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

Percentage Accurate: 99.9% → 99.9%

Time: 9.0s

Alternatives: 9

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. Final simplification 99.8%

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

Alternative 2: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+89} \lor \neg \left(z \leq 1.15 \cdot 10^{+95}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2e+89) (not (<= z 1.15e+95)))
   (- (log t) (+ y z))
   (- (+ (log t) (* x (log y))) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2e+89) || !(z <= 1.15e+95)) {
		tmp = log(t) - (y + z);
	} else {
		tmp = (log(t) + (x * log(y))) - 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 <= (-2d+89)) .or. (.not. (z <= 1.15d+95))) then
        tmp = log(t) - (y + z)
    else
        tmp = (log(t) + (x * log(y))) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2e+89) || !(z <= 1.15e+95)) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = (Math.log(t) + (x * Math.log(y))) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2e+89) or not (z <= 1.15e+95):
		tmp = math.log(t) - (y + z)
	else:
		tmp = (math.log(t) + (x * math.log(y))) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2e+89) || !(z <= 1.15e+95))
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(Float64(log(t) + Float64(x * log(y))) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2e+89) || ~((z <= 1.15e+95)))
		tmp = log(t) - (y + z);
	else
		tmp = (log(t) + (x * log(y))) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2e+89], N[Not[LessEqual[z, 1.15e+95]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999999e89 or 1.14999999999999999e95 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 85.3%

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

   if -1.99999999999999999e89 < z < 1.14999999999999999e95

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 95.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+89} \lor \neg \left(z \leq 1.15 \cdot 10^{+95}\right):\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 3: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t + x \cdot \log y\\ \mathbf{if}\;y \leq 9 \cdot 10^{+80}:\\ \;\;\;\;t_1 - z\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+183}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (log t) (* x (log y)))))
   (if (<= y 9e+80)
     (- t_1 z)
     (if (<= y 1.75e+183) (- (log t) (+ y z)) (- t_1 y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) + (x * log(y));
	double tmp;
	if (y <= 9e+80) {
		tmp = t_1 - z;
	} else if (y <= 1.75e+183) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(t) + (x * log(y))
    if (y <= 9d+80) then
        tmp = t_1 - z
    else if (y <= 1.75d+183) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(t) + (x * Math.log(y));
	double tmp;
	if (y <= 9e+80) {
		tmp = t_1 - z;
	} else if (y <= 1.75e+183) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(t) + (x * math.log(y))
	tmp = 0
	if y <= 9e+80:
		tmp = t_1 - z
	elif y <= 1.75e+183:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1 - y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) + Float64(x * log(y)))
	tmp = 0.0
	if (y <= 9e+80)
		tmp = Float64(t_1 - z);
	elseif (y <= 1.75e+183)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = Float64(t_1 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(t) + (x * log(y));
	tmp = 0.0;
	if (y <= 9e+80)
		tmp = t_1 - z;
	elseif (y <= 1.75e+183)
		tmp = log(t) - (y + z);
	else
		tmp = t_1 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e+80], N[(t$95$1 - z), $MachinePrecision], If[LessEqual[y, 1.75e+183], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.00000000000000013e80

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 95.3%

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

   if 9.00000000000000013e80 < y < 1.74999999999999994e183

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 91.5%

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

   if 1.74999999999999994e183 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 91.1%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+80}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+183}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - y\\ \end{array} \]

Alternative 4: 60.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \log t - y\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+94}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+82}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (log t) y)))
   (if (<= z -1.2e+94)
     (- z)
     (if (<= z -3.2e-40)
       t_2
       (if (<= z -3.2e-58)
         t_1
         (if (<= z 3.95e-10)
           t_2
           (if (<= z 2.4e+52) t_1 (if (<= z 1.08e+82) (- y) (- z)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = log(t) - y;
	double tmp;
	if (z <= -1.2e+94) {
		tmp = -z;
	} else if (z <= -3.2e-40) {
		tmp = t_2;
	} else if (z <= -3.2e-58) {
		tmp = t_1;
	} else if (z <= 3.95e-10) {
		tmp = t_2;
	} else if (z <= 2.4e+52) {
		tmp = t_1;
	} else if (z <= 1.08e+82) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = log(t) - y
    if (z <= (-1.2d+94)) then
        tmp = -z
    else if (z <= (-3.2d-40)) then
        tmp = t_2
    else if (z <= (-3.2d-58)) then
        tmp = t_1
    else if (z <= 3.95d-10) then
        tmp = t_2
    else if (z <= 2.4d+52) then
        tmp = t_1
    else if (z <= 1.08d+82) then
        tmp = -y
    else
        tmp = -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 = Math.log(t) - y;
	double tmp;
	if (z <= -1.2e+94) {
		tmp = -z;
	} else if (z <= -3.2e-40) {
		tmp = t_2;
	} else if (z <= -3.2e-58) {
		tmp = t_1;
	} else if (z <= 3.95e-10) {
		tmp = t_2;
	} else if (z <= 2.4e+52) {
		tmp = t_1;
	} else if (z <= 1.08e+82) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = math.log(t) - y
	tmp = 0
	if z <= -1.2e+94:
		tmp = -z
	elif z <= -3.2e-40:
		tmp = t_2
	elif z <= -3.2e-58:
		tmp = t_1
	elif z <= 3.95e-10:
		tmp = t_2
	elif z <= 2.4e+52:
		tmp = t_1
	elif z <= 1.08e+82:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(log(t) - y)
	tmp = 0.0
	if (z <= -1.2e+94)
		tmp = Float64(-z);
	elseif (z <= -3.2e-40)
		tmp = t_2;
	elseif (z <= -3.2e-58)
		tmp = t_1;
	elseif (z <= 3.95e-10)
		tmp = t_2;
	elseif (z <= 2.4e+52)
		tmp = t_1;
	elseif (z <= 1.08e+82)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = log(t) - y;
	tmp = 0.0;
	if (z <= -1.2e+94)
		tmp = -z;
	elseif (z <= -3.2e-40)
		tmp = t_2;
	elseif (z <= -3.2e-58)
		tmp = t_1;
	elseif (z <= 3.95e-10)
		tmp = t_2;
	elseif (z <= 2.4e+52)
		tmp = t_1;
	elseif (z <= 1.08e+82)
		tmp = -y;
	else
		tmp = -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[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[z, -1.2e+94], (-z), If[LessEqual[z, -3.2e-40], t$95$2, If[LessEqual[z, -3.2e-58], t$95$1, If[LessEqual[z, 3.95e-10], t$95$2, If[LessEqual[z, 2.4e+52], t$95$1, If[LessEqual[z, 1.08e+82], (-y), (-z)]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.19999999999999991e94 or 1.08e82 < z

   1. Initial program 99.9%

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

   2. Taylor expanded in z around inf 67.2%

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

      1. neg-mul-1 67.2%

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

   4. Simplified 67.2%

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

   if -1.19999999999999991e94 < z < -3.20000000000000002e-40 or -3.2000000000000001e-58 < z < 3.9499999999999998e-10

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 97.0%

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

   3. Taylor expanded in x around 0 63.1%

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

   if -3.20000000000000002e-40 < z < -3.2000000000000001e-58 or 3.9499999999999998e-10 < z < 2.4e52

   1. Initial program 99.3%

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

      1. associate--l- 99.3%

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

      2. associate-+l- 99.3%

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

      3. *-commutative 99.3%

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

      4. add-cube-cbrt 98.6%

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

      5. associate-*l* 98.8%

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

      6. fma-neg 98.8%

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

      7. pow2 98.8%

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

   3. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. sub-neg 98.8%

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

      3. log-rec 98.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      4. distribute-neg-in 98.8%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      5. log-rec 98.8%

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

      6. remove-double-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. sub-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 68.8%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 68.8%

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

      2. *-lft-identity 68.8%

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

   8. Simplified 68.8%

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

   if 2.4e52 < z < 1.08e82

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 75.4%

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

      1. neg-mul-1 75.4%

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

   4. Simplified 75.4%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+94}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-40}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 3.95 \cdot 10^{-10}:\\ \;\;\;\;\log t - y\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+52}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+82}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 5: 49.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 2.4 \cdot 10^{-198}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+39}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= y 2.4e-198)
     t_1
     (if (<= y 3.9e+39) (- z) (if (<= y 2.5e+81) t_1 (- y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (y <= 2.4e-198) {
		tmp = t_1;
	} else if (y <= 3.9e+39) {
		tmp = -z;
	} else if (y <= 2.5e+81) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (y <= 2.4d-198) then
        tmp = t_1
    else if (y <= 3.9d+39) then
        tmp = -z
    else if (y <= 2.5d+81) then
        tmp = t_1
    else
        tmp = -y
    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 (y <= 2.4e-198) {
		tmp = t_1;
	} else if (y <= 3.9e+39) {
		tmp = -z;
	} else if (y <= 2.5e+81) {
		tmp = t_1;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if y <= 2.4e-198:
		tmp = t_1
	elif y <= 3.9e+39:
		tmp = -z
	elif y <= 2.5e+81:
		tmp = t_1
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 2.4e-198)
		tmp = t_1;
	elseif (y <= 3.9e+39)
		tmp = Float64(-z);
	elseif (y <= 2.5e+81)
		tmp = t_1;
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (y <= 2.4e-198)
		tmp = t_1;
	elseif (y <= 3.9e+39)
		tmp = -z;
	elseif (y <= 2.5e+81)
		tmp = t_1;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.4e-198], t$95$1, If[LessEqual[y, 3.9e+39], (-z), If[LessEqual[y, 2.5e+81], t$95$1, (-y)]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.39999999999999986e-198 or 3.9000000000000001e39 < y < 2.4999999999999999e81

   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%

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

      2. associate-+l- 99.7%

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

      3. *-commutative 99.7%

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

      4. add-cube-cbrt 99.1%

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

      5. associate-*l* 99.1%

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

      6. fma-neg 99.1%

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

      7. pow2 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. *-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. log-rec 99.1%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      4. distribute-neg-in 99.1%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      5. log-rec 99.1%

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

      6. remove-double-neg 99.1%

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

      7. +-commutative 99.1%

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

      8. sub-neg 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in x around inf 53.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 53.5%

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

      2. *-lft-identity 53.5%

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

   8. Simplified 53.5%

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

   if 2.39999999999999986e-198 < y < 3.9000000000000001e39

   1. Initial program 99.8%

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

   2. Taylor expanded in z around inf 43.3%

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

      1. neg-mul-1 43.3%

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

   4. Simplified 43.3%

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

   if 2.4999999999999999e81 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 68.9%

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

      1. neg-mul-1 68.9%

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

   4. Simplified 68.9%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-198}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+39}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 6: 83.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+188} \lor \neg \left(x \leq 6.1 \cdot 10^{+187}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e+188) (not (<= x 6.1e+187)))
   (* x (log y))
   (- (log t) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e+188) || !(x <= 6.1e+187)) {
		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 <= (-2.5d+188)) .or. (.not. (x <= 6.1d+187))) 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 <= -2.5e+188) || !(x <= 6.1e+187)) {
		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 <= -2.5e+188) or not (x <= 6.1e+187):
		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 <= -2.5e+188) || !(x <= 6.1e+187))
		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 <= -2.5e+188) || ~((x <= 6.1e+187)))
		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, -2.5e+188], N[Not[LessEqual[x, 6.1e+187]], $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 < -2.5000000000000001e188 or 6.0999999999999996e187 < 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%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. add-cube-cbrt 98.6%

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

      5. associate-*l* 98.6%

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

      6. fma-neg 98.6%

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

      7. pow2 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. *-commutative 98.6%

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

      2. sub-neg 98.6%

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

      3. log-rec 98.6%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      4. distribute-neg-in 98.6%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      5. log-rec 98.6%

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

      6. remove-double-neg 98.6%

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

      7. +-commutative 98.6%

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

      8. sub-neg 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in x around inf 76.3%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 76.3%

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

      2. *-lft-identity 76.3%

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

   8. Simplified 76.3%

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

   if -2.5000000000000001e188 < x < 6.0999999999999996e187

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 83.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+188} \lor \neg \left(x \leq 6.1 \cdot 10^{+187}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \end{array} \]

Alternative 7: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+42}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.2e+42) (- (log t) z) (if (<= y 1.6e+81) (* x (log y)) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.2e+42) {
		tmp = log(t) - z;
	} else if (y <= 1.6e+81) {
		tmp = x * log(y);
	} 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 <= 6.2d+42) then
        tmp = log(t) - z
    else if (y <= 1.6d+81) then
        tmp = x * log(y)
    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 <= 6.2e+42) {
		tmp = Math.log(t) - z;
	} else if (y <= 1.6e+81) {
		tmp = x * Math.log(y);
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 6.2e+42:
		tmp = math.log(t) - z
	elif y <= 1.6e+81:
		tmp = x * math.log(y)
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.2e+42)
		tmp = Float64(log(t) - z);
	elseif (y <= 1.6e+81)
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 6.2e+42)
		tmp = log(t) - z;
	elseif (y <= 1.6e+81)
		tmp = x * log(y);
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6.2e+42], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], If[LessEqual[y, 1.6e+81], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-y)]]

Derivation

1. Split input into 3 regimes

2. if y < 6.2000000000000003e42

   1. Initial program 99.8%

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

   2. Taylor expanded in y around 0 97.0%

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

   3. Taylor expanded in x around 0 62.0%

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

   if 6.2000000000000003e42 < y < 1.6e81

   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%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. add-cube-cbrt 99.0%

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

      5. associate-*l* 98.9%

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

      6. fma-neg 98.9%

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

      7. pow2 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. *-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. log-rec 98.9%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, -\left(\left(y + z\right) + \color{blue}{\log \left(\frac{1}{t}\right)}\right)\right) \]

      4. distribute-neg-in 98.9%

         \[\leadsto \mathsf{fma}\left({\left(\sqrt[3]{\log y}\right)}^{2}, x \cdot \sqrt[3]{\log y}, \color{blue}{\left(-\left(y + z\right)\right) + \left(-\log \left(\frac{1}{t}\right)\right)}\right) \]

      5. log-rec 98.9%

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

      6. remove-double-neg 98.9%

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

      7. +-commutative 98.9%

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

      8. sub-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in x around inf 57.5%

      \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \left(\log y \cdot x\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 57.5%

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

      2. *-lft-identity 57.5%

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

   8. Simplified 57.5%

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

   if 1.6e81 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 68.9%

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

      1. neg-mul-1 68.9%

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

   4. Simplified 68.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+42}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+81}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 8: 48.4% accurate, 51.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.05e+86) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.05e+86) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 2.05d+86) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.05e+86) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.05e+86:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.05e+86)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 2.05e+86)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.05e+86], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.05e86

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 37.4%

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

      1. neg-mul-1 37.4%

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

   4. Simplified 37.4%

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

   if 2.05e86 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 70.2%

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

      1. neg-mul-1 70.2%

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

   4. Simplified 70.2%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.05 \cdot 10^{+86}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \]

Alternative 9: 30.2% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in y around inf 30.3%

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

   1. neg-mul-1 30.3%

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

4. Simplified 30.3%

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

5. Final simplification 30.3%

   \[\leadsto -y \]

Reproduce

herbie shell --seed 2023176 
(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)))