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

Percentage Accurate: 99.9% → 99.9%

Time: 12.6s

Alternatives: 11

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $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. 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. Final simplification 99.9%

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

Alternative 2: 89.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.1 \cdot 10^{-16}:\\ \;\;\;\;\left(t\_1 + \log t\right) - y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5.1e-16)
     (- (+ t_1 (log t)) y)
     (if (<= x 1.2e+46) (- (- (log t) z) y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -5.1e-16) {
		tmp = (t_1 + log(t)) - y;
	} else if (x <= 1.2e+46) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-5.1d-16)) then
        tmp = (t_1 + log(t)) - y
    else if (x <= 1.2d+46) then
        tmp = (log(t) - z) - y
    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 tmp;
	if (x <= -5.1e-16) {
		tmp = (t_1 + Math.log(t)) - y;
	} else if (x <= 1.2e+46) {
		tmp = (Math.log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -5.1e-16:
		tmp = (t_1 + math.log(t)) - y
	elif x <= 1.2e+46:
		tmp = (math.log(t) - z) - y
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -5.1e-16)
		tmp = Float64(Float64(t_1 + log(t)) - y);
	elseif (x <= 1.2e+46)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -5.1e-16)
		tmp = (t_1 + log(t)) - y;
	elseif (x <= 1.2e+46)
		tmp = (log(t) - z) - y;
	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]}, If[LessEqual[x, -5.1e-16], N[(N[(t$95$1 + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 1.2e+46], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1e-16

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate--l+ 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 95.1%

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

   if -5.1e-16 < x < 1.20000000000000004e46

   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- 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. Step-by-step derivation

      1. add-log-exp 26.9%

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

      2. associate-+r- 26.9%

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

      3. exp-diff 26.9%

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

      4. add-exp-log 26.9%

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

   6. Applied egg-rr 26.9%

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

   7. Taylor expanded in x around 0 25.6%

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

      1. mul-1-neg 25.6%

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

   9. Simplified 25.6%

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

   10. Taylor expanded in t around 0 98.7%

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

       1. neg-mul-1 98.7%

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

       2. sub-neg 98.7%

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

   12. Simplified 98.7%

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

   if 1.20000000000000004e46 < 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 z around inf 87.2%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{-16}:\\ \;\;\;\;\left(x \cdot \log y + \log t\right) - y\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -260 \lor \neg \left(x \leq 1.55 \cdot 10^{+42}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -260.0) (not (<= x 1.55e+42)))
   (- (* x (log y)) y)
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -260.0) || !(x <= 1.55e+42)) {
		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 <= (-260.0d0)) .or. (.not. (x <= 1.55d+42))) 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 <= -260.0) || !(x <= 1.55e+42)) {
		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 <= -260.0) or not (x <= 1.55e+42):
		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 <= -260.0) || !(x <= 1.55e+42))
		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 <= -260.0) || ~((x <= 1.55e+42)))
		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, -260.0], N[Not[LessEqual[x, 1.55e+42]], $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 < -260 or 1.5500000000000001e42 < 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 87.4%

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

   if -260 < x < 1.5500000000000001e42

   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 x around 0 98.6%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -260 \lor \neg \left(x \leq 1.55 \cdot 10^{+42}\right):\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -260:\\ \;\;\;\;t\_1 - y\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+45}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -260.0)
     (- t_1 y)
     (if (<= x 1.18e+45) (- (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 tmp;
	if (x <= -260.0) {
		tmp = t_1 - y;
	} else if (x <= 1.18e+45) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-260.0d0)) then
        tmp = t_1 - y
    else if (x <= 1.18d+45) 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 tmp;
	if (x <= -260.0) {
		tmp = t_1 - y;
	} else if (x <= 1.18e+45) {
		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)
	tmp = 0
	if x <= -260.0:
		tmp = t_1 - y
	elif x <= 1.18e+45:
		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))
	tmp = 0.0
	if (x <= -260.0)
		tmp = Float64(t_1 - y);
	elseif (x <= 1.18e+45)
		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);
	tmp = 0.0;
	if (x <= -260.0)
		tmp = t_1 - y;
	elseif (x <= 1.18e+45)
		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]}, If[LessEqual[x, -260.0], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 1.18e+45], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -260

   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 93.6%

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

   if -260 < x < 1.17999999999999993e45

   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 x around 0 98.6%

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

   if 1.17999999999999993e45 < 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 z around inf 87.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -260:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{+45}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -260:\\ \;\;\;\;t\_1 - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -260.0)
     (- t_1 y)
     (if (<= x 5.6e+45) (- (- (log t) z) y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -260.0) {
		tmp = t_1 - y;
	} else if (x <= 5.6e+45) {
		tmp = (log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-260.0d0)) then
        tmp = t_1 - y
    else if (x <= 5.6d+45) then
        tmp = (log(t) - z) - y
    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 tmp;
	if (x <= -260.0) {
		tmp = t_1 - y;
	} else if (x <= 5.6e+45) {
		tmp = (Math.log(t) - z) - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -260.0:
		tmp = t_1 - y
	elif x <= 5.6e+45:
		tmp = (math.log(t) - z) - y
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -260.0)
		tmp = Float64(t_1 - y);
	elseif (x <= 5.6e+45)
		tmp = Float64(Float64(log(t) - z) - y);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -260.0)
		tmp = t_1 - y;
	elseif (x <= 5.6e+45)
		tmp = (log(t) - z) - y;
	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]}, If[LessEqual[x, -260.0], N[(t$95$1 - y), $MachinePrecision], If[LessEqual[x, 5.6e+45], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -260

   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 93.6%

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

   if -260 < x < 5.5999999999999999e45

   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- 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. Step-by-step derivation

      1. add-log-exp 27.2%

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

      2. associate-+r- 27.2%

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

      3. exp-diff 27.2%

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

      4. add-exp-log 27.2%

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

   6. Applied egg-rr 27.2%

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

   7. Taylor expanded in x around 0 25.9%

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

      1. mul-1-neg 25.9%

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

   9. Simplified 25.9%

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

   10. Taylor expanded in t around 0 98.7%

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

       1. neg-mul-1 98.7%

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

       2. sub-neg 98.7%

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

   12. Simplified 98.7%

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

   if 5.5999999999999999e45 < 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 z around inf 87.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -260:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+45}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+54}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2.1e+54) (- (log t) z) (- (log t) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.1e+54) {
		tmp = log(t) - z;
	} else {
		tmp = log(t) - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.1d+54) then
        tmp = log(t) - z
    else
        tmp = log(t) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.1e+54) {
		tmp = Math.log(t) - z;
	} else {
		tmp = Math.log(t) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.1e+54:
		tmp = math.log(t) - z
	else:
		tmp = math.log(t) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.1e+54)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(log(t) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.1e+54)
		tmp = log(t) - z;
	else
		tmp = log(t) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.1e+54], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.09999999999999986e54

   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 x around 0 66.1%

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

   6. Taylor expanded in y around 0 59.3%

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

   if 2.09999999999999986e54 < y

   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 68.4%

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

      1. mul-1-neg 68.4%

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

   7. Simplified 68.4%

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

   8. Taylor expanded in y around 0 68.4%

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

      1. mul-1-neg 68.4%

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

      2. sub-neg 68.4%

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

   10. Simplified 68.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+54}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log t - y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[t], $MachinePrecision] - N[(y + z), $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. Taylor expanded in x around 0 72.1%

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

6. Final simplification 72.1%

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

Alternative 9: 42.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[t], $MachinePrecision] - y), $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. Taylor expanded in y around inf 46.1%

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

   1. mul-1-neg 46.1%

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

7. Simplified 46.1%

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

8. Taylor expanded in y around 0 46.1%

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

   1. mul-1-neg 46.1%

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

   2. sub-neg 46.1%

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

10. Simplified 46.1%

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

11. Final simplification 46.1%

    \[\leadsto \log t - y \]
12. Add Preprocessing

Alternative 10: 14.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \log t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Log[t], $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. Taylor expanded in y around inf 46.1%

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

   1. mul-1-neg 46.1%

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

7. Simplified 46.1%

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

8. Taylor expanded in y around 0 14.3%

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

9. Final simplification 14.3%

   \[\leadsto \log t \]
10. Add Preprocessing

Alternative 11: 2.2% 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 46.1%

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

   1. mul-1-neg 46.1%

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

7. Simplified 46.1%

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

   1. *-un-lft-identity 46.1%

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

   2. add-sqr-sqrt 0.0%

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

   3. sqrt-unprod 13.5%

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

   4. sqr-neg 13.5%

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

   5. sqrt-unprod 13.5%

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

   6. add-sqr-sqrt 13.5%

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

9. Applied egg-rr 13.5%

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

    1. *-lft-identity 13.5%

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

11. Simplified 13.5%

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

12. Taylor expanded in y around inf 2.2%

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

13. Final simplification 2.2%

    \[\leadsto y \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(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)))