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

Percentage Accurate: 99.9% → 99.9%

Time: 12.2s

Alternatives: 13

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. associate--l+ 99.9%

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

   3. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+21} \lor \neg \left(t\_2 \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;t\_1 - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (or (<= t_2 -1e+21) (not (<= t_2 2e-34)))
     (- t_1 (+ y z))
     (- (log t) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if ((t_2 <= -1e+21) || !(t_2 <= 2e-34)) {
		tmp = t_1 - (y + z);
	} else {
		tmp = log(t) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if ((t_2 <= (-1d+21)) .or. (.not. (t_2 <= 2d-34))) then
        tmp = t_1 - (y + z)
    else
        tmp = log(t) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if ((t_2 <= -1e+21) || !(t_2 <= 2e-34)) {
		tmp = t_1 - (y + z);
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if (t_2 <= -1e+21) or not (t_2 <= 2e-34):
		tmp = t_1 - (y + z)
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if ((t_2 <= -1e+21) || !(t_2 <= 2e-34))
		tmp = Float64(t_1 - Float64(y + z));
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if ((t_2 <= -1e+21) || ~((t_2 <= 2e-34)))
		tmp = t_1 - (y + z);
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -1e+21], N[Not[LessEqual[t$95$2, 2e-34]], $MachinePrecision]], N[(t$95$1 - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e21 or 1.99999999999999986e-34 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 99.9%

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

   if -1e21 < (-.f64 (*.f64 x (log.f64 y)) y) < 1.99999999999999986e-34

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -1 \cdot 10^{+21} \lor \neg \left(x \cdot \log y - y \leq 2 \cdot 10^{-34}\right):\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+215}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-34}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -1e+215)
     t_2
     (if (<= t_2 2e-34) (- (log t) (+ y z)) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -1e+215) {
		tmp = t_2;
	} else if (t_2 <= 2e-34) {
		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) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-1d+215)) then
        tmp = t_2
    else if (t_2 <= 2d-34) 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 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -1e+215) {
		tmp = t_2;
	} else if (t_2 <= 2e-34) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -1e+215:
		tmp = t_2
	elif t_2 <= 2e-34:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -1e+215)
		tmp = t_2;
	elseif (t_2 <= 2e-34)
		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 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -1e+215)
		tmp = t_2;
	elseif (t_2 <= 2e-34)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+215], t$95$2, If[LessEqual[t$95$2, 2e-34], 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 (-.f64 (*.f64 x (log.f64 y)) y) < -9.99999999999999907e214

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 92.9%

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

   if -9.99999999999999907e214 < (-.f64 (*.f64 x (log.f64 y)) y) < 1.99999999999999986e-34

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 91.8%

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

   if 1.99999999999999986e-34 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.8%

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

   6. Taylor expanded in y around 0 99.4%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 5: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - z\\ \mathbf{if}\;y \leq 1.4 \cdot 10^{-201}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) z)))
   (if (<= y 1.4e-201)
     t_1
     (if (<= y 4e-54) (* x (log y)) (if (<= y 3.8e+21) t_1 (- (- y) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - z;
	double tmp;
	if (y <= 1.4e-201) {
		tmp = t_1;
	} else if (y <= 4e-54) {
		tmp = x * log(y);
	} else if (y <= 3.8e+21) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = log(t) - z
    if (y <= 1.4d-201) then
        tmp = t_1
    else if (y <= 4d-54) then
        tmp = x * log(y)
    else if (y <= 3.8d+21) then
        tmp = t_1
    else
        tmp = -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(t) - z;
	double tmp;
	if (y <= 1.4e-201) {
		tmp = t_1;
	} else if (y <= 4e-54) {
		tmp = x * Math.log(y);
	} else if (y <= 3.8e+21) {
		tmp = t_1;
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) - z
	tmp = 0
	if y <= 1.4e-201:
		tmp = t_1
	elif y <= 4e-54:
		tmp = x * math.log(y)
	elif y <= 3.8e+21:
		tmp = t_1
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - z)
	tmp = 0.0
	if (y <= 1.4e-201)
		tmp = t_1;
	elseif (y <= 4e-54)
		tmp = Float64(x * log(y));
	elseif (y <= 3.8e+21)
		tmp = t_1;
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) - z;
	tmp = 0.0;
	if (y <= 1.4e-201)
		tmp = t_1;
	elseif (y <= 4e-54)
		tmp = x * log(y);
	elseif (y <= 3.8e+21)
		tmp = t_1;
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[y, 1.4e-201], t$95$1, If[LessEqual[y, 4e-54], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+21], t$95$1, N[((-y) - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.4e-201 or 4.0000000000000001e-54 < y < 3.8e21

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 80.7%

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

      1. neg-mul-1 80.7%

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

   7. Simplified 80.7%

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

   if 1.4e-201 < y < 4.0000000000000001e-54

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. unsub-neg 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 56.8%

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

   if 3.8e21 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 99.9%

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

   6. Taylor expanded in x around 0 84.5%

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

      1. neg-mul-1 84.5%

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

      2. distribute-neg-in 84.5%

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

      3. sub-neg 84.5%

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

   8. Simplified 84.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.4 \cdot 10^{-201}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+30} \lor \neg \left(x \leq 1.15 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.42e+30) (not (<= x 1.15e+70)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.42e+30) || !(x <= 1.15e+70)) {
		tmp = (x * log(y)) - y;
	} else {
		tmp = log(t) - (y + z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.42d+30)) .or. (.not. (x <= 1.15d+70))) then
        tmp = (x * log(y)) - y
    else
        tmp = log(t) - (y + z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.42e+30) || !(x <= 1.15e+70)) {
		tmp = (x * Math.log(y)) - y;
	} else {
		tmp = Math.log(t) - (y + z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.42e+30) or not (x <= 1.15e+70):
		tmp = (x * math.log(y)) - y
	else:
		tmp = math.log(t) - (y + z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.42e+30) || !(x <= 1.15e+70))
		tmp = Float64(Float64(x * log(y)) - y);
	else
		tmp = Float64(log(t) - Float64(y + z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.42e+30) || ~((x <= 1.15e+70)))
		tmp = (x * log(y)) - y;
	else
		tmp = log(t) - (y + z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.42e+30], N[Not[LessEqual[x, 1.15e+70]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.41999999999999991e30 or 1.14999999999999997e70 < x

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

      2. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 84.0%

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

   if -1.41999999999999991e30 < x < 1.14999999999999997e70

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+30} \lor \neg \left(x \leq 1.15 \cdot 10^{+70}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+186} \lor \neg \left(x \leq 3.6 \cdot 10^{+223}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.15e+186) (not (<= x 3.6e+223)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.15e+186) || !(x <= 3.6e+223)) {
		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+186)) .or. (.not. (x <= 3.6d+223))) 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+186) || !(x <= 3.6e+223)) {
		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+186) or not (x <= 3.6e+223):
		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+186) || !(x <= 3.6e+223))
		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+186) || ~((x <= 3.6e+223)))
		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+186], N[Not[LessEqual[x, 3.6e+223]], $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 < -2.15e186 or 3.59999999999999991e223 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 89.8%

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

   if -2.15e186 < x < 3.59999999999999991e223

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. associate--l+ 100.0%

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

      3. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 85.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+186} \lor \neg \left(x \leq 3.6 \cdot 10^{+223}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+185} \lor \neg \left(x \leq 3.6 \cdot 10^{+223}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.1e+185) (not (<= x 3.6e+223))) (* x (log y)) (- (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e+185) || !(x <= 3.6e+223)) {
		tmp = x * log(y);
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-3.1d+185)) .or. (.not. (x <= 3.6d+223))) then
        tmp = x * log(y)
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.1e+185) || !(x <= 3.6e+223)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.1e+185) or not (x <= 3.6e+223):
		tmp = x * math.log(y)
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.1e+185) || !(x <= 3.6e+223))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.1e+185) || ~((x <= 3.6e+223)))
		tmp = x * log(y);
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.1e+185], N[Not[LessEqual[x, 3.6e+223]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e185 or 3.59999999999999991e223 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 89.8%

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

   if -3.1e185 < x < 3.59999999999999991e223

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 83.0%

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

   6. Taylor expanded in x around 0 68.6%

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

      1. neg-mul-1 68.6%

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

      2. distribute-neg-in 68.6%

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

      3. sub-neg 68.6%

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

   8. Simplified 68.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+185} \lor \neg \left(x \leq 3.6 \cdot 10^{+223}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-151} \lor \neg \left(y \leq 1.35 \cdot 10^{-51}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y 4.9e-151) (not (<= y 1.35e-51))) (- (- y) z) (log t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= 4.9e-151) || !(y <= 1.35e-51)) {
		tmp = -y - z;
	} else {
		tmp = log(t);
	}
	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 <= 4.9d-151) .or. (.not. (y <= 1.35d-51))) then
        tmp = -y - z
    else
        tmp = log(t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= 4.9e-151) || !(y <= 1.35e-51)) {
		tmp = -y - z;
	} else {
		tmp = Math.log(t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= 4.9e-151) or not (y <= 1.35e-51):
		tmp = -y - z
	else:
		tmp = math.log(t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= 4.9e-151) || !(y <= 1.35e-51))
		tmp = Float64(Float64(-y) - z);
	else
		tmp = log(t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= 4.9e-151) || ~((y <= 1.35e-51)))
		tmp = -y - z;
	else
		tmp = log(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, 4.9e-151], N[Not[LessEqual[y, 1.35e-51]], $MachinePrecision]], N[((-y) - z), $MachinePrecision], N[Log[t], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.89999999999999966e-151 or 1.3499999999999999e-51 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 89.7%

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

   6. Taylor expanded in x around 0 68.1%

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

      1. neg-mul-1 68.1%

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

      2. distribute-neg-in 68.1%

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

      3. sub-neg 68.1%

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

   8. Simplified 68.1%

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

   if 4.89999999999999966e-151 < y < 1.3499999999999999e-51

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. associate--l+ 99.8%

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

      3. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 33.3%

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

      1. mul-1-neg 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in y around 0 33.3%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.9 \cdot 10^{-151} \lor \neg \left(y \leq 1.35 \cdot 10^{-51}\right):\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.3% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{+84}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.3e+84) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.3e+84) {
		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 <= 4.3d+84) 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 <= 4.3e+84) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.3e+84:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.3e+84)
		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 <= 4.3e+84)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.3e+84], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 4.2999999999999996e84

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf 37.8%

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

      1. neg-mul-1 37.8%

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

   10. Simplified 37.8%

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

   if 4.2999999999999996e84 < y

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 73.5%

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

      1. neg-mul-1 73.5%

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

   10. Simplified 73.5%

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

Alternative 11: 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 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 85.9%

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

6. Taylor expanded in x around 0 58.1%

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

   1. neg-mul-1 58.1%

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

   2. distribute-neg-in 58.1%

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

   3. sub-neg 58.1%

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

8. Simplified 58.1%

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

Alternative 12: 29.9% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l- 99.9%

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

   2. associate--l- 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in z around inf 99.9%

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

   1. mul-1-neg 99.9%

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

   2. unsub-neg 99.9%

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

7. Simplified 99.9%

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

8. Taylor expanded in y around inf 31.6%

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

   1. neg-mul-1 31.6%

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

10. Simplified 31.6%

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

Alternative 13: 2.3% accurate, 209.0× 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 y
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := y

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. associate--l+ 99.9%

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

   3. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 45.2%

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

   1. mul-1-neg 45.2%

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

7. Simplified 45.2%

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

   1. *-un-lft-identity 45.2%

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

   2. +-commutative 45.2%

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

   3. add-sqr-sqrt 0.0%

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

   4. sqrt-unprod 15.7%

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

   5. sqr-neg 15.7%

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

   6. sqrt-unprod 15.7%

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

   7. add-sqr-sqrt 15.7%

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

9. Applied egg-rr 15.7%

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

    1. *-lft-identity 15.7%

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

11. Simplified 15.7%

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

12. Taylor expanded in y around inf 2.3%

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

Reproduce

herbie shell --seed 2024116 
(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)))