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

Percentage Accurate: 99.9% → 99.9%

Time: 13.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

   2. sub-neg 99.8%

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

   3. sub-neg 99.8%

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

   4. +-commutative 99.8%

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

   5. associate-+l+ 99.8%

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

   6. +-commutative 99.8%

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

   7. unsub-neg 99.8%

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

   8. fma-udef 99.8%

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

   9. neg-sub0 99.8%

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

   10. associate-+l- 99.8%

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

   11. neg-sub0 99.8%

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

   12. +-commutative 99.8%

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

   13. unsub-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - y\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq -2 \cdot 10^{+118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq -3.5 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq -1400000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) y)) (t_2 (- (- y) z)))
   (if (<= t_1 -2e+164)
     t_1
     (if (<= t_1 -2e+118)
       t_2
       (if (<= t_1 -3.5e+79)
         t_1
         (if (<= t_1 -1400000000.0)
           t_2
           (if (<= t_1 5e+61) (- (log t) z) t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - y;
	double t_2 = -y - z;
	double tmp;
	if (t_1 <= -2e+164) {
		tmp = t_1;
	} else if (t_1 <= -2e+118) {
		tmp = t_2;
	} else if (t_1 <= -3.5e+79) {
		tmp = t_1;
	} else if (t_1 <= -1400000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+61) {
		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)) - y
    t_2 = -y - z
    if (t_1 <= (-2d+164)) then
        tmp = t_1
    else if (t_1 <= (-2d+118)) then
        tmp = t_2
    else if (t_1 <= (-3.5d+79)) then
        tmp = t_1
    else if (t_1 <= (-1400000000.0d0)) then
        tmp = t_2
    else if (t_1 <= 5d+61) 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)) - y;
	double t_2 = -y - z;
	double tmp;
	if (t_1 <= -2e+164) {
		tmp = t_1;
	} else if (t_1 <= -2e+118) {
		tmp = t_2;
	} else if (t_1 <= -3.5e+79) {
		tmp = t_1;
	} else if (t_1 <= -1400000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e+61) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - y
	t_2 = -y - z
	tmp = 0
	if t_1 <= -2e+164:
		tmp = t_1
	elif t_1 <= -2e+118:
		tmp = t_2
	elif t_1 <= -3.5e+79:
		tmp = t_1
	elif t_1 <= -1400000000.0:
		tmp = t_2
	elif t_1 <= 5e+61:
		tmp = math.log(t) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - y)
	t_2 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (t_1 <= -2e+164)
		tmp = t_1;
	elseif (t_1 <= -2e+118)
		tmp = t_2;
	elseif (t_1 <= -3.5e+79)
		tmp = t_1;
	elseif (t_1 <= -1400000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+61)
		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)) - y;
	t_2 = -y - z;
	tmp = 0.0;
	if (t_1 <= -2e+164)
		tmp = t_1;
	elseif (t_1 <= -2e+118)
		tmp = t_2;
	elseif (t_1 <= -3.5e+79)
		tmp = t_1;
	elseif (t_1 <= -1400000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e+61)
		tmp = log(t) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]}, Block[{t$95$2 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+164], t$95$1, If[LessEqual[t$95$1, -2e+118], t$95$2, If[LessEqual[t$95$1, -3.5e+79], t$95$1, If[LessEqual[t$95$1, -1400000000.0], t$95$2, If[LessEqual[t$95$1, 5e+61], 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) < -2e164 or -1.99999999999999993e118 < (-.f64 (*.f64 x (log.f64 y)) y) < -3.4999999999999998e79 or 5.00000000000000018e61 < (-.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. Step-by-step derivation

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in z around 0 92.2%

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

   if -2e164 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.99999999999999993e118 or -3.4999999999999998e79 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.4e9

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 98.1%

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

   6. Taylor expanded in x around 0 87.1%

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

      1. neg-mul-1 87.1%

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

      2. distribute-neg-in 87.1%

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

      3. sub-neg 87.1%

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

   8. Simplified 87.1%

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

   if -1.4e9 < (-.f64 (*.f64 x (log.f64 y)) y) < 5.00000000000000018e61

   1. Initial program 99.9%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.7%

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

   6. Taylor expanded in x around 0 95.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+164}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \cdot \log y - y \leq -3.5 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq -1400000000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \cdot \log y - y \leq 5 \cdot 10^{+61}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t_1 - y\\ t_3 := \left(-y\right) - z\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -2 \cdot 10^{+118}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq -3.5 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_2 \leq -1400000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t_1 - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)) (t_3 (- (- y) z)))
   (if (<= t_2 -2e+164)
     t_2
     (if (<= t_2 -2e+118)
       t_3
       (if (<= t_2 -3.5e+79)
         t_2
         (if (<= t_2 -1400000000.0)
           t_3
           (if (<= t_2 2e-24) (- (log t) z) (- t_1 z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double t_3 = -y - z;
	double tmp;
	if (t_2 <= -2e+164) {
		tmp = t_2;
	} else if (t_2 <= -2e+118) {
		tmp = t_3;
	} else if (t_2 <= -3.5e+79) {
		tmp = t_2;
	} else if (t_2 <= -1400000000.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-24) {
		tmp = log(t) - z;
	} else {
		tmp = t_1 - 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    t_3 = -y - z
    if (t_2 <= (-2d+164)) then
        tmp = t_2
    else if (t_2 <= (-2d+118)) then
        tmp = t_3
    else if (t_2 <= (-3.5d+79)) then
        tmp = t_2
    else if (t_2 <= (-1400000000.0d0)) then
        tmp = t_3
    else if (t_2 <= 2d-24) then
        tmp = log(t) - z
    else
        tmp = t_1 - 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);
	double t_2 = t_1 - y;
	double t_3 = -y - z;
	double tmp;
	if (t_2 <= -2e+164) {
		tmp = t_2;
	} else if (t_2 <= -2e+118) {
		tmp = t_3;
	} else if (t_2 <= -3.5e+79) {
		tmp = t_2;
	} else if (t_2 <= -1400000000.0) {
		tmp = t_3;
	} else if (t_2 <= 2e-24) {
		tmp = Math.log(t) - z;
	} else {
		tmp = t_1 - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	t_3 = -y - z
	tmp = 0
	if t_2 <= -2e+164:
		tmp = t_2
	elif t_2 <= -2e+118:
		tmp = t_3
	elif t_2 <= -3.5e+79:
		tmp = t_2
	elif t_2 <= -1400000000.0:
		tmp = t_3
	elif t_2 <= 2e-24:
		tmp = math.log(t) - z
	else:
		tmp = t_1 - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	t_3 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (t_2 <= -2e+164)
		tmp = t_2;
	elseif (t_2 <= -2e+118)
		tmp = t_3;
	elseif (t_2 <= -3.5e+79)
		tmp = t_2;
	elseif (t_2 <= -1400000000.0)
		tmp = t_3;
	elseif (t_2 <= 2e-24)
		tmp = Float64(log(t) - z);
	else
		tmp = Float64(t_1 - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - y;
	t_3 = -y - z;
	tmp = 0.0;
	if (t_2 <= -2e+164)
		tmp = t_2;
	elseif (t_2 <= -2e+118)
		tmp = t_3;
	elseif (t_2 <= -3.5e+79)
		tmp = t_2;
	elseif (t_2 <= -1400000000.0)
		tmp = t_3;
	elseif (t_2 <= 2e-24)
		tmp = log(t) - z;
	else
		tmp = t_1 - z;
	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]}, Block[{t$95$3 = N[((-y) - z), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+164], t$95$2, If[LessEqual[t$95$2, -2e+118], t$95$3, If[LessEqual[t$95$2, -3.5e+79], t$95$2, If[LessEqual[t$95$2, -1400000000.0], t$95$3, If[LessEqual[t$95$2, 2e-24], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], N[(t$95$1 - z), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -2e164 or -1.99999999999999993e118 < (-.f64 (*.f64 x (log.f64 y)) y) < -3.4999999999999998e79

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.8%

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

   6. Taylor expanded in z around 0 93.4%

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

   if -2e164 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.99999999999999993e118 or -3.4999999999999998e79 < (-.f64 (*.f64 x (log.f64 y)) y) < -1.4e9

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 98.1%

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

   6. Taylor expanded in x around 0 87.1%

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

      1. neg-mul-1 87.1%

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

      2. distribute-neg-in 87.1%

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

      3. sub-neg 87.1%

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

   8. Simplified 87.1%

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

   if -1.4e9 < (-.f64 (*.f64 x (log.f64 y)) y) < 1.99999999999999985e-24

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.7%

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

   6. Taylor expanded in x around 0 99.5%

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

   if 1.99999999999999985e-24 < (-.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. Step-by-step derivation

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 98.5%

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

   6. Taylor expanded in y around 0 98.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \log y - y \leq -2 \cdot 10^{+164}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \cdot \log y - y \leq -3.5 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \cdot \log y - y \leq -1400000000:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \cdot \log y - y \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - z\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;y \leq 0.0003:\\ \;\;\;\;\left(\log t + t_1\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= y 0.0003) (- (+ (log t) t_1) z) (- (- t_1 y) z))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (y <= 0.0003) {
		tmp = (log(t) + t_1) - z;
	} else {
		tmp = (t_1 - y) - 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)
    if (y <= 0.0003d0) then
        tmp = (log(t) + t_1) - z
    else
        tmp = (t_1 - y) - 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);
	double tmp;
	if (y <= 0.0003) {
		tmp = (Math.log(t) + t_1) - z;
	} else {
		tmp = (t_1 - y) - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if y <= 0.0003:
		tmp = (math.log(t) + t_1) - z
	else:
		tmp = (t_1 - y) - z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (y <= 0.0003)
		tmp = Float64(Float64(log(t) + t_1) - z);
	else
		tmp = Float64(Float64(t_1 - y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (y <= 0.0003)
		tmp = (log(t) + t_1) - z;
	else
		tmp = (t_1 - y) - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.0003], N[(N[(N[Log[t], $MachinePrecision] + t$95$1), $MachinePrecision] - z), $MachinePrecision], N[(N[(t$95$1 - y), $MachinePrecision] - z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999974e-4

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 99.6%

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

   if 2.99999999999999974e-4 < y

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.0003:\\ \;\;\;\;\left(\log t + x \cdot \log y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

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

Alternative 7: 70.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := \left(-y\right) - z\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-292}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-230}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+67}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- (- y) z)))
   (if (<= x -5.2e+105)
     t_1
     (if (<= x -1.55e-292)
       t_2
       (if (<= x 4.2e-230) (- (log t) z) (if (<= x 2.7e+67) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = -y - z;
	double tmp;
	if (x <= -5.2e+105) {
		tmp = t_1;
	} else if (x <= -1.55e-292) {
		tmp = t_2;
	} else if (x <= 4.2e-230) {
		tmp = log(t) - z;
	} else if (x <= 2.7e+67) {
		tmp = t_2;
	} 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 = -y - z
    if (x <= (-5.2d+105)) then
        tmp = t_1
    else if (x <= (-1.55d-292)) then
        tmp = t_2
    else if (x <= 4.2d-230) then
        tmp = log(t) - z
    else if (x <= 2.7d+67) then
        tmp = t_2
    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 = -y - z;
	double tmp;
	if (x <= -5.2e+105) {
		tmp = t_1;
	} else if (x <= -1.55e-292) {
		tmp = t_2;
	} else if (x <= 4.2e-230) {
		tmp = Math.log(t) - z;
	} else if (x <= 2.7e+67) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = -y - z
	tmp = 0
	if x <= -5.2e+105:
		tmp = t_1
	elif x <= -1.55e-292:
		tmp = t_2
	elif x <= 4.2e-230:
		tmp = math.log(t) - z
	elif x <= 2.7e+67:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(Float64(-y) - z)
	tmp = 0.0
	if (x <= -5.2e+105)
		tmp = t_1;
	elseif (x <= -1.55e-292)
		tmp = t_2;
	elseif (x <= 4.2e-230)
		tmp = Float64(log(t) - z);
	elseif (x <= 2.7e+67)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = -y - z;
	tmp = 0.0;
	if (x <= -5.2e+105)
		tmp = t_1;
	elseif (x <= -1.55e-292)
		tmp = t_2;
	elseif (x <= 4.2e-230)
		tmp = log(t) - z;
	elseif (x <= 2.7e+67)
		tmp = t_2;
	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[((-y) - z), $MachinePrecision]}, If[LessEqual[x, -5.2e+105], t$95$1, If[LessEqual[x, -1.55e-292], t$95$2, If[LessEqual[x, 4.2e-230], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], If[LessEqual[x, 2.7e+67], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.2000000000000004e105 or 2.6999999999999999e67 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 98.0%

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

      3. associate--l- 98.0%

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

      4. associate-+r- 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in x around inf 73.6%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \log y}}\right)}^{3} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. *-commutative 74.7%

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

   9. Applied egg-rr 74.7%

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

   if -5.2000000000000004e105 < x < -1.55e-292 or 4.1999999999999997e-230 < x < 2.6999999999999999e67

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 85.9%

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

   6. Taylor expanded in x around 0 79.7%

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

      1. neg-mul-1 79.7%

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

      2. distribute-neg-in 79.7%

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

      3. sub-neg 79.7%

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

   8. Simplified 79.7%

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

   if -1.55e-292 < x < 4.1999999999999997e-230

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 92.0%

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

   6. Taylor expanded in x around 0 92.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-292}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-230}:\\ \;\;\;\;\log t - z\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+67}:\\ \;\;\;\;\left(-y\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t_1 - y\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;t_1 - z\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= x -1.9e+62)
     t_2
     (if (<= x 1.3e-7)
       (- (- (log t) z) y)
       (if (<= x 5.6e+146) (- t_1 z) t_2)))))

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 (x <= -1.9e+62) {
		tmp = t_2;
	} else if (x <= 1.3e-7) {
		tmp = (log(t) - z) - y;
	} else if (x <= 5.6e+146) {
		tmp = t_1 - z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - y
    if (x <= (-1.9d+62)) then
        tmp = t_2
    else if (x <= 1.3d-7) then
        tmp = (log(t) - z) - y
    else if (x <= 5.6d+146) then
        tmp = t_1 - z
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (x <= -1.9e+62) {
		tmp = t_2;
	} else if (x <= 1.3e-7) {
		tmp = (Math.log(t) - z) - y;
	} else if (x <= 5.6e+146) {
		tmp = t_1 - z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - y
	tmp = 0
	if x <= -1.9e+62:
		tmp = t_2
	elif x <= 1.3e-7:
		tmp = (math.log(t) - z) - y
	elif x <= 5.6e+146:
		tmp = t_1 - z
	else:
		tmp = t_2
	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 (x <= -1.9e+62)
		tmp = t_2;
	elseif (x <= 1.3e-7)
		tmp = Float64(Float64(log(t) - z) - y);
	elseif (x <= 5.6e+146)
		tmp = Float64(t_1 - z);
	else
		tmp = t_2;
	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 (x <= -1.9e+62)
		tmp = t_2;
	elseif (x <= 1.3e-7)
		tmp = (log(t) - z) - y;
	elseif (x <= 5.6e+146)
		tmp = t_1 - z;
	else
		tmp = t_2;
	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[x, -1.9e+62], t$95$2, If[LessEqual[x, 1.3e-7], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 5.6e+146], N[(t$95$1 - z), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.89999999999999992e62 or 5.6000000000000002e146 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.7%

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

   6. Taylor expanded in z around 0 91.9%

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

   if -1.89999999999999992e62 < x < 1.29999999999999999e-7

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. unsub-neg 100.0%

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

      8. fma-udef 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 97.7%

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

   if 1.29999999999999999e-7 < x < 5.6000000000000002e146

   1. Initial program 99.8%

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

      1. associate-+l- 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.1%

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

   6. Taylor expanded in y around 0 90.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \log y - y\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;x \cdot \log y - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5 \lor \neg \left(x \leq 8.8 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.5) (not (<= x 8.8e-14)))
   (- (- (* x (log y)) y) z)
   (- (- (log t) z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.5) || !(x <= 8.8e-14)) {
		tmp = ((x * log(y)) - y) - z;
	} else {
		tmp = (log(t) - z) - 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 ((x <= (-0.5d0)) .or. (.not. (x <= 8.8d-14))) then
        tmp = ((x * log(y)) - y) - z
    else
        tmp = (log(t) - z) - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.5) || !(x <= 8.8e-14)) {
		tmp = ((x * Math.log(y)) - y) - z;
	} else {
		tmp = (Math.log(t) - z) - y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.5) or not (x <= 8.8e-14):
		tmp = ((x * math.log(y)) - y) - z
	else:
		tmp = (math.log(t) - z) - y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.5) || !(x <= 8.8e-14))
		tmp = Float64(Float64(Float64(x * log(y)) - y) - z);
	else
		tmp = Float64(Float64(log(t) - z) - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.5) || ~((x <= 8.8e-14)))
		tmp = ((x * log(y)) - y) - z;
	else
		tmp = (log(t) - z) - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.5], N[Not[LessEqual[x, 8.8e-14]], $MachinePrecision]], N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision] - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.5 or 8.8000000000000004e-14 < x

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 99.1%

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

   if -0.5 < x < 8.8000000000000004e-14

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

      7. unsub-neg 100.0%

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

      8. fma-udef 100.0%

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

      9. neg-sub0 100.0%

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

      10. associate-+l- 100.0%

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

      11. neg-sub0 100.0%

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

      12. +-commutative 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.5 \lor \neg \left(x \leq 8.8 \cdot 10^{-14}\right):\\ \;\;\;\;\left(x \cdot \log y - y\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(\log t - z\right) - y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+105} \lor \neg \left(x \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.8e+105) (not (<= x 1.8e+67))) (* x (log y)) (- (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e+105) || !(x <= 1.8e+67)) {
		tmp = x * log(y);
	} else {
		tmp = -y - 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 <= (-6.8d+105)) .or. (.not. (x <= 1.8d+67))) then
        tmp = x * log(y)
    else
        tmp = -y - z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.8e+105) || !(x <= 1.8e+67)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -y - z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.8e+105) or not (x <= 1.8e+67):
		tmp = x * math.log(y)
	else:
		tmp = -y - z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.8e+105) || !(x <= 1.8e+67))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(-y) - z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -6.8e+105) || ~((x <= 1.8e+67)))
		tmp = x * log(y);
	else
		tmp = -y - z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.8e+105], N[Not[LessEqual[x, 1.8e+67]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[((-y) - z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7999999999999999e105 or 1.7999999999999999e67 < x

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 98.0%

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

      3. associate--l- 98.0%

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

      4. associate-+r- 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in x around inf 73.6%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \log y}}\right)}^{3} \]
   8. Step-by-step derivation

      1. rem-cube-cbrt 74.7%

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

      2. *-commutative 74.7%

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

   9. Applied egg-rr 74.7%

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

   if -6.7999999999999999e105 < x < 1.7999999999999999e67

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 82.1%

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

   6. Taylor expanded in x around 0 76.8%

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

      1. neg-mul-1 76.8%

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

      2. distribute-neg-in 76.8%

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

      3. sub-neg 76.8%

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

   8. Simplified 76.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+105} \lor \neg \left(x \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) - z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.7% accurate, 52.3× 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 99.8%

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

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 89.5%

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

6. Taylor expanded in x around 0 55.0%

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

   1. neg-mul-1 55.0%

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

   2. distribute-neg-in 55.0%

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

   3. sub-neg 55.0%

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

8. Simplified 55.0%

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

9. Final simplification 55.0%

   \[\leadsto \left(-y\right) - z \]
10. Add Preprocessing

Alternative 12: 30.4% accurate, 104.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. associate-+l- 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 28.1%

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

   1. neg-mul-1 28.1%

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

7. Simplified 28.1%

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

8. Final simplification 28.1%

   \[\leadsto -z \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(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)))