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

Percentage Accurate: 99.9% → 99.9%

Time: 11.1s

Alternatives: 12

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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ t_2 := t\_1 - z\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)) (t_2 (- t_1 z)))
   (if (<= t_1 -2e+16) t_2 (if (<= t_1 2e-27) (- (log t) (+ y z)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double t_2 = t_1 - z;
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = t_2;
	} else if (t_1 <= 2e-27) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x * log(y)) - y
    t_2 = t_1 - z
    if (t_1 <= (-2d+16)) then
        tmp = t_2
    else if (t_1 <= 2d-27) then
        tmp = log(t) - (y + z)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - y;
	double t_2 = t_1 - z;
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = t_2;
	} else if (t_1 <= 2e-27) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	t_2 = t_1 - z
	tmp = 0
	if t_1 <= -2e+16:
		tmp = t_2
	elif t_1 <= 2e-27:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	t_2 = Float64(t_1 - z)
	tmp = 0.0
	if (t_1 <= -2e+16)
		tmp = t_2;
	elseif (t_1 <= 2e-27)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - y;
	t_2 = t_1 - z;
	tmp = 0.0;
	if (t_1 <= -2e+16)
		tmp = t_2;
	elseif (t_1 <= 2e-27)
		tmp = log(t) - (y + z);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+16], t$95$2, If[LessEqual[t$95$1, 2e-27], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2e16 or 2.0000000000000001e-27 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. 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 99.5

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

   7. Applied rewrites 99.5%

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

   if -2e16 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.0000000000000001e-27

   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 97.5

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

   5. Applied rewrites 97.5%

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

Alternative 3: 69.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10000000:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)) (t_2 (- (- y) z)))
   (if (<= t_1 -2e+16) t_2 (if (<= t_1 10000000.0) (log t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double t_2 = -y - z;
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = t_2;
	} else if (t_1 <= 10000000.0) {
		tmp = log(t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((x * log(y)) - y) - z
    t_2 = -y - z
    if (t_1 <= (-2d+16)) then
        tmp = t_2
    else if (t_1 <= 10000000.0d0) then
        tmp = log(t)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x * Math.log(y)) - y) - z;
	double t_2 = -y - z;
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = t_2;
	} else if (t_1 <= 10000000.0) {
		tmp = Math.log(t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	t_2 = -y - z
	tmp = 0
	if t_1 <= -2e+16:
		tmp = t_2
	elif t_1 <= 10000000.0:
		tmp = math.log(t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	t_2 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (t_1 <= -2e+16)
		tmp = t_2;
	elseif (t_1 <= 10000000.0)
		tmp = log(t);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * log(y)) - y) - z;
	t_2 = -y - z;
	tmp = 0.0;
	if (t_1 <= -2e+16)
		tmp = t_2;
	elseif (t_1 <= 10000000.0)
		tmp = log(t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+16], t$95$2, If[LessEqual[t$95$1, 10000000.0], N[Log[t], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -2e16 or 1e7 < (-.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 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 62.2

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

   7. Applied rewrites 62.2%

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

   if -2e16 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 1e7

   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 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.9%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 89.1%

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

Alternative 4: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 5e+33) (- (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 <= 5e+33) {
		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 <= 5d+33) 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 <= 5e+33) {
		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 <= 5e+33:
		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 <= 5e+33)
		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 <= 5e+33)
		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, 5e+33], 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 69.2

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

   7. Applied rewrites 69.2%

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

   if -2e16 < (-.f64 (*.f64 x (log.f64 y)) y) < 4.99999999999999973e33

   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 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   if 4.99999999999999973e33 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 80.8

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

   5. Applied rewrites 80.8%

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

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;z \leq -410:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= z -410.0)
     t_1
     (if (<= z 1.9e-26) (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 <= -410.0) {
		tmp = t_1;
	} else if (z <= 1.9e-26) {
		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 <= -410.0)
		tmp = t_1;
	elseif (z <= 1.9e-26)
		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, -410.0], t$95$1, If[LessEqual[z, 1.9e-26], 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 < -410 or 1.90000000000000007e-26 < 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 99.4

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

   7. Applied rewrites 99.4%

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

   if -410 < z < 1.90000000000000007e-26

   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 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 6: 69.9% 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 69.2

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

   7. Applied rewrites 69.2%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. 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 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 63.8%

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

Alternative 7: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00016:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 0.00016) (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 <= 0.00016) {
		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 <= 0.00016)
		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, 0.00016], 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 < 1.60000000000000013e-4

   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 1.60000000000000013e-4 < 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 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 8: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - z\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+31}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) z)))
   (if (<= x -4.5e+34) t_1 (if (<= x 2.6e+31) (- (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 <= -4.5e+34) {
		tmp = t_1;
	} else if (x <= 2.6e+31) {
		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 <= (-4.5d+34)) then
        tmp = t_1
    else if (x <= 2.6d+31) 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 <= -4.5e+34) {
		tmp = t_1;
	} else if (x <= 2.6e+31) {
		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 <= -4.5e+34:
		tmp = t_1
	elif x <= 2.6e+31:
		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 <= -4.5e+34)
		tmp = t_1;
	elseif (x <= 2.6e+31)
		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 <= -4.5e+34)
		tmp = t_1;
	elseif (x <= 2.6e+31)
		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, -4.5e+34], t$95$1, If[LessEqual[x, 2.6e+31], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e34 or 2.6e31 < x

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-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

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 79.8

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

   7. Applied rewrites 79.8%

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

   if -4.5e34 < x < 2.6e31

   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 97.6

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

   5. Applied rewrites 97.6%

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

Alternative 9: 82.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+39}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.25e+90) t_1 (if (<= x 8.5e+39) (- (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 <= -1.25e+90) {
		tmp = t_1;
	} else if (x <= 8.5e+39) {
		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 <= (-1.25d+90)) then
        tmp = t_1
    else if (x <= 8.5d+39) 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 <= -1.25e+90) {
		tmp = t_1;
	} else if (x <= 8.5e+39) {
		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 <= -1.25e+90:
		tmp = t_1
	elif x <= 8.5e+39:
		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 <= -1.25e+90)
		tmp = t_1;
	elseif (x <= 8.5e+39)
		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 <= -1.25e+90)
		tmp = t_1;
	elseif (x <= 8.5e+39)
		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, -1.25e+90], t$95$1, If[LessEqual[x, 8.5e+39], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2500000000000001e90 or 8.49999999999999971e39 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -1.2500000000000001e90 < x < 8.49999999999999971e39

   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 93.2

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

   5. Applied rewrites 93.2%

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

Alternative 10: 46.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+98}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) -5e+98) (- y) (- z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= -5e+98) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x * log(y)) - y) <= (-5d+98)) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * Math.log(y)) - y) <= -5e+98) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x * math.log(y)) - y) <= -5e+98:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(x * log(y)) - y) <= -5e+98)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x * log(y)) - y) <= -5e+98)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision], -5e+98], (-y), (-z)]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -4.9999999999999998e98

   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 57.3

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

   5. Applied rewrites 57.3%

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

   if -4.9999999999999998e98 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 37.2

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

   5. Applied rewrites 37.2%

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

Alternative 11: 57.7% 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 53.6

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

7. Applied rewrites 53.6%

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

Alternative 12: 29.3% 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 30.2

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

5. Applied rewrites 30.2%

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

Reproduce

herbie shell --seed 2024219 
(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)))