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

Percentage Accurate: 99.9% → 99.9%

Time: 9.7s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+238}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 + \log t\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -2e+238)
     t_2
     (if (<= t_2 -5e-9) (- (log t) (+ y z)) (- (+ t_1 (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 <= -2e+238) {
		tmp = t_2;
	} else if (t_2 <= -5e-9) {
		tmp = log(t) - (y + z);
	} else {
		tmp = (t_1 + 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 <= (-2d+238)) then
        tmp = t_2
    else if (t_2 <= (-5d-9)) then
        tmp = log(t) - (y + z)
    else
        tmp = (t_1 + 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 <= -2e+238) {
		tmp = t_2;
	} else if (t_2 <= -5e-9) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = (t_1 + 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 <= -2e+238:
		tmp = t_2
	elif t_2 <= -5e-9:
		tmp = math.log(t) - (y + z)
	else:
		tmp = (t_1 + 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 <= -2e+238)
		tmp = t_2;
	elseif (t_2 <= -5e-9)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(Float64(t_1 + 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 <= -2e+238)
		tmp = t_2;
	elseif (t_2 <= -5e-9)
		tmp = log(t) - (y + z);
	else
		tmp = (t_1 + 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[LessEqual[t$95$2, -2e+238], t$95$2, If[LessEqual[t$95$2, -5e-9], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[Log[t], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2.0000000000000001e238

   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 y around inf 95.2%

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

   if -2.0000000000000001e238 < (-.f64 (*.f64 x (log.f64 y)) y) < -5.0000000000000001e-9

   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 x around 0 82.5%

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

   if -5.0000000000000001e-9 < (-.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 y around 0 98.1%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+238}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq -5 \cdot 10^{-9}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+238}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -2e+238)
     t_2
     (if (<= t_2 2e-26) (- (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 <= -2e+238) {
		tmp = t_2;
	} else if (t_2 <= 2e-26) {
		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 <= (-2d+238)) then
        tmp = t_2
    else if (t_2 <= 2d-26) 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 <= -2e+238) {
		tmp = t_2;
	} else if (t_2 <= 2e-26) {
		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 <= -2e+238:
		tmp = t_2
	elif t_2 <= 2e-26:
		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 <= -2e+238)
		tmp = t_2;
	elseif (t_2 <= 2e-26)
		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 <= -2e+238)
		tmp = t_2;
	elseif (t_2 <= 2e-26)
		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, -2e+238], t$95$2, If[LessEqual[t$95$2, 2e-26], 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) < -2.0000000000000001e238

   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 y around inf 95.2%

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

   if -2.0000000000000001e238 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.0000000000000001e-26

   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 x around 0 88.8%

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

   if 2.0000000000000001e-26 < (-.f64 (*.f64 x (log.f64 y)) y)

   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. associate--l- 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in z around inf 95.5%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+238}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{-26}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+234} \lor \neg \left(x \leq 4.1 \cdot 10^{+116}\right) \land \left(x \leq 5.5 \cdot 10^{+173} \lor \neg \left(x \leq 2.7 \cdot 10^{+216}\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.12e+234)
         (and (not (<= x 4.1e+116))
              (or (<= x 5.5e+173) (not (<= x 2.7e+216)))))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.12e+234) || (!(x <= 4.1e+116) && ((x <= 5.5e+173) || !(x <= 2.7e+216)))) {
		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 <= (-1.12d+234)) .or. (.not. (x <= 4.1d+116)) .and. (x <= 5.5d+173) .or. (.not. (x <= 2.7d+216))) 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 <= -1.12e+234) || (!(x <= 4.1e+116) && ((x <= 5.5e+173) || !(x <= 2.7e+216)))) {
		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 <= -1.12e+234) or (not (x <= 4.1e+116) and ((x <= 5.5e+173) or not (x <= 2.7e+216))):
		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 <= -1.12e+234) || (!(x <= 4.1e+116) && ((x <= 5.5e+173) || !(x <= 2.7e+216))))
		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 <= -1.12e+234) || (~((x <= 4.1e+116)) && ((x <= 5.5e+173) || ~((x <= 2.7e+216)))))
		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, -1.12e+234], And[N[Not[LessEqual[x, 4.1e+116]], $MachinePrecision], Or[LessEqual[x, 5.5e+173], N[Not[LessEqual[x, 2.7e+216]], $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 < -1.12000000000000004e234 or 4.0999999999999998e116 < x < 5.50000000000000049e173 or 2.7000000000000001e216 < 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. associate--l- 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. add-cube-cbrt 98.4%

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

      3. associate-*r* 98.5%

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

      4. fma-neg 98.4%

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

      5. pow2 98.4%

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

      6. associate-+r- 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around inf 80.1%

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

   if -1.12000000000000004e234 < x < 4.0999999999999998e116 or 5.50000000000000049e173 < x < 2.7000000000000001e216

   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 x around 0 87.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+234} \lor \neg \left(x \leq 4.1 \cdot 10^{+116}\right) \land \left(x \leq 5.5 \cdot 10^{+173} \lor \neg \left(x \leq 2.7 \cdot 10^{+216}\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(-1 - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-201}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 0.00031:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- -1.0 (/ y z)))))
   (if (<= z -1.15e-43)
     t_1
     (if (<= z -1.6e-201) (- y) (if (<= z 0.00031) (* x (log y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (-1.0 - (y / z));
	double tmp;
	if (z <= -1.15e-43) {
		tmp = t_1;
	} else if (z <= -1.6e-201) {
		tmp = -y;
	} else if (z <= 0.00031) {
		tmp = x * log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * ((-1.0d0) - (y / z))
    if (z <= (-1.15d-43)) then
        tmp = t_1
    else if (z <= (-1.6d-201)) then
        tmp = -y
    else if (z <= 0.00031d0) then
        tmp = x * log(y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (-1.0 - (y / z));
	double tmp;
	if (z <= -1.15e-43) {
		tmp = t_1;
	} else if (z <= -1.6e-201) {
		tmp = -y;
	} else if (z <= 0.00031) {
		tmp = x * Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (-1.0 - (y / z))
	tmp = 0
	if z <= -1.15e-43:
		tmp = t_1
	elif z <= -1.6e-201:
		tmp = -y
	elif z <= 0.00031:
		tmp = x * math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(-1.0 - Float64(y / z)))
	tmp = 0.0
	if (z <= -1.15e-43)
		tmp = t_1;
	elseif (z <= -1.6e-201)
		tmp = Float64(-y);
	elseif (z <= 0.00031)
		tmp = Float64(x * log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (-1.0 - (y / z));
	tmp = 0.0;
	if (z <= -1.15e-43)
		tmp = t_1;
	elseif (z <= -1.6e-201)
		tmp = -y;
	elseif (z <= 0.00031)
		tmp = x * log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(-1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e-43], t$95$1, If[LessEqual[z, -1.6e-201], (-y), If[LessEqual[z, 0.00031], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1499999999999999e-43 or 3.1e-4 < z

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. add-cube-cbrt 99.6%

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

      3. associate-*r* 99.6%

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

      4. fma-neg 99.6%

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

      5. pow2 99.6%

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

      6. associate-+r- 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in z around inf 99.8%

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

      1. +-commutative 99.8%

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

      2. associate--r+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. associate-/l* 99.8%

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

      6. div-sub 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in y around inf 80.1%

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

       1. associate-*r/ 80.1%

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

       2. neg-mul-1 80.1%

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

   12. Simplified 80.1%

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

   if -1.1499999999999999e-43 < z < -1.6000000000000001e-201

   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 51.4%

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

      1. mul-1-neg 51.4%

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

   7. Simplified 51.4%

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

   if -1.6000000000000001e-201 < z < 3.1e-4

   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. Step-by-step derivation

      1. *-commutative 99.7%

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

      2. add-cube-cbrt 99.1%

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

      3. associate-*r* 99.1%

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

      4. fma-neg 99.1%

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

      5. pow2 99.1%

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

      6. associate-+r- 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in x around inf 46.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \left(-1 - \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-201}:\\ \;\;\;\;-y\\ \mathbf{elif}\;z \leq 0.00031:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-1 - \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+61} \lor \neg \left(x \leq 2.55 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.15e+61) (not (<= x 2.55e+54)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.15e+61) || !(x <= 2.55e+54)) {
		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 <= (-2.15d+61)) .or. (.not. (x <= 2.55d+54))) 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 <= -2.15e+61) || !(x <= 2.55e+54)) {
		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 <= -2.15e+61) or not (x <= 2.55e+54):
		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 <= -2.15e+61) || !(x <= 2.55e+54))
		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 <= -2.15e+61) || ~((x <= 2.55e+54)))
		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, -2.15e+61], N[Not[LessEqual[x, 2.55e+54]], $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 < -2.1500000000000001e61 or 2.55000000000000005e54 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 78.8%

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

   if -2.1500000000000001e61 < x < 2.55000000000000005e54

   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 x around 0 96.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+61} \lor \neg \left(x \leq 2.55 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 12.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-42} \lor \neg \left(z \leq 2.7 \cdot 10^{-100}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.2e-42) (not (<= z 2.7e-100))) (* z (- -1.0 (/ y z))) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.2e-42) || !(z <= 2.7e-100)) {
		tmp = z * (-1.0 - (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 ((z <= (-6.2d-42)) .or. (.not. (z <= 2.7d-100))) then
        tmp = z * ((-1.0d0) - (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 ((z <= -6.2e-42) || !(z <= 2.7e-100)) {
		tmp = z * (-1.0 - (y / z));
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.2e-42) or not (z <= 2.7e-100):
		tmp = z * (-1.0 - (y / z))
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.2e-42) || !(z <= 2.7e-100))
		tmp = Float64(z * Float64(-1.0 - Float64(y / z)));
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.2e-42) || ~((z <= 2.7e-100)))
		tmp = z * (-1.0 - (y / z));
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.2e-42], N[Not[LessEqual[z, 2.7e-100]], $MachinePrecision]], N[(z * N[(-1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if z < -6.2000000000000005e-42 or 2.70000000000000016e-100 < z

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. associate--l- 99.9%

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

   3. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. add-cube-cbrt 99.6%

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

      3. associate-*r* 99.6%

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

      4. fma-neg 99.5%

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

      5. pow2 99.5%

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

      6. associate-+r- 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in z around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. associate--r+ 98.7%

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

      3. +-commutative 98.7%

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

      4. associate--l+ 98.7%

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

      5. associate-/l* 98.7%

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

      6. div-sub 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in y around inf 76.1%

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

       1. associate-*r/ 76.1%

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

       2. neg-mul-1 76.1%

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

   12. Simplified 76.1%

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

   if -6.2000000000000005e-42 < z < 2.70000000000000016e-100

   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 37.4%

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

      1. mul-1-neg 37.4%

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

   7. Simplified 37.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-42} \lor \neg \left(z \leq 2.7 \cdot 10^{-100}\right):\\ \;\;\;\;z \cdot \left(-1 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.5% accurate, 29.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.2e+14) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+14) {
		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 <= 4.2d+14) 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 <= 4.2e+14) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.2e+14:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.2e+14)
		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 <= 4.2e+14)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.2e+14], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 4.2e14

   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 48.0%

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

      1. mul-1-neg 48.0%

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

   7. Simplified 48.0%

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

   if 4.2e14 < 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 y around inf 60.7%

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

      1. mul-1-neg 60.7%

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

   7. Simplified 60.7%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+14}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.5% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. associate--l- 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in y around inf 30.8%

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

   1. mul-1-neg 30.8%

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

7. Simplified 30.8%

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

8. Final simplification 30.8%

   \[\leadsto -y \]
9. Add Preprocessing

Alternative 10: 2.2% accurate, 209.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
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 60.0%

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

   1. sub-neg 60.0%

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

   2. add-sqr-sqrt 28.3%

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

   3. sqrt-unprod 30.2%

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

   4. sqr-neg 30.2%

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

   5. sqrt-unprod 14.1%

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

   6. add-sqr-sqrt 27.0%

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

   7. expm1-log1p-u 14.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \log y + z\right)\right)} \]

   8. fma-define 14.3%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, \log y, z\right)}\right)\right) \]

7. Applied egg-rr 14.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, \log y, z\right)\right)\right)} \]

8. Taylor expanded in x around 0 2.2%

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

9. Final simplification 2.2%

   \[\leadsto z \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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)))