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

Percentage Accurate: 99.9% → 99.9%

Time: 9.4s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\log y \cdot x - y\right) - z\\ t_2 := \left(-z\right) - y\\ \mathbf{if}\;t\_1 \leq -100000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* (log y) x) y) z)) (t_2 (- (- z) y)))
   (if (<= t_1 -100000.0) t_2 (if (<= t_1 0.02) (log t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((log(y) * x) - y) - z;
	double t_2 = -z - y;
	double tmp;
	if (t_1 <= -100000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.02) {
		tmp = log(t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((log(y) * x) - y) - z
    t_2 = -z - y
    if (t_1 <= (-100000.0d0)) then
        tmp = t_2
    else if (t_1 <= 0.02d0) then
        tmp = log(t)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((Math.log(y) * x) - y) - z;
	double t_2 = -z - y;
	double tmp;
	if (t_1 <= -100000.0) {
		tmp = t_2;
	} else if (t_1 <= 0.02) {
		tmp = Math.log(t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((math.log(y) * x) - y) - z
	t_2 = -z - y
	tmp = 0
	if t_1 <= -100000.0:
		tmp = t_2
	elif t_1 <= 0.02:
		tmp = math.log(t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(log(y) * x) - y) - z)
	t_2 = Float64(Float64(-z) - y)
	tmp = 0.0
	if (t_1 <= -100000.0)
		tmp = t_2;
	elseif (t_1 <= 0.02)
		tmp = log(t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((log(y) * x) - y) - z;
	t_2 = -z - y;
	tmp = 0.0;
	if (t_1 <= -100000.0)
		tmp = t_2;
	elseif (t_1 <= 0.02)
		tmp = log(t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[((-z) - y), $MachinePrecision]}, If[LessEqual[t$95$1, -100000.0], t$95$2, If[LessEqual[t$95$1, 0.02], N[Log[t], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -1e5 or 0.0200000000000000004 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 65.8

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

   7. Applied rewrites 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 65.9

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

   9. Applied rewrites 65.9%

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

   if -1e5 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 0.0200000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \log t \]
   10. Step-by-step derivation

   11. Applied rewrites 94.3%

       \[\leadsto \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\log y \cdot x - y\right) - z \leq -100000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;\left(\log y \cdot x - y\right) - z \leq 0.02:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -100000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)) (t_2 (- t_1 y)))
   (if (<= t_2 -100000.0) (- (- z) y) (if (<= t_2 4e+19) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -100000.0) {
		tmp = -z - y;
	} else if (t_2 <= 4e+19) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = log(y) * x
    t_2 = t_1 - y
    if (t_2 <= (-100000.0d0)) then
        tmp = -z - y
    else if (t_2 <= 4d+19) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -100000.0) {
		tmp = -z - y;
	} else if (t_2 <= 4e+19) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -100000.0:
		tmp = -z - y
	elif t_2 <= 4e+19:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -100000.0)
		tmp = Float64(Float64(-z) - y);
	elseif (t_2 <= 4e+19)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -100000.0)
		tmp = -z - y;
	elseif (t_2 <= 4e+19)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -100000.0], N[((-z) - y), $MachinePrecision], If[LessEqual[t$95$2, 4e+19], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e5

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.7

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

   7. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 77.9

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

   9. Applied rewrites 77.9%

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

   if -1e5 < (-.f64 (*.f64 x (log.f64 y)) y) < 4e19

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

   8. Applied rewrites 98.3%

      \[\leadsto \log t - z \]

   if 4e19 < (-.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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 83.2

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

   5. Applied rewrites 83.2%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -100000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{elif}\;\log y \cdot x - y \leq 4 \cdot 10^{+19}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -100000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* (log y) x) y) -100000.0) (- (- z) y) (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((log(y) * x) - y) <= -100000.0) {
		tmp = -z - y;
	} else {
		tmp = log(t) - z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((log(y) * x) - y) <= (-100000.0d0)) then
        tmp = -z - y
    else
        tmp = log(t) - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((Math.log(y) * x) - y) <= -100000.0) {
		tmp = -z - y;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((math.log(y) * x) - y) <= -100000.0:
		tmp = -z - y
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(log(y) * x) - y) <= -100000.0)
		tmp = Float64(Float64(-z) - y);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((log(y) * x) - y) <= -100000.0)
		tmp = -z - y;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - y), $MachinePrecision], -100000.0], N[((-z) - y), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -1e5

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 77.7

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

   7. Applied rewrites 77.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 77.9

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

   9. Applied rewrites 77.9%

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

   if -1e5 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - z \]
   7. Step-by-step derivation

   8. Applied rewrites 61.7%

      \[\leadsto \log t - z \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log y \cdot x - y \leq -100000:\\ \;\;\;\;\left(-z\right) - y\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;\log y \cdot x - z\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+146}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.2e+18)
   (- (* (log y) x) z)
   (if (<= x 3.4e+146) (- (- (log t) y) z) (fma (log y) x (- y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e+18) {
		tmp = (log(y) * x) - z;
	} else if (x <= 3.4e+146) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = fma(log(y), x, -y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.2e+18)
		tmp = Float64(Float64(log(y) * x) - z);
	elseif (x <= 3.4e+146)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = fma(log(y), x, Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.2e+18], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 3.4e+146], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2e18

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y - z \]
   7. Step-by-step derivation

   8. Applied rewrites 85.4%

      \[\leadsto \log y \cdot x - z \]

   if -4.2e18 < x < 3.39999999999999991e146

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 3.39999999999999991e146 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate--l- N/A

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

      5. associate-+l- N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. associate-+r- N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 94.1

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

   7. Applied rewrites 94.1%

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

Alternative 6: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - z\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+136}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) z)))
   (if (<= x -4.2e+18) t_1 (if (<= x 8.2e+136) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - z;
	double tmp;
	if (x <= -4.2e+18) {
		tmp = t_1;
	} else if (x <= 8.2e+136) {
		tmp = (log(t) - y) - z;
	} 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 = (log(y) * x) - z
    if (x <= (-4.2d+18)) then
        tmp = t_1
    else if (x <= 8.2d+136) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.log(y) * x) - z;
	double tmp;
	if (x <= -4.2e+18) {
		tmp = t_1;
	} else if (x <= 8.2e+136) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - z
	tmp = 0
	if x <= -4.2e+18:
		tmp = t_1
	elif x <= 8.2e+136:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - z)
	tmp = 0.0
	if (x <= -4.2e+18)
		tmp = t_1;
	elseif (x <= 8.2e+136)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - z;
	tmp = 0.0;
	if (x <= -4.2e+18)
		tmp = t_1;
	elseif (x <= 8.2e+136)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -4.2e+18], t$95$1, If[LessEqual[x, 8.2e+136], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e18 or 8.1999999999999995e136 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \log y - z \]
   7. Step-by-step derivation

   8. Applied rewrites 85.1%

      \[\leadsto \log y \cdot x - z \]

   if -4.2e18 < x < 8.1999999999999995e136

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 97.4

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

   5. Applied rewrites 97.4%

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

Alternative 7: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.28 \cdot 10^{+107}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+146}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -1.28e+107) t_1 (if (<= x 3.8e+146) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -1.28e+107) {
		tmp = t_1;
	} else if (x <= 3.8e+146) {
		tmp = (log(t) - y) - z;
	} 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 = log(y) * x
    if (x <= (-1.28d+107)) then
        tmp = t_1
    else if (x <= 3.8d+146) then
        tmp = (log(t) - y) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -1.28e+107) {
		tmp = t_1;
	} else if (x <= 3.8e+146) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -1.28e+107:
		tmp = t_1
	elif x <= 3.8e+146:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -1.28e+107)
		tmp = t_1;
	elseif (x <= 3.8e+146)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -1.28e+107)
		tmp = t_1;
	elseif (x <= 3.8e+146)
		tmp = (log(t) - y) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.28e+107], t$95$1, If[LessEqual[x, 3.8e+146], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2799999999999999e107 or 3.79999999999999979e146 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.9

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

   5. Applied rewrites 77.9%

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

   if -1.2799999999999999e107 < x < 3.79999999999999979e146

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--r+ N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 8: 48.1% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.15 \cdot 10^{+93}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 2.15e+93) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2.15e+93) {
		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 <= 2.15d+93) 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 <= 2.15e+93) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 2.15e+93:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2.15e+93)
		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 <= 2.15e+93)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2.15e+93], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 2.15e93

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 37.8

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

   5. Applied rewrites 37.8%

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

   if 2.15e93 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 68.3

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

   5. Applied rewrites 68.3%

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

Alternative 9: 58.3% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 57.7

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

7. Applied rewrites 57.7%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. remove-double-div 57.8

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

9. Applied rewrites 57.8%

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

Alternative 10: 29.6% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. lower-neg.f64 29.8

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

5. Applied rewrites 29.8%

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

Reproduce

herbie shell --seed 2024255 
(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)))