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

Percentage Accurate: 99.9% → 99.9%

Time: 12.2s

Alternatives: 15

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: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(t\_1 - y\right) - z\\ \mathbf{if}\;t\_2 \leq -1000000:\\ \;\;\;\;\frac{1}{\frac{1}{t\_1 - \left(y + z\right)}}\\ \mathbf{elif}\;t\_2 \leq 50000000:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- t_1 y) z)))
   (if (<= t_2 -1000000.0)
     (/ 1.0 (/ 1.0 (- t_1 (+ y z))))
     (if (<= t_2 50000000.0) (fma x (log y) (log t)) (fma x (log y) (- z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = (t_1 - y) - z;
	double tmp;
	if (t_2 <= -1000000.0) {
		tmp = 1.0 / (1.0 / (t_1 - (y + z)));
	} else if (t_2 <= 50000000.0) {
		tmp = fma(x, log(y), log(t));
	} else {
		tmp = fma(x, log(y), -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(t_1 - y) - z)
	tmp = 0.0
	if (t_2 <= -1000000.0)
		tmp = Float64(1.0 / Float64(1.0 / Float64(t_1 - Float64(y + z))));
	elseif (t_2 <= 50000000.0)
		tmp = fma(x, log(y), log(t));
	else
		tmp = fma(x, log(y), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$2, -1000000.0], N[(1.0 / N[(1.0 / N[(t$95$1 - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 50000000.0], N[(x * N[Log[y], $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision], N[(x * N[Log[y], $MachinePrecision] + (-z)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -1e6

   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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.2

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

   7. Simplified 98.2%

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

   if -1e6 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 5e7

   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. 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 99.8

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

   5. Simplified 99.8%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. log-lowering-log.f64 99.8

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

   8. Simplified 99.8%

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

   if 5e7 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z)

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 99.7

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

   5. Simplified 99.7%

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

   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-lowering-neg.f64 99.7

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

   8. Simplified 99.7%

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

Alternative 3: 68.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \log y - y\right) - z\\ \mathbf{if}\;t\_1 \leq -1000000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+28}:\\ \;\;\;\;\log t\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- (* x (log y)) y) z)))
   (if (<= t_1 -1000000.0) (- (- y) z) (if (<= t_1 4e+28) (log t) (- z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * log(y)) - y) - z;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = -y - z;
	} else if (t_1 <= 4e+28) {
		tmp = log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x * log(y)) - y) - z
    if (t_1 <= (-1000000.0d0)) then
        tmp = -y - z
    else if (t_1 <= 4d+28) then
        tmp = log(t)
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((x * Math.log(y)) - y) - z;
	double tmp;
	if (t_1 <= -1000000.0) {
		tmp = -y - z;
	} else if (t_1 <= 4e+28) {
		tmp = Math.log(t);
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x * math.log(y)) - y) - z
	tmp = 0
	if t_1 <= -1000000.0:
		tmp = -y - z
	elif t_1 <= 4e+28:
		tmp = math.log(t)
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * log(y)) - y) - z)
	tmp = 0.0
	if (t_1 <= -1000000.0)
		tmp = Float64(Float64(-y) - z);
	elseif (t_1 <= 4e+28)
		tmp = log(t);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x * log(y)) - y) - z;
	tmp = 0.0;
	if (t_1 <= -1000000.0)
		tmp = -y - z;
	elseif (t_1 <= 4e+28)
		tmp = log(t);
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[t$95$1, -1000000.0], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$1, 4e+28], N[Log[t], $MachinePrecision], (-z)]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < -1e6

   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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.2

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

   7. Simplified 98.2%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 75.1

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

   10. Simplified 75.1%

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

   if -1e6 < (-.f64 (-.f64 (*.f64 x (log.f64 y)) y) z) < 3.99999999999999983e28

   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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 87.6

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

   5. Simplified 87.6%

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

   6. Taylor expanded in z around 0

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

      1. log-lowering-log.f64 87.6

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

   8. Simplified 87.6%

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

   if 3.99999999999999983e28 < (-.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. 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-lowering-neg.f64 52.4

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

   5. Simplified 52.4%

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

Alternative 4: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -1500000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;t\_2 \leq 10^{+14}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -1500000.0) (- (- y) z) (if (<= t_2 1e+14) (- (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 <= -1500000.0) {
		tmp = -y - z;
	} else if (t_2 <= 1e+14) {
		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 <= (-1500000.0d0)) then
        tmp = -y - z
    else if (t_2 <= 1d+14) 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 <= -1500000.0) {
		tmp = -y - z;
	} else if (t_2 <= 1e+14) {
		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 <= -1500000.0:
		tmp = -y - z
	elif t_2 <= 1e+14:
		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 <= -1500000.0)
		tmp = Float64(Float64(-y) - z);
	elseif (t_2 <= 1e+14)
		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 <= -1500000.0)
		tmp = -y - z;
	elseif (t_2 <= 1e+14)
		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, -1500000.0], N[((-y) - z), $MachinePrecision], If[LessEqual[t$95$2, 1e+14], 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) < -1.5e6

   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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.0

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 74.9

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

   10. Simplified 74.9%

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

   if -1.5e6 < (-.f64 (*.f64 x (log.f64 y)) y) < 1e14

   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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 95.5

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

   5. Simplified 95.5%

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

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

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

      4. log-lowering-log.f64 95.5

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

   7. Applied egg-rr 95.5%

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

   if 1e14 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 86.9

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

   5. Simplified 86.9%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\frac{1}{x \cdot \log y - \left(y + z\right)}}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(x, \log y, -y\right)}{z}, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.5e+30)
   (/ 1.0 (/ 1.0 (- (* x (log y)) (+ y z))))
   (if (<= z 2.4e-20)
     (fma x (log y) (- (log t) y))
     (fma z (/ (fma x (log y) (- y)) z) (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.5e+30) {
		tmp = 1.0 / (1.0 / ((x * log(y)) - (y + z)));
	} else if (z <= 2.4e-20) {
		tmp = fma(x, log(y), (log(t) - y));
	} else {
		tmp = fma(z, (fma(x, log(y), -y) / z), -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.5e+30)
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(x * log(y)) - Float64(y + z))));
	elseif (z <= 2.4e-20)
		tmp = fma(x, log(y), Float64(log(t) - y));
	else
		tmp = fma(z, Float64(fma(x, log(y), Float64(-y)) / z), Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.5e+30], N[(1.0 / N[(1.0 / N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e-20], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(x * N[Log[y], $MachinePrecision] + (-y)), $MachinePrecision] / z), $MachinePrecision] + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.50000000000000025e30

   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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.8

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

   7. Simplified 99.8%

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

   if -5.50000000000000025e30 < z < 2.39999999999999993e-20

   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 99.8

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

   5. Simplified 99.8%

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

   if 2.39999999999999993e-20 < 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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.1

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

   7. Simplified 98.1%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{x \cdot \log y}{z} - \left(1 + \frac{y}{z}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-frac-neg2 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

   10. Simplified 98.2%

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

Alternative 6: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -1500000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\log t - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- (* x (log y)) y) -1500000.0) (- (- y) z) (- (log t) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x * log(y)) - y) <= -1500000.0) {
		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) <= (-1500000.0d0)) 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) <= -1500000.0) {
		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) <= -1500000.0:
		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) <= -1500000.0)
		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) <= -1500000.0)
		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], -1500000.0], 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) < -1.5e6

   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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.0

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 74.9

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

   10. Simplified 74.9%

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

   if -1.5e6 < (-.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

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 67.6

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

   5. Simplified 67.6%

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

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

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

      4. log-lowering-log.f64 67.6

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

   7. Applied egg-rr 67.6%

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

Alternative 7: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x \cdot \log y - \left(y + z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 1.42e-5)
   (fma x (log y) (- (log t) z))
   (/ 1.0 (/ 1.0 (- (* x (log y)) (+ y z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.42e-5) {
		tmp = fma(x, log(y), (log(t) - z));
	} else {
		tmp = 1.0 / (1.0 / ((x * log(y)) - (y + z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.42e-5)
		tmp = fma(x, log(y), Float64(log(t) - z));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(x * log(y)) - Float64(y + z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.42e-5], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.42e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. 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 99.8

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

   5. Simplified 99.8%

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

   if 1.42e-5 < y

   1. Initial program 100.0%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.0

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

   7. Simplified 99.0%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{x \cdot \log y} - \left(y + z\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\log y - \frac{y + z}{x}\right)\\ \mathbf{if}\;x \leq -41000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1300000:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (log y) (/ (+ y z) x)))))
   (if (<= x -41000.0) t_1 (if (<= x 1300000.0) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (log(y) - ((y + z) / x));
	double tmp;
	if (x <= -41000.0) {
		tmp = t_1;
	} else if (x <= 1300000.0) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (log(y) - ((y + z) / x))
    if (x <= (-41000.0d0)) then
        tmp = t_1
    else if (x <= 1300000.0d0) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (Math.log(y) - ((y + z) / x));
	double tmp;
	if (x <= -41000.0) {
		tmp = t_1;
	} else if (x <= 1300000.0) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (math.log(y) - ((y + z) / x))
	tmp = 0
	if x <= -41000.0:
		tmp = t_1
	elif x <= 1300000.0:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(log(y) - Float64(Float64(y + z) / x)))
	tmp = 0.0
	if (x <= -41000.0)
		tmp = t_1;
	elseif (x <= 1300000.0)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (log(y) - ((y + z) / x));
	tmp = 0.0;
	if (x <= -41000.0)
		tmp = t_1;
	elseif (x <= 1300000.0)
		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[(N[Log[y], $MachinePrecision] - N[(N[(y + z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -41000.0], t$95$1, If[LessEqual[x, 1300000.0], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -41000 or 1.3e6 < 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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.1

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

   7. Simplified 99.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\log y + -1 \cdot \frac{y + z}{x}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. distribute-lft-in N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. div-sub N/A

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

      6. associate-*r/ N/A

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

      7. associate--l+ N/A

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

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

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

      9. associate--l+ N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-in N/A

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

      15. associate-*r/ N/A

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

      16. mul-1-neg N/A

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

      17. unsub-neg N/A

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

      18. --lowering--.f64 N/A

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

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

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

   10. Simplified 99.2%

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

   if -41000 < x < 1.3e6

   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 98.3

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

   5. Simplified 98.3%

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

Alternative 9: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -z\right)\\ \mathbf{if}\;x \leq -3.25 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- z))))
   (if (<= x -3.25e+22) t_1 (if (<= x 1.4e+25) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, log(y), -z);
	double tmp;
	if (x <= -3.25e+22) {
		tmp = t_1;
	} else if (x <= 1.4e+25) {
		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(-z))
	tmp = 0.0
	if (x <= -3.25e+22)
		tmp = t_1;
	elseif (x <= 1.4e+25)
		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] + (-z)), $MachinePrecision]}, If[LessEqual[x, -3.25e+22], t$95$1, If[LessEqual[x, 1.4e+25], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2499999999999999e22 or 1.4000000000000001e25 < 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 81.7

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

   5. Simplified 81.7%

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

   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-lowering-neg.f64 81.7

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

   8. Simplified 81.7%

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

   if -3.2499999999999999e22 < x < 1.4000000000000001e25

   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 97.8

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

   5. Simplified 97.8%

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

Alternative 10: 88.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \log y, -y\right)\\ \mathbf{if}\;x \leq -2.45 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{+44}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (log y) (- y))))
   (if (<= x -2.45e+120) t_1 (if (<= x 6.1e+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), -y);
	double tmp;
	if (x <= -2.45e+120) {
		tmp = t_1;
	} else if (x <= 6.1e+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(-y))
	tmp = 0.0
	if (x <= -2.45e+120)
		tmp = t_1;
	elseif (x <= 6.1e+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] + (-y)), $MachinePrecision]}, If[LessEqual[x, -2.45e+120], t$95$1, If[LessEqual[x, 6.1e+44], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.45000000000000005e120 or 6.09999999999999983e44 < 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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 99.6

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

   7. Simplified 99.6%

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

   8. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto x \cdot \log y + \color{blue}{-1 \cdot y} \]

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

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 84.0

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

   10. Simplified 84.0%

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

   if -2.45000000000000005e120 < x < 6.09999999999999983e44

   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 94.0

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

   5. Simplified 94.0%

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

Alternative 11: 82.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -6 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+217}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -6e+120) t_1 (if (<= x 9.5e+217) (- (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 <= -6e+120) {
		tmp = t_1;
	} else if (x <= 9.5e+217) {
		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 <= (-6d+120)) then
        tmp = t_1
    else if (x <= 9.5d+217) 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 <= -6e+120) {
		tmp = t_1;
	} else if (x <= 9.5e+217) {
		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 <= -6e+120:
		tmp = t_1
	elif x <= 9.5e+217:
		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 <= -6e+120)
		tmp = t_1;
	elseif (x <= 9.5e+217)
		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 <= -6e+120)
		tmp = t_1;
	elseif (x <= 9.5e+217)
		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, -6e+120], t$95$1, If[LessEqual[x, 9.5e+217], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6e120 or 9.5000000000000003e217 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 84.2

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

   5. Simplified 84.2%

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

   if -6e120 < x < 9.5000000000000003e217

   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 86.3

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

   5. Simplified 86.3%

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

Alternative 12: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) - z\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-32}:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- y) z)))
   (if (<= z -1.7e+37) t_1 (if (<= z 4.6e-32) (- (log t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -y - z;
	double tmp;
	if (z <= -1.7e+37) {
		tmp = t_1;
	} else if (z <= 4.6e-32) {
		tmp = log(t) - y;
	} 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 = -y - z
    if (z <= (-1.7d+37)) then
        tmp = t_1
    else if (z <= 4.6d-32) then
        tmp = log(t) - y
    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 = -y - z;
	double tmp;
	if (z <= -1.7e+37) {
		tmp = t_1;
	} else if (z <= 4.6e-32) {
		tmp = Math.log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -y - z
	tmp = 0
	if z <= -1.7e+37:
		tmp = t_1
	elif z <= 4.6e-32:
		tmp = math.log(t) - y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (z <= -1.7e+37)
		tmp = t_1;
	elseif (z <= 4.6e-32)
		tmp = Float64(log(t) - y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -y - z;
	tmp = 0.0;
	if (z <= -1.7e+37)
		tmp = t_1;
	elseif (z <= 4.6e-32)
		tmp = log(t) - y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[z, -1.7e+37], t$95$1, If[LessEqual[z, 4.6e-32], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.70000000000000003e37 or 4.6000000000000001e-32 < 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. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

      4. clear-num N/A

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

      5. flip3-+ N/A

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

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

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

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

      2. log-lowering-log.f64 98.8

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

      2. mul-1-neg N/A

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

      3. sub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

         \[\leadsto \color{blue}{-1 \cdot y - z} \]

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 72.7

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

   10. Simplified 72.7%

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

   if -1.70000000000000003e37 < z < 4.6000000000000001e-32

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \left(\color{blue}{y \cdot \left(-1 \cdot \frac{x \cdot \log \left(\frac{1}{y}\right)}{y} - 1\right)} - z\right) + \log t \]
   4. Step-by-step derivation

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. metadata-eval N/A

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

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

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

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

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

      14. /-lowering-/.f64 88.6

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

   5. Simplified 88.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. log-lowering-log.f64 88.7

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

   8. Simplified 88.7%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

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

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

       4. log-lowering-log.f64 70.9

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

   11. Simplified 70.9%

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

Alternative 13: 47.8% accurate, 23.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;-y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= y 1.1e+35) (- z) (- y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.1e+35) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= 1.1d+35) then
        tmp = -z
    else
        tmp = -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 1.1e+35) {
		tmp = -z;
	} else {
		tmp = -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= 1.1e+35:
		tmp = -z
	else:
		tmp = -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 1.1e+35)
		tmp = Float64(-z);
	else
		tmp = Float64(-y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= 1.1e+35)
		tmp = -z;
	else
		tmp = -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 1.1e+35], (-z), (-y)]

Derivation

1. Split input into 2 regimes

2. if y < 1.0999999999999999e35

   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-lowering-neg.f64 35.5

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

   5. Simplified 35.5%

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

   if 1.0999999999999999e35 < y

   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 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-lowering-neg.f64 68.5

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

   5. Simplified 68.5%

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

Alternative 14: 57.5% 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. flip3-+ N/A

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

   2. clear-num N/A

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

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

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

   4. clear-num N/A

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

   5. flip3-+ N/A

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in x around inf

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

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

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

   2. log-lowering-log.f64 84.5

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

7. Simplified 84.5%

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

8. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{-1 \cdot y + -1 \cdot z} \]

   2. mul-1-neg N/A

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

   3. sub-neg N/A

      \[\leadsto \color{blue}{-1 \cdot y - z} \]

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

      \[\leadsto \color{blue}{-1 \cdot y - z} \]

   5. mul-1-neg N/A

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

   6. neg-lowering-neg.f64 58.0

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

10. Simplified 58.0%

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

Alternative 15: 29.2% accurate, 71.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

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

   2. neg-lowering-neg.f64 32.4

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

5. Simplified 32.4%

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

Reproduce

herbie shell --seed 2024201 
(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)))