Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 11.0s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.8163294102678397e+175
  2. y = 1.4375187394242394e+203

Specification

Math

\[\begin{array}{l} \\ \log \left(1 + e^{x}\right) - x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + e^{x}\right) - x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(1 + e^{x}\right) - x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

Julia

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := x \cdot \left(0 - y\right)\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (* x (- 0.0 y))))
   (if (<= t_0 0.1) t_1 (if (<= t_0 100.0) (fma x 0.5 (log 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = x * (0.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 100.0) {
		tmp = fma(x, 0.5, log(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(x * Float64(0.0 - y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = t_1;
	elseif (t_0 <= 100.0)
		tmp = fma(x, 0.5, log(2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], t$95$1, If[LessEqual[t$95$0, 100.0], N[(x * 0.5 + N[Log[2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.10000000000000001 or 100 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.5%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 97.2

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

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

         \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      8. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      9. cube-neg N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

      11. +-lft-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}} \]

      12. mul0-lft N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{y \cdot y + \color{blue}{0}} \]

      13. +-rgt-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot y} \cdot x} \]

      15. div-inv N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y}\right)} \cdot x \]

      16. +-rgt-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{y \cdot y + 0}}\right) \cdot x \]

      17. mul0-lft N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y + \color{blue}{0 \cdot y}}\right) \cdot x \]

      18. +-lft-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}}\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)}\right) \cdot x \]

      20. associate-*l* N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \left(\frac{1}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x\right)} \]

   7. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, 0\right), 0\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, y, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{y \cdot y + 0}\right) \cdot x} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{\color{blue}{y \cdot y}}\right) \cdot x \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y}} \cdot x \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{y \cdot y + 0}} \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y + \color{blue}{0 \cdot y}} \cdot x \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{0}^{3}} - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{y \cdot \left(y \cdot y + 0\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      11. cube-unmult N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      12. flip3-- N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      13. sub0-neg N/A

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

   9. Applied egg-rr 97.2%

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

   if 0.10000000000000001 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 100

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\frac{1}{2} - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} - y\right) + \log 2} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} - y, \log 2\right)} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} - y}, \log 2\right) \]

      4. log-lowering-log.f64 99.4

         \[\leadsto \mathsf{fma}\left(x, 0.5 - y, \color{blue}{\log 2}\right) \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 - y, \log 2\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log 2 + \frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \log 2} \]

      2. remove-double-neg N/A

         \[\leadsto \frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 2\right)\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{2} \cdot x + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log 2}\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot x - -1 \cdot \log 2} \]

      5. *-rgt-identity N/A

         \[\leadsto \frac{1}{2} \cdot x - \color{blue}{\left(-1 \cdot \log 2\right) \cdot 1} \]

      6. *-inverses N/A

         \[\leadsto \frac{1}{2} \cdot x - \left(-1 \cdot \log 2\right) \cdot \color{blue}{\frac{x}{x}} \]

      7. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot x - \color{blue}{\frac{\left(-1 \cdot \log 2\right) \cdot x}{x}} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{1}{2} \cdot x - \color{blue}{\frac{-1 \cdot \log 2}{x} \cdot x} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{1}{2} \cdot x - \color{blue}{\left(-1 \cdot \frac{\log 2}{x}\right)} \cdot x \]

      10. unsub-neg N/A

          \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\log 2}{x}\right) \cdot x\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\log 2}{x}\right) \cdot x\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto x \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\log 2}{x}\right)\right)} \cdot x\right)\right) \]

      13. distribute-lft-neg-out N/A

          \[\leadsto x \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\log 2}{x} \cdot x\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto x \cdot \frac{1}{2} + \color{blue}{\frac{\log 2}{x} \cdot x} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2}, \frac{\log 2}{x} \cdot x\right)} \]

      16. associate-*l/ N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\frac{\log 2 \cdot x}{x}}\right) \]

      17. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\log 2 \cdot \frac{x}{x}}\right) \]

      18. *-inverses N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \log 2 \cdot \color{blue}{1}\right) \]

      19. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log 2\right)\right)\right)\right)} \cdot 1\right) \]

      20. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log 2}\right)\right) \cdot 1\right) \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\left(-1 \cdot \log 2\right) \cdot 1\right)}\right) \]

      22. *-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \mathsf{neg}\left(\color{blue}{-1 \cdot \log 2}\right)\right) \]

      23. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(x, \frac{1}{2}, \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log 2\right)\right)}\right)\right) \]

   8. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \log 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.1:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 100:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := x \cdot \left(0 - y\right)\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;\mathsf{log1p}\left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (* x (- 0.0 y))))
   (if (<= t_0 0.1) t_1 (if (<= t_0 100.0) (log1p (+ 1.0 x)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = x * (0.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 100.0) {
		tmp = log1p((1.0 + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double t_1 = x * (0.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 100.0) {
		tmp = Math.log1p((1.0 + x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	t_1 = x * (0.0 - y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = t_1
	elif t_0 <= 100.0:
		tmp = math.log1p((1.0 + x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(x * Float64(0.0 - y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = t_1;
	elseif (t_0 <= 100.0)
		tmp = log1p(Float64(1.0 + x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], t$95$1, If[LessEqual[t$95$0, 100.0], N[Log[1 + N[(1.0 + x), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.10000000000000001 or 100 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.5%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 97.2

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

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

         \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      8. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      9. cube-neg N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

      11. +-lft-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}} \]

      12. mul0-lft N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{y \cdot y + \color{blue}{0}} \]

      13. +-rgt-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot y} \cdot x} \]

      15. div-inv N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y}\right)} \cdot x \]

      16. +-rgt-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{y \cdot y + 0}}\right) \cdot x \]

      17. mul0-lft N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y + \color{blue}{0 \cdot y}}\right) \cdot x \]

      18. +-lft-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}}\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)}\right) \cdot x \]

      20. associate-*l* N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \left(\frac{1}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x\right)} \]

   7. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, 0\right), 0\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, y, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{y \cdot y + 0}\right) \cdot x} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{\color{blue}{y \cdot y}}\right) \cdot x \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y}} \cdot x \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{y \cdot y + 0}} \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y + \color{blue}{0 \cdot y}} \cdot x \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{0}^{3}} - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{y \cdot \left(y \cdot y + 0\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      11. cube-unmult N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      12. flip3-- N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      13. sub0-neg N/A

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

   9. Applied egg-rr 97.2%

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

   if 0.10000000000000001 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 100

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
   4. Step-by-step derivation

      1. accelerator-lowering-log1p.f64 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

      2. exp-lowering-exp.f64 99.5

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{x}}\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\color{blue}{x + 1}\right) \]

      2. +-lowering-+.f64 98.8

         \[\leadsto \mathsf{log1p}\left(\color{blue}{x + 1}\right) \]

   8. Simplified 98.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x + 1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.1:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 100:\\ \;\;\;\;\mathsf{log1p}\left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := x \cdot \left(0 - y\right)\\ \mathbf{if}\;t\_0 \leq 0.1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (* x (- 0.0 y))))
   (if (<= t_0 0.1) t_1 (if (<= t_0 100.0) (log 2.0) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = x * (0.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 100.0) {
		tmp = log(2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = log((1.0d0 + exp(x))) - (x * y)
    t_1 = x * (0.0d0 - y)
    if (t_0 <= 0.1d0) then
        tmp = t_1
    else if (t_0 <= 100.0d0) then
        tmp = log(2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double t_1 = x * (0.0 - y);
	double tmp;
	if (t_0 <= 0.1) {
		tmp = t_1;
	} else if (t_0 <= 100.0) {
		tmp = Math.log(2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	t_1 = x * (0.0 - y)
	tmp = 0
	if t_0 <= 0.1:
		tmp = t_1
	elif t_0 <= 100.0:
		tmp = math.log(2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(x * Float64(0.0 - y))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = t_1;
	elseif (t_0 <= 100.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((1.0 + exp(x))) - (x * y);
	t_1 = x * (0.0 - y);
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = t_1;
	elseif (t_0 <= 100.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.1], t$95$1, If[LessEqual[t$95$0, 100.0], N[Log[2.0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.10000000000000001 or 100 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.5%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 97.2

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

   5. Simplified 97.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

         \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      8. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      9. cube-neg N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

      11. +-lft-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}} \]

      12. mul0-lft N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{y \cdot y + \color{blue}{0}} \]

      13. +-rgt-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot y} \cdot x} \]

      15. div-inv N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y}\right)} \cdot x \]

      16. +-rgt-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{y \cdot y + 0}}\right) \cdot x \]

      17. mul0-lft N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y + \color{blue}{0 \cdot y}}\right) \cdot x \]

      18. +-lft-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}}\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)}\right) \cdot x \]

      20. associate-*l* N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \left(\frac{1}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x\right)} \]

   7. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, 0\right), 0\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, y, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{y \cdot y + 0}\right) \cdot x} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{\color{blue}{y \cdot y}}\right) \cdot x \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y}} \cdot x \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{y \cdot y + 0}} \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y + \color{blue}{0 \cdot y}} \cdot x \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{0}^{3}} - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{y \cdot \left(y \cdot y + 0\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      11. cube-unmult N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      12. flip3-- N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      13. sub0-neg N/A

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

   9. Applied egg-rr 97.2%

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

   if 0.10000000000000001 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 100

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} \]
   4. Step-by-step derivation

      1. log-lowering-log.f64 98.7

         \[\leadsto \color{blue}{\log 2} \]

   5. Simplified 98.7%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.1:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 100:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.36) (* x (- 0.0 y)) (fma x (- 0.5 y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.36) {
		tmp = x * (0.0 - y);
	} else {
		tmp = fma(x, (0.5 - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.36)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = fma(x, Float64(0.5 - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.36], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3600000000000001

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

         \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      8. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      9. cube-neg N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

      11. +-lft-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}} \]

      12. mul0-lft N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{y \cdot y + \color{blue}{0}} \]

      13. +-rgt-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot y} \cdot x} \]

      15. div-inv N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y}\right)} \cdot x \]

      16. +-rgt-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{y \cdot y + 0}}\right) \cdot x \]

      17. mul0-lft N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y + \color{blue}{0 \cdot y}}\right) \cdot x \]

      18. +-lft-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}}\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)}\right) \cdot x \]

      20. associate-*l* N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \left(\frac{1}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x\right)} \]

   7. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, 0\right), 0\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, y, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{y \cdot y + 0}\right) \cdot x} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{\color{blue}{y \cdot y}}\right) \cdot x \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y}} \cdot x \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{y \cdot y + 0}} \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y + \color{blue}{0 \cdot y}} \cdot x \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{0}^{3}} - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{y \cdot \left(y \cdot y + 0\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      11. cube-unmult N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      12. flip3-- N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      13. sub0-neg N/A

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

   9. Applied egg-rr 100.0%

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

   if -1.3600000000000001 < x

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\frac{1}{2} - y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} - y\right) + \log 2} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} - y, \log 2\right)} \]

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{1}{2} - y}, \log 2\right) \]

      4. log-lowering-log.f64 98.6

         \[\leadsto \mathsf{fma}\left(x, 0.5 - y, \color{blue}{\log 2}\right) \]

   5. Simplified 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 - y, \log 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -125:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -125.0) (* x (- 0.0 y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -125.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = log(2.0) - (x * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-125.0d0)) then
        tmp = x * (0.0d0 - y)
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -125.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -125.0:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -125.0)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -125.0)
		tmp = x * (0.0 - y);
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -125.0], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -125

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

      5. neg-sub0 N/A

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

      6. --lowering--.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

      2. sub0-neg N/A

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

      3. *-commutative N/A

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

      4. sub0-neg N/A

         \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      5. flip3-- N/A

         \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      8. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      9. cube-neg N/A

         \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

      10. metadata-eval N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

      11. +-lft-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}} \]

      12. mul0-lft N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{y \cdot y + \color{blue}{0}} \]

      13. +-rgt-identity N/A

          \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y}} \]

      14. associate-*l/ N/A

          \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot y} \cdot x} \]

      15. div-inv N/A

          \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y}\right)} \cdot x \]

      16. +-rgt-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{y \cdot y + 0}}\right) \cdot x \]

      17. mul0-lft N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y + \color{blue}{0 \cdot y}}\right) \cdot x \]

      18. +-lft-identity N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}}\right) \cdot x \]

      19. metadata-eval N/A

          \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)}\right) \cdot x \]

      20. associate-*l* N/A

          \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \left(\frac{1}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x\right)} \]

   7. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, 0\right), 0\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, y, 0\right)} \cdot x\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{y \cdot y + 0}\right) \cdot x} \]

      2. +-rgt-identity N/A

         \[\leadsto \left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{\color{blue}{y \cdot y}}\right) \cdot x \]

      3. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y}} \cdot x \]

      4. +-rgt-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{y \cdot y + 0}} \cdot x \]

      5. mul0-lft N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y + \color{blue}{0 \cdot y}} \cdot x \]

      6. +-lft-identity N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

      7. metadata-eval N/A

         \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      8. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{0}^{3}} - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      9. +-rgt-identity N/A

         \[\leadsto \frac{{0}^{3} - \color{blue}{y \cdot \left(y \cdot y + 0\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      10. +-rgt-identity N/A

          \[\leadsto \frac{{0}^{3} - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      11. cube-unmult N/A

          \[\leadsto \frac{{0}^{3} - \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

      12. flip3-- N/A

          \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

      13. sub0-neg N/A

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

   9. Applied egg-rr 100.0%

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

   if -125 < x

   1. Initial program 98.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]
   4. Step-by-step derivation

      1. log-lowering-log.f64 98.5

         \[\leadsto \color{blue}{\log 2} - x \cdot y \]

   5. Simplified 98.5%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -125:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.1% accurate, 23.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0 - y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- 0.0 y)))

C

double code(double x, double y) {
	return x * (0.0 - y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (0.0d0 - y)
end function

Java

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

Python

def code(x, y):
	return x * (0.0 - y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 51.3

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

5. Simplified 51.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]

   2. sub0-neg N/A

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

   3. *-commutative N/A

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

   4. sub0-neg N/A

      \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

   5. flip3-- N/A

      \[\leadsto \color{blue}{\frac{{0}^{3} - {y}^{3}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\left({0}^{3} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)}} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\left(\color{blue}{0} - {y}^{3}\right) \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   8. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left({y}^{3}\right)\right)} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   9. cube-neg N/A

      \[\leadsto \frac{\color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3}} \cdot x}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{0} + \left(y \cdot y + 0 \cdot y\right)} \]

   11. +-lft-identity N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y + 0 \cdot y}} \]

   12. mul0-lft N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{y \cdot y + \color{blue}{0}} \]

   13. +-rgt-identity N/A

       \[\leadsto \frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot x}{\color{blue}{y \cdot y}} \]

   14. associate-*l/ N/A

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{neg}\left(y\right)\right)}^{3}}{y \cdot y} \cdot x} \]

   15. div-inv N/A

       \[\leadsto \color{blue}{\left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y}\right)} \cdot x \]

   16. +-rgt-identity N/A

       \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{y \cdot y + 0}}\right) \cdot x \]

   17. mul0-lft N/A

       \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{y \cdot y + \color{blue}{0 \cdot y}}\right) \cdot x \]

   18. +-lft-identity N/A

       \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}}\right) \cdot x \]

   19. metadata-eval N/A

       \[\leadsto \left({\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \frac{1}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)}\right) \cdot x \]

   20. associate-*l* N/A

       \[\leadsto \color{blue}{{\left(\mathsf{neg}\left(y\right)\right)}^{3} \cdot \left(\frac{1}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x\right)} \]

7. Applied egg-rr 13.2%

   \[\leadsto \color{blue}{\left(0 - \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, 0\right), 0\right)\right) \cdot \left(\frac{1}{\mathsf{fma}\left(y, y, 0\right)} \cdot x\right)} \]
8. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{y \cdot y + 0}\right) \cdot x} \]

   2. +-rgt-identity N/A

      \[\leadsto \left(\left(0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)\right) \cdot \frac{1}{\color{blue}{y \cdot y}}\right) \cdot x \]

   3. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y}} \cdot x \]

   4. +-rgt-identity N/A

      \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{y \cdot y + 0}} \cdot x \]

   5. mul0-lft N/A

      \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{y \cdot y + \color{blue}{0 \cdot y}} \cdot x \]

   6. +-lft-identity N/A

      \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 + \left(y \cdot y + 0 \cdot y\right)}} \cdot x \]

   7. metadata-eval N/A

      \[\leadsto \frac{0 - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{\color{blue}{0 \cdot 0} + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{{0}^{3}} - \left(y \cdot \left(y \cdot y + 0\right) + 0\right)}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

   9. +-rgt-identity N/A

      \[\leadsto \frac{{0}^{3} - \color{blue}{y \cdot \left(y \cdot y + 0\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

   10. +-rgt-identity N/A

       \[\leadsto \frac{{0}^{3} - y \cdot \color{blue}{\left(y \cdot y\right)}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

   11. cube-unmult N/A

       \[\leadsto \frac{{0}^{3} - \color{blue}{{y}^{3}}}{0 \cdot 0 + \left(y \cdot y + 0 \cdot y\right)} \cdot x \]

   12. flip3-- N/A

       \[\leadsto \color{blue}{\left(0 - y\right)} \cdot x \]

   13. sub0-neg N/A

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

9. Applied egg-rr 51.3%

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

10. Final simplification 51.3%

    \[\leadsto x \cdot \left(0 - y\right) \]
11. Add Preprocessing

Alternative 8: 2.3% accurate, 35.3× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x y))

C

double code(double x, double y) {
	return x * y;
}

Fortran

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

Java

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

Python

def code(x, y):
	return x * y

Julia

function code(x, y)
	return Float64(x * y)
end

MATLAB

function tmp = code(x, y)
	tmp = x * y;
end

Wolfram

code[x_, y_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right) + 0} \]

   2. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot y\right)\right)} + 0 \]

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

      \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} + 0 \]

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

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

   5. neg-sub0 N/A

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

   6. --lowering--.f64 51.3

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

5. Simplified 51.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0 - y, 0\right)} \]
6. Step-by-step derivation

   1. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(x \cdot \left(0 - y\right)\right)}^{3} + {0}^{3}}{\left(x \cdot \left(0 - y\right)\right) \cdot \left(x \cdot \left(0 - y\right)\right) + \left(0 \cdot 0 - \left(x \cdot \left(0 - y\right)\right) \cdot 0\right)}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left({\left(x \cdot \left(0 - y\right)\right)}^{3} + {0}^{3}\right) \cdot \frac{1}{\left(x \cdot \left(0 - y\right)\right) \cdot \left(x \cdot \left(0 - y\right)\right) + \left(0 \cdot 0 - \left(x \cdot \left(0 - y\right)\right) \cdot 0\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \left({\left(x \cdot \left(0 - y\right)\right)}^{3} + \color{blue}{0}\right) \cdot \frac{1}{\left(x \cdot \left(0 - y\right)\right) \cdot \left(x \cdot \left(0 - y\right)\right) + \left(0 \cdot 0 - \left(x \cdot \left(0 - y\right)\right) \cdot 0\right)} \]

   4. +-rgt-identity N/A

      \[\leadsto \color{blue}{{\left(x \cdot \left(0 - y\right)\right)}^{3}} \cdot \frac{1}{\left(x \cdot \left(0 - y\right)\right) \cdot \left(x \cdot \left(0 - y\right)\right) + \left(0 \cdot 0 - \left(x \cdot \left(0 - y\right)\right) \cdot 0\right)} \]

   5. sqr-pow N/A

      \[\leadsto \color{blue}{\left({\left(x \cdot \left(0 - y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x \cdot \left(0 - y\right)\right)}^{\left(\frac{3}{2}\right)}\right)} \cdot \frac{1}{\left(x \cdot \left(0 - y\right)\right) \cdot \left(x \cdot \left(0 - y\right)\right) + \left(0 \cdot 0 - \left(x \cdot \left(0 - y\right)\right) \cdot 0\right)} \]

7. Applied egg-rr 2.4%

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

8. Final simplification 2.4%

   \[\leadsto x \cdot y \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0)
		tmp = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y));
	else
		tmp = Float64(log(Float64(1.0 + exp(Float64(-x)))) - Float64(Float64(-x) * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = log((1.0 + exp(x))) - (x * y);
	else
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

  (- (log (+ 1.0 (exp x))) (* x y)))