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

Percentage Accurate: 99.9% → 99.9%

Time: 13.5s

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 68.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+78}:\\ \;\;\;\;0 - y\\ \mathbf{elif}\;t\_2 \leq 100:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+78) (- 0.0 y) (if (<= t_2 100.0) (- (log t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+78) {
		tmp = 0.0 - y;
	} else if (t_2 <= 100.0) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (t_2 <= (-5d+78)) then
        tmp = 0.0d0 - y
    else if (t_2 <= 100.0d0) then
        tmp = log(t) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+78) {
		tmp = 0.0 - y;
	} else if (t_2 <= 100.0) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if t_2 <= -5e+78:
		tmp = 0.0 - y
	elif t_2 <= 100.0:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -5e+78)
		tmp = Float64(0.0 - y);
	elseif (t_2 <= 100.0)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	tmp = 0.0;
	if (t_2 <= -5e+78)
		tmp = 0.0 - y;
	elseif (t_2 <= 100.0)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+78], N[(0.0 - y), $MachinePrecision], If[LessEqual[t$95$2, 100.0], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -4.99999999999999984e78

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y} \]

      3. --lowering--.f64 58.0

         \[\leadsto \color{blue}{0 - y} \]

   5. Simplified 58.0%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 58.0

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

   7. Applied egg-rr 58.0%

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

   if -4.99999999999999984e78 < (-.f64 (*.f64 x (log.f64 y)) y) < 100

   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. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-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. --lowering--.f64 N/A

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

      8. log-lowering-log.f64 90.2

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

   5. Simplified 90.2%

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

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 84.2

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

   8. Simplified 84.2%

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

   if 100 < (-.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. Step-by-step derivation

      1. flip-- N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(x \cdot \log y + \color{blue}{\left(0 - y\right)}\right) \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      11. associate-+r- N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\color{blue}{\left(\left(x \cdot \log y + 0\right) - y\right)} \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      12. --lowering--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\color{blue}{\left(\left(x \cdot \log y + 0\right) - y\right)} \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\color{blue}{\mathsf{fma}\left(x, \log y, 0\right)} - y\right) \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \color{blue}{\log y}, 0\right) - y\right) \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \color{blue}{\frac{1}{x \cdot \log y + y}}\right) - z\right) + \log t \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x, \log y, y\right)}}\right) - z\right) + \log t \]

      17. log-lowering-log.f64 96.9

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \frac{1}{\mathsf{fma}\left(x, \color{blue}{\log y}, y\right)}\right) - z\right) + \log t \]

   4. Applied egg-rr 96.9%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \frac{1}{\mathsf{fma}\left(x, \log y, y\right)}\right)} - z\right) + \log t \]

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 81.3

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

   7. Simplified 81.3%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -5 \cdot 10^{+78}:\\ \;\;\;\;0 - y\\ \mathbf{elif}\;x \cdot \log y - y \leq 100:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t - \left(y + z\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (log t) (+ y z))))
   (if (<= z -5e+27)
     t_1
     (if (<= z 2.4e+58) (fma x (log y) (- (log t) y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(t) - (y + z);
	double tmp;
	if (z <= -5e+27) {
		tmp = t_1;
	} else if (z <= 2.4e+58) {
		tmp = fma(x, log(y), (log(t) - y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(t) - Float64(y + z))
	tmp = 0.0
	if (z <= -5e+27)
		tmp = t_1;
	elseif (z <= 2.4e+58)
		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[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5e+27], t$95$1, If[LessEqual[z, 2.4e+58], 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 < -4.99999999999999979e27 or 2.4e58 < z

   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. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-lowering-+.f64 87.4

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

   5. Simplified 87.4%

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

   if -4.99999999999999979e27 < z < 2.4e58

   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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. log-lowering-log.f64 97.4

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

   5. Simplified 97.4%

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

Alternative 4: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\log t + \mathsf{fma}\left(x, \log y, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 5.4e+85)
   (+ (log t) (fma x (log y) (- 0.0 z)))
   (fma x (log y) (- (log t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 5.4e+85) {
		tmp = log(t) + fma(x, log(y), (0.0 - z));
	} else {
		tmp = fma(x, log(y), (log(t) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 5.4e+85)
		tmp = Float64(log(t) + fma(x, log(y), Float64(0.0 - z)));
	else
		tmp = fma(x, log(y), Float64(log(t) - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 5.4e+85], N[(N[Log[t], $MachinePrecision] + N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.39999999999999966e85

   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

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - z}\right) + \log t \]

      5. --lowering--.f64 93.5

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - z}\right) + \log t \]

   5. Simplified 93.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, 0 - z\right)} + \log t \]

   if 5.39999999999999966e85 < 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 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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. log-lowering-log.f64 89.9

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

   5. Simplified 89.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{+85}:\\ \;\;\;\;\log t + \mathsf{fma}\left(x, \log y, 0 - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e+85)
   (fma x (log y) (- (log t) z))
   (fma x (log y) (- (log t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+85) {
		tmp = fma(x, log(y), (log(t) - z));
	} else {
		tmp = fma(x, log(y), (log(t) - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+85)
		tmp = fma(x, log(y), Float64(log(t) - z));
	else
		tmp = fma(x, log(y), Float64(log(t) - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+85], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e85

   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. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-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. --lowering--.f64 N/A

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

      8. log-lowering-log.f64 93.4

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

   5. Simplified 93.4%

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

   if 3.2999999999999999e85 < 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 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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. log-lowering-log.f64 89.9

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

   5. Simplified 89.9%

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

Alternative 6: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - y\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+83}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, 0 - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9e+64)
   (fma x (log y) (- 0.0 y))
   (if (<= x 1.15e+83) (- (log t) (+ y z)) (fma x (log y) (- 0.0 z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9e+64) {
		tmp = fma(x, log(y), (0.0 - y));
	} else if (x <= 1.15e+83) {
		tmp = log(t) - (y + z);
	} else {
		tmp = fma(x, log(y), (0.0 - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9e+64)
		tmp = fma(x, log(y), Float64(0.0 - y));
	elseif (x <= 1.15e+83)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = fma(x, log(y), Float64(0.0 - z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9e+64], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e+83], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.99999999999999946e64

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. log-lowering-log.f64 87.9

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

   5. Simplified 87.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - y}\right) \]

      3. --lowering--.f64 87.9

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - y}\right) \]

   8. Simplified 87.9%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - y}\right) \]

   if -8.99999999999999946e64 < x < 1.14999999999999997e83

   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. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-lowering-+.f64 94.3

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

   5. Simplified 94.3%

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

   if 1.14999999999999997e83 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. 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. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-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. --lowering--.f64 N/A

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

      8. log-lowering-log.f64 86.1

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

   5. Simplified 86.1%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - z}\right) \]

      3. --lowering--.f64 86.1

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - z}\right) \]

   8. Simplified 86.1%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, 0 - y\right)\\ \mathbf{if}\;x \leq -9 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+44}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- 0.0 y))))
   (if (<= x -9e+63) t_1 (if (<= x 2.2e+44) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), (0.0 - y));
	double tmp;
	if (x <= -9e+63) {
		tmp = t_1;
	} else if (x <= 2.2e+44) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, log(y), Float64(0.0 - y))
	tmp = 0.0
	if (x <= -9e+63)
		tmp = t_1;
	elseif (x <= 2.2e+44)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision] + N[(0.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9e+63], t$95$1, If[LessEqual[x, 2.2e+44], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.00000000000000034e63 or 2.19999999999999996e44 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in 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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. log-lowering-log.f64 84.9

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

   5. Simplified 84.9%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - y}\right) \]

      3. --lowering--.f64 84.9

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - y}\right) \]

   8. Simplified 84.9%

      \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{0 - y}\right) \]

   if -9.00000000000000034e63 < x < 2.19999999999999996e44

   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. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-lowering-+.f64 95.2

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

   5. Simplified 95.2%

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

Alternative 8: 83.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+97}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -1.4e+65) t_1 (if (<= x 5.2e+97) (- (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 <= -1.4e+65) {
		tmp = t_1;
	} else if (x <= 5.2e+97) {
		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 <= (-1.4d+65)) then
        tmp = t_1
    else if (x <= 5.2d+97) 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 <= -1.4e+65) {
		tmp = t_1;
	} else if (x <= 5.2e+97) {
		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 <= -1.4e+65:
		tmp = t_1
	elif x <= 5.2e+97:
		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 <= -1.4e+65)
		tmp = t_1;
	elseif (x <= 5.2e+97)
		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 <= -1.4e+65)
		tmp = t_1;
	elseif (x <= 5.2e+97)
		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, -1.4e+65], t$95$1, If[LessEqual[x, 5.2e+97], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999e65 or 5.2e97 < 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. flip-- N/A

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

      2. div-inv N/A

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

      3. difference-of-squares N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. log-lowering-log.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. sub-neg N/A

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

      10. neg-sub0 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(x \cdot \log y + \color{blue}{\left(0 - y\right)}\right) \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      11. associate-+r- N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\color{blue}{\left(\left(x \cdot \log y + 0\right) - y\right)} \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      12. --lowering--.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\color{blue}{\left(\left(x \cdot \log y + 0\right) - y\right)} \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\color{blue}{\mathsf{fma}\left(x, \log y, 0\right)} - y\right) \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \color{blue}{\log y}, 0\right) - y\right) \cdot \frac{1}{x \cdot \log y + y}\right) - z\right) + \log t \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \color{blue}{\frac{1}{x \cdot \log y + y}}\right) - z\right) + \log t \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(x, \log y, y\right)}}\right) - z\right) + \log t \]

      17. log-lowering-log.f64 98.4

          \[\leadsto \left(\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \frac{1}{\mathsf{fma}\left(x, \color{blue}{\log y}, y\right)}\right) - z\right) + \log t \]

   4. Applied egg-rr 98.4%

      \[\leadsto \left(\color{blue}{\mathsf{fma}\left(x, \log y, y\right) \cdot \left(\left(\mathsf{fma}\left(x, \log y, 0\right) - y\right) \cdot \frac{1}{\mathsf{fma}\left(x, \log y, y\right)}\right)} - z\right) + \log t \]

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 68.6

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

   7. Simplified 68.6%

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

   if -1.3999999999999999e65 < x < 5.2e97

   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. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. +-lowering-+.f64 92.7

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

   5. Simplified 92.7%

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

Alternative 9: 59.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+126}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+68}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.3e+126) (- 0.0 z) (if (<= z 3.5e+68) (- (log t) y) (- 0.0 z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.3e+126) {
		tmp = 0.0 - z;
	} else if (z <= 3.5e+68) {
		tmp = log(t) - y;
	} else {
		tmp = 0.0 - 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 (z <= (-6.3d+126)) then
        tmp = 0.0d0 - z
    else if (z <= 3.5d+68) then
        tmp = log(t) - y
    else
        tmp = 0.0d0 - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.3e+126) {
		tmp = 0.0 - z;
	} else if (z <= 3.5e+68) {
		tmp = Math.log(t) - y;
	} else {
		tmp = 0.0 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6.3e+126:
		tmp = 0.0 - z
	elif z <= 3.5e+68:
		tmp = math.log(t) - y
	else:
		tmp = 0.0 - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.3e+126)
		tmp = Float64(0.0 - z);
	elseif (z <= 3.5e+68)
		tmp = Float64(log(t) - y);
	else
		tmp = Float64(0.0 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6.3e+126)
		tmp = 0.0 - z;
	elseif (z <= 3.5e+68)
		tmp = log(t) - y;
	else
		tmp = 0.0 - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.3e+126], N[(0.0 - z), $MachinePrecision], If[LessEqual[z, 3.5e+68], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], N[(0.0 - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.29999999999999975e126 or 3.49999999999999977e68 < z

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - z} \]

      3. --lowering--.f64 74.4

         \[\leadsto \color{blue}{0 - z} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.4

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

   7. Applied egg-rr 74.4%

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

   if -6.29999999999999975e126 < z < 3.49999999999999977e68

   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. accelerator-lowering-fma.f64 N/A

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

      4. log-lowering-log.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. log-lowering-log.f64 94.2

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

   5. Simplified 94.2%

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

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 60.4

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

   8. Simplified 60.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+126}:\\ \;\;\;\;0 - z\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+68}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;0 - z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+88}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 3.3e+88) (- (log t) z) (- 0.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+88) {
		tmp = log(t) - z;
	} else {
		tmp = 0.0 - 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 <= 3.3d+88) then
        tmp = log(t) - z
    else
        tmp = 0.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+88) {
		tmp = Math.log(t) - z;
	} else {
		tmp = 0.0 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.3e+88:
		tmp = math.log(t) - z
	else:
		tmp = 0.0 - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+88)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(0.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.3e+88)
		tmp = log(t) - z;
	else
		tmp = 0.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+88], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(0.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000003e88

   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. accelerator-lowering-fma.f64 N/A

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

      5. log-lowering-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. --lowering--.f64 N/A

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

      8. log-lowering-log.f64 93.5

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

   5. Simplified 93.5%

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

   6. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

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

      2. log-lowering-log.f64 59.3

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

   8. Simplified 59.3%

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

   if 3.3000000000000003e88 < 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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y} \]

      3. --lowering--.f64 76.0

         \[\leadsto \color{blue}{0 - y} \]

   5. Simplified 76.0%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 76.0

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

   7. Applied egg-rr 76.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+88}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.9% accurate, 21.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+88}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 3.3e+88) (- 0.0 z) (- 0.0 y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+88) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - 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 <= 3.3d+88) then
        tmp = 0.0d0 - z
    else
        tmp = 0.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 3.3e+88) {
		tmp = 0.0 - z;
	} else {
		tmp = 0.0 - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 3.3e+88:
		tmp = 0.0 - z
	else:
		tmp = 0.0 - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 3.3e+88)
		tmp = Float64(0.0 - z);
	else
		tmp = Float64(0.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 3.3e+88)
		tmp = 0.0 - z;
	else
		tmp = 0.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 3.3e+88], N[(0.0 - z), $MachinePrecision], N[(0.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.3000000000000003e88

   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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - z} \]

      3. --lowering--.f64 36.2

         \[\leadsto \color{blue}{0 - z} \]

   5. Simplified 36.2%

      \[\leadsto \color{blue}{0 - z} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 36.2

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

   7. Applied egg-rr 36.2%

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

   if 3.3000000000000003e88 < 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. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - y} \]

      3. --lowering--.f64 76.0

         \[\leadsto \color{blue}{0 - y} \]

   5. Simplified 76.0%

      \[\leadsto \color{blue}{0 - y} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 76.0

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

   7. Applied egg-rr 76.0%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+88}:\\ \;\;\;\;0 - z\\ \mathbf{else}:\\ \;\;\;\;0 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 29.9% accurate, 53.8× speedup

Math

\[\begin{array}{l} \\ 0 - y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 y))

C

double code(double x, double y, double z, double t) {
	return 0.0 - 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 = 0.0d0 - y
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - y;
}

Python

def code(x, y, z, t):
	return 0.0 - y

Julia

function code(x, y, z, t)
	return Float64(0.0 - y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - y;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - y), $MachinePrecision]

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. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - y} \]

   3. --lowering--.f64 31.5

      \[\leadsto \color{blue}{0 - y} \]

5. Simplified 31.5%

   \[\leadsto \color{blue}{0 - y} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 31.5

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

7. Applied egg-rr 31.5%

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

8. Final simplification 31.5%

   \[\leadsto 0 - y \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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)))