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

Percentage Accurate: 99.9% → 99.9%

Time: 11.7s

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lower--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -4e+19) t_1 (if (<= t_1 5e+17) (fma x (log y) (log t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -4e+19) {
		tmp = t_1;
	} else if (t_1 <= 5e+17) {
		tmp = fma(x, log(y), log(t));
	} else {
		tmp = t_1;
	}
	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 <= -4e+19)
		tmp = t_1;
	elseif (t_1 <= 5e+17)
		tmp = fma(x, log(y), log(t));
	else
		tmp = t_1;
	end
	return 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, -4e+19], t$95$1, If[LessEqual[t$95$1, 5e+17], N[(x * N[Log[y], $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -4e19 or 5e17 < (-.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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -4e19 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 5e17

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.1%

      \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{-5}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -50000.0) t_1 (if (<= t_1 1e-5) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_1;
	} else if (t_1 <= 1e-5) {
		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)) - y) - z
    if (t_1 <= (-50000.0d0)) then
        tmp = t_1
    else if (t_1 <= 1d-5) 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)) - y) - z;
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_1;
	} else if (t_1 <= 1e-5) {
		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)) - y) - z
	tmp = 0
	if t_1 <= -50000.0:
		tmp = t_1
	elif t_1 <= 1e-5:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	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 <= -50000.0)
		tmp = t_1;
	elseif (t_1 <= 1e-5)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	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 <= -50000.0)
		tmp = t_1;
	elseif (t_1 <= 1e-5)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -50000.0], t$95$1, If[LessEqual[t$95$1, 1e-5], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -5e4 or 1.00000000000000008e-5 < (-.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. 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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if -5e4 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 1.00000000000000008e-5

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

Alternative 4: 79.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_2 \leq 6 \cdot 10^{+15}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -2e+16) (- (- y) z) (if (<= t_2 6e+15) (- (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 <= -2e+16) {
		tmp = -y - z;
	} else if (t_2 <= 6e+15) {
		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 <= (-2d+16)) then
        tmp = -y - z
    else if (t_2 <= 6d+15) 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 <= -2e+16) {
		tmp = -y - z;
	} else if (t_2 <= 6e+15) {
		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 <= -2e+16:
		tmp = -y - z
	elif t_2 <= 6e+15:
		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 <= -2e+16)
		tmp = Float64(Float64(-y) - z);
	elseif (t_2 <= 6e+15)
		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 <= -2e+16)
		tmp = -y - z;
	elseif (t_2 <= 6e+15)
		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, -2e+16], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$2, 6e+15], 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) < -2e16

   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2e16 < (-.f64 (*.f64 x (log.f64 y)) y) < 6e15

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - \color{blue}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.9%

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

   if 6e15 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.5%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 73.5

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

   5. Applied rewrites 73.5%

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

Alternative 5: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 220:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= z -4.5e+31)
     t_1
     (if (<= z 220.0) (fma x (log y) (- (log t) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (z <= -4.5e+31) {
		tmp = t_1;
	} else if (z <= 220.0) {
		tmp = fma(x, log(y), (log(t) - y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	tmp = 0.0
	if (z <= -4.5e+31)
		tmp = t_1;
	elseif (z <= 220.0)
		tmp = fma(x, log(y), Float64(log(t) - y));
	else
		tmp = t_1;
	end
	return 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[z, -4.5e+31], t$95$1, If[LessEqual[z, 220.0], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4999999999999996e31 or 220 < 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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -4.4999999999999996e31 < z < 220

   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. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-log.f64 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 6: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) -2e+16) (- (- y) z) (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= -2e+16) {
		tmp = -y - z;
	} 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 (((x * log(y)) - y) <= (-2d+16)) then
        tmp = -y - z
    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 (((x * Math.log(y)) - y) <= -2e+16) {
		tmp = -y - z;
	} else {
		tmp = Math.log(t) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * math.log(y)) - y) <= -2e+16:
		tmp = -y - z
	else:
		tmp = math.log(t) - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * log(y)) - y) <= -2e+16)
		tmp = Float64(Float64(-y) - z);
	else
		tmp = Float64(log(t) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * log(y)) - y) <= -2e+16)
		tmp = -y - z;
	else
		tmp = log(t) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], -2e+16], N[((-y) - z), $MachinePrecision], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2e16

   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2e16 < (-.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 y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log t - \color{blue}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.2%

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

Alternative 7: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 2e-5) (fma x (log y) (- (log t) z)) (- (- (* x (log y)) y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 2e-5) {
		tmp = fma(x, log(y), (log(t) - z));
	} else {
		tmp = ((x * log(y)) - y) - z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 2e-5)
		tmp = fma(x, log(y), Float64(log(t) - z));
	else
		tmp = Float64(Float64(Float64(x * log(y)) - y) - z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 2e-5], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000016e-5

   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. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if 2.00000000000000016e-5 < y

   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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 99.9

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

   7. Applied rewrites 99.9%

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

Alternative 8: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - z\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) z)))
   (if (<= x -1.8e+93) t_1 (if (<= x 1.8e+65) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - z;
	double tmp;
	if (x <= -1.8e+93) {
		tmp = t_1;
	} else if (x <= 1.8e+65) {
		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)) - z
    if (x <= (-1.8d+93)) then
        tmp = t_1
    else if (x <= 1.8d+65) 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)) - z;
	double tmp;
	if (x <= -1.8e+93) {
		tmp = t_1;
	} else if (x <= 1.8e+65) {
		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)) - z
	tmp = 0
	if x <= -1.8e+93:
		tmp = t_1
	elif x <= 1.8e+65:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - z)
	tmp = 0.0
	if (x <= -1.8e+93)
		tmp = t_1;
	elseif (x <= 1.8e+65)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - z;
	tmp = 0.0;
	if (x <= -1.8e+93)
		tmp = t_1;
	elseif (x <= 1.8e+65)
		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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[x, -1.8e+93], t$95$1, If[LessEqual[x, 1.8e+65], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e93 or 1.79999999999999989e65 < 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. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 84.5

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

   7. Applied rewrites 84.5%

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

   if -1.8e93 < x < 1.79999999999999989e65

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 9: 83.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+200}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{+155}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -2.1e+200) t_1 (if (<= x 1e+155) (- (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 <= -2.1e+200) {
		tmp = t_1;
	} else if (x <= 1e+155) {
		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 <= (-2.1d+200)) then
        tmp = t_1
    else if (x <= 1d+155) 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 <= -2.1e+200) {
		tmp = t_1;
	} else if (x <= 1e+155) {
		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 <= -2.1e+200:
		tmp = t_1
	elif x <= 1e+155:
		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 <= -2.1e+200)
		tmp = t_1;
	elseif (x <= 1e+155)
		tmp = Float64(log(t) - Float64(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 <= -2.1e+200)
		tmp = t_1;
	elseif (x <= 1e+155)
		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, -2.1e+200], t$95$1, If[LessEqual[x, 1e+155], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999997e200 or 1.00000000000000001e155 < x

   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. lower-*.f64 N/A

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

      2. lower-log.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -2.09999999999999997e200 < x < 1.00000000000000001e155

   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. lower--.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-+.f64 86.8

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

   5. Applied rewrites 86.8%

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

Alternative 10: 48.1% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+19}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 3.5e+19) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.5e+19) {
		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 <= 3.5d+19) 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 <= 3.5e+19) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.5e+19:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.5e+19)
		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 <= 3.5e+19)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.5e+19], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 3.5e19

   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 39.6

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

   5. Applied rewrites 39.6%

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

   if 3.5e19 < 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 61.3

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

   5. Applied rewrites 61.3%

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

Alternative 11: 58.2% accurate, 35.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[((-y) - z), $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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lower--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 56.9

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

7. Applied rewrites 56.9%

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

Alternative 12: 30.2% 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 30.5

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

5. Applied rewrites 30.5%

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

Alternative 13: 2.2% accurate, 215.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
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. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lower--.f64 N/A

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

   6. lift--.f64 N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 28.7

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

7. Applied rewrites 28.7%

   \[\leadsto \color{blue}{-z} \]
8. Step-by-step derivation

9. Applied rewrites 4.1%

   \[\leadsto \frac{0 - z \cdot \left(z \cdot z\right)}{\color{blue}{0 + \mathsf{fma}\left(z, z, 0 \cdot z\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 2.4%

    \[\leadsto z \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))