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

Percentage Accurate: 99.9% → 99.9%

Time: 10.0s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 55.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_1 \leq 10^{-11}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -1000.0) (- y) (if (<= t_1 1e-11) (log t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = -y;
	} else if (t_1 <= 1e-11) {
		tmp = log(t);
	} 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) :: tmp
    t_1 = ((x * log(y)) - y) - z
    if (t_1 <= (-1000.0d0)) then
        tmp = -y
    else if (t_1 <= 1d-11) then
        tmp = log(t)
    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)) - y) - z;
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = -y;
	} else if (t_1 <= 1e-11) {
		tmp = Math.log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	tmp = 0
	if t_1 <= -1000.0:
		tmp = -y
	elif t_1 <= 1e-11:
		tmp = math.log(t)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = Float64(-y);
	elseif (t_1 <= 1e-11)
		tmp = log(t);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * log(y)) - y) - z;
	tmp = 0.0;
	if (t_1 <= -1000.0)
		tmp = -y;
	elseif (t_1 <= 1e-11)
		tmp = log(t);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], (-y), If[LessEqual[t$95$1, 1e-11], N[Log[t], $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -1e3

   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 47.1

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

   5. Applied rewrites 47.1%

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

   if -1e3 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 9.99999999999999939e-12

   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 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 96.2%

       \[\leadsto \log t \]

   if 9.99999999999999939e-12 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 45.0

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

   5. Applied rewrites 45.0%

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

Alternative 3: 68.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\left(-y\right) + \log t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+81)
     (+ (- y) (log t))
     (if (<= t_2 2e+92) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+81) {
		tmp = -y + log(t);
	} else if (t_2 <= 2e+92) {
		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 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-5d+81)) then
        tmp = -y + log(t)
    else if (t_2 <= 2d+92) 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 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+81) {
		tmp = -y + Math.log(t);
	} else if (t_2 <= 2e+92) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -5e+81:
		tmp = -y + math.log(t)
	elif t_2 <= 2e+92:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -5e+81)
		tmp = Float64(Float64(-y) + log(t));
	elseif (t_2 <= 2e+92)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -5e+81)
		tmp = -y + log(t);
	elseif (t_2 <= 2e+92)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+81], N[((-y) + N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+92], 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) < -4.9999999999999998e81

   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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 56.3

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

   5. Applied rewrites 56.3%

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

   if -4.9999999999999998e81 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.0000000000000001e92

   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 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.9%

      \[\leadsto \log t - z \]

   if 2.0000000000000001e92 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+81}:\\ \;\;\;\;\left(-y\right) + \log t\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+81}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+81) (- y) (if (<= t_2 2e+92) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+81) {
		tmp = -y;
	} else if (t_2 <= 2e+92) {
		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 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-5d+81)) then
        tmp = -y
    else if (t_2 <= 2d+92) 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 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+81) {
		tmp = -y;
	} else if (t_2 <= 2e+92) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -5e+81:
		tmp = -y
	elif t_2 <= 2e+92:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -5e+81)
		tmp = Float64(-y);
	elseif (t_2 <= 2e+92)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -5e+81)
		tmp = -y;
	elseif (t_2 <= 2e+92)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+81], (-y), If[LessEqual[t$95$2, 2e+92], 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) < -4.9999999999999998e81

   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 56.2

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

   5. Applied rewrites 56.2%

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

   if -4.9999999999999998e81 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.0000000000000001e92

   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 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 75.9%

      \[\leadsto \log t - z \]

   if 2.0000000000000001e92 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 85.7

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

   5. Applied rewrites 85.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+81}:\\ \;\;\;\;-y\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{+92}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, \log t\right)\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;z \leq 660000000:\\ \;\;\;\;t\_1 - y\\ \mathbf{else}:\\ \;\;\;\;t\_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (log t))))
   (if (<= z -1.2e+68)
     (- (- (log t) y) z)
     (if (<= z 660000000.0) (- t_1 y) (- t_1 z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, log(t));
	double tmp;
	if (z <= -1.2e+68) {
		tmp = (log(t) - y) - z;
	} else if (z <= 660000000.0) {
		tmp = t_1 - y;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, log(t))
	tmp = 0.0
	if (z <= -1.2e+68)
		tmp = Float64(Float64(log(t) - y) - z);
	elseif (z <= 660000000.0)
		tmp = Float64(t_1 - y);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e+68], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 660000000.0], N[(t$95$1 - y), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000004e68

   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 89.4

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

   5. Applied rewrites 89.4%

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

   if -1.20000000000000004e68 < z < 6.6e8

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 6.6e8 < z

   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 90.0

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

   5. Applied rewrites 90.0%

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

Alternative 6: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;z \leq 880000000:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \log t\right) - y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.2e+68)
   (- (- (log t) y) z)
   (if (<= z 880000000.0)
     (- (fma (log y) x (log t)) y)
     (fma (log y) x (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.2e+68) {
		tmp = (log(t) - y) - z;
	} else if (z <= 880000000.0) {
		tmp = fma(log(y), x, log(t)) - y;
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.2e+68)
		tmp = Float64(Float64(log(t) - y) - z);
	elseif (z <= 880000000.0)
		tmp = Float64(fma(log(y), x, log(t)) - y);
	else
		tmp = fma(log(y), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.2e+68], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[z, 880000000.0], N[(N[(N[Log[y], $MachinePrecision] * x + N[Log[t], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000004e68

   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 89.4

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

   5. Applied rewrites 89.4%

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

   if -1.20000000000000004e68 < z < 8.8e8

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-log.f64 98.3

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

   5. Applied rewrites 98.3%

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

   if 8.8e8 < z

   1. Initial program 99.7%

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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.8

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

   7. Applied rewrites 88.8%

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

Alternative 7: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 8: 89.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -z\right)\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 820000000000:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- z))))
   (if (<= x -4.6e+21)
     t_1
     (if (<= x 820000000000.0)
       (- (- (log t) y) z)
       (if (<= x 7.8e+129) (fma (log y) x (- y)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -z);
	double tmp;
	if (x <= -4.6e+21) {
		tmp = t_1;
	} else if (x <= 820000000000.0) {
		tmp = (log(t) - y) - z;
	} else if (x <= 7.8e+129) {
		tmp = fma(log(y), x, -y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-z))
	tmp = 0.0
	if (x <= -4.6e+21)
		tmp = t_1;
	elseif (x <= 820000000000.0)
		tmp = Float64(Float64(log(t) - y) - z);
	elseif (x <= 7.8e+129)
		tmp = fma(log(y), x, Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]}, If[LessEqual[x, -4.6e+21], t$95$1, If[LessEqual[x, 820000000000.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 7.8e+129], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e21 or 7.7999999999999994e129 < x

   1. Initial program 99.7%

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

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.9

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

   7. Applied rewrites 91.9%

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

   if -4.6e21 < x < 8.2e11

   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 98.6

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

   5. Applied rewrites 98.6%

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

   if 8.2e11 < x < 7.7999999999999994e129

   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. 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 99.9

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

   4. Applied rewrites 99.9%

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

   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 90.5

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

   7. Applied rewrites 90.5%

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

Alternative 9: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 820000000000:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- y))))
   (if (<= x -1.15e+38)
     t_1
     (if (<= x 820000000000.0) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -y);
	double tmp;
	if (x <= -1.15e+38) {
		tmp = t_1;
	} else if (x <= 820000000000.0) {
		tmp = (log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-y))
	tmp = 0.0
	if (x <= -1.15e+38)
		tmp = t_1;
	elseif (x <= 820000000000.0)
		tmp = Float64(Float64(log(t) - y) - z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision]}, If[LessEqual[x, -1.15e+38], t$95$1, If[LessEqual[x, 820000000000.0], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1500000000000001e38 or 8.2e11 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 83.6

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

   7. Applied rewrites 83.6%

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

   if -1.1500000000000001e38 < x < 8.2e11

   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 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 10: 83.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5e+125) t_1 (if (<= x 5.2e+57) (- (- (log t) y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -5e+125) {
		tmp = t_1;
	} else if (x <= 5.2e+57) {
		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 = x * log(y)
    if (x <= (-5d+125)) then
        tmp = t_1
    else if (x <= 5.2d+57) 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 = x * Math.log(y);
	double tmp;
	if (x <= -5e+125) {
		tmp = t_1;
	} else if (x <= 5.2e+57) {
		tmp = (Math.log(t) - y) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -5e+125:
		tmp = t_1
	elif x <= 5.2e+57:
		tmp = (math.log(t) - y) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -5e+125)
		tmp = t_1;
	elseif (x <= 5.2e+57)
		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 = x * log(y);
	tmp = 0.0;
	if (x <= -5e+125)
		tmp = t_1;
	elseif (x <= 5.2e+57)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+125], t$95$1, If[LessEqual[x, 5.2e+57], N[(N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999962e125 or 5.2e57 < 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 76.9

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

   5. Applied rewrites 76.9%

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

   if -4.99999999999999962e125 < x < 5.2e57

   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 91.2

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

   5. Applied rewrites 91.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+125}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\left(\log t - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{+105}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 5.8e+105) (- (log t) z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.8e+105) {
		tmp = log(t) - 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 <= 5.8d+105) then
        tmp = log(t) - 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 <= 5.8e+105) {
		tmp = Math.log(t) - z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 5.8e+105:
		tmp = math.log(t) - z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.8e+105)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 5.8e+105)
		tmp = log(t) - z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.8e+105], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 5.8000000000000002e105

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.6%

      \[\leadsto \log t - z \]

   if 5.8000000000000002e105 < 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 69.0

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

   5. Applied rewrites 69.0%

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

Alternative 12: 47.7% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{+80}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 4.2e+80) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+80) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 4.2d+80) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 4.2e+80) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 4.2e+80:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 4.2e+80)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 4.2e+80)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 4.2e+80], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 4.20000000000000003e80

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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 35.3

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

   5. Applied rewrites 35.3%

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

   if 4.20000000000000003e80 < 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 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 13: 30.1% 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 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
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 27.0

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

5. Applied rewrites 27.0%

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

Reproduce

herbie shell --seed 2024244 
(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)))