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

   7. Taylor expanded in y around inf 44.6%

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

   if 2.6000000000000001e152 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 82.2%

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

      1. neg-mul-1 82.2%

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

   4. Simplified 82.2%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+152}:\\ \;\;\;\;y - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 13: 44.9% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+152}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.9e+152) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.9e+152) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.9d+152) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.9e+152) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.9e+152:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.9e+152)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.9e+152)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.9e+152], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e152

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 44.6%

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

      1. mul-1-neg 44.6%

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

   4. Simplified 44.6%

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

   if 2.8999999999999998e152 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 82.2%

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

      1. neg-mul-1 82.2%

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

   4. Simplified 82.2%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{+152}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 14: 57.8% accurate, 52.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. sub-neg 99.8%

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

   3. associate--l+ 99.8%

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

   4. *-commutative 99.8%

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

   5. fma-def 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in z around inf 88.4%

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

5. Taylor expanded in x around 0 62.5%

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

   1. mul-1-neg 62.5%

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

7. Simplified 62.5%

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

8. Final simplification 62.5%

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

Alternative 15: 30.8% 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 28.0%

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

   1. neg-mul-1 28.0%

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

4. Simplified 28.0%

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

5. Final simplification 28.0%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023187 
(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)))