Logarithmic Transform

Percentage Accurate: 42.2% → 99.4%

Time: 7.6s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. c = -2.7965141841383366e-67
  2. x = 3.3433761288704576e+24
  3. y = 1.056693917686406e+223

Specification

Math

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

FPCore

(FPCore (c x y)
 :precision binary64
 (* c (log (+ 1.0 (* (- (pow E x) 1.0) y)))))

C

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

Java

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

Python

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

Julia

function code(c, x, y)
	return Float64(c * log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))))
end

MATLAB

function tmp = code(c, x, y)
	tmp = c * log((1.0 + (((2.71828182845904523536 ^ x) - 1.0) * y)));
end

Wolfram

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

Initial Program: 42.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (c x y)
 :precision binary64
 (* c (log (+ 1.0 (* (- (pow E x) 1.0) y)))))

C

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

Java

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

Python

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

Julia

function code(c, x, y)
	return Float64(c * log(Float64(1.0 + Float64(Float64((exp(1) ^ x) - 1.0) * y))))
end

MATLAB

function tmp = code(c, x, y)
	tmp = c * log((1.0 + (((2.71828182845904523536 ^ x) - 1.0) * y)));
end

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{-42}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y (expm1 x))))))
   (if (<= y -2.1e-8)
     t_0
     (if (<= y 1e-42)
       (* y (fma -0.5 (* c (* y (pow (expm1 x) 2.0))) (* c (expm1 x))))
       t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * expm1(x)));
	double tmp;
	if (y <= -2.1e-8) {
		tmp = t_0;
	} else if (y <= 1e-42) {
		tmp = y * fma(-0.5, (c * (y * pow(expm1(x), 2.0))), (c * expm1(x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(y * expm1(x))))
	tmp = 0.0
	if (y <= -2.1e-8)
		tmp = t_0;
	elseif (y <= 1e-42)
		tmp = Float64(y * fma(-0.5, Float64(c * Float64(y * (expm1(x) ^ 2.0))), Float64(c * expm1(x))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.1e-8], t$95$0, If[LessEqual[y, 1e-42], N[(y * N[(-0.5 * N[(c * N[(y * N[Power[N[(Exp[x] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.09999999999999994e-8 or 1.00000000000000004e-42 < y

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   if -2.09999999999999994e-8 < y < 1.00000000000000004e-42

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.6

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.6%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (log1p (* y (expm1 x)))))

C

double code(double c, double x, double y) {
	return c * log1p((y * expm1(x)));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log1p((y * Math.expm1(x)));
}

Python

def code(c, x, y):
	return c * math.log1p((y * math.expm1(x)))

Julia

function code(c, x, y)
	return Float64(c * log1p(Float64(y * expm1(x))))
end

Wolfram

code[c_, x_, y_] := N[(c * N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.2%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
2. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

   3. lower-log1p.f64 56.6

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

   5. *-commutative N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

   6. *-rgt-identity N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

   8. *-rgt-identity 56.6

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

   10. sub-negate-rev N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

   11. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

   12. sub-negate N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

   13. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   15. lift-pow.f64 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   16. lift-E.f64 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   17. e-exp-1 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   18. pow-exp N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   19. *-lft-identity N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   20. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

   21. lower-expm1.f64 93.7

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

3. Applied rewrites 93.7%

   \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
4. Add Preprocessing

Alternative 3: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* c (* y (expm1 x)))))
   (if (<= t_0 -2e-308)
     t_1
     (if (<= t_0 0.0)
       (* c (log1p (* y x)))
       (if (<= t_0 2e-29) t_1 (* (log (fma y (expm1 x) 1.0)) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double t_1 = c * (y * expm1(x));
	double tmp;
	if (t_0 <= -2e-308) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((y * x));
	} else if (t_0 <= 2e-29) {
		tmp = t_1;
	} else {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	t_1 = Float64(c * Float64(y * expm1(x)))
	tmp = 0.0
	if (t_0 <= -2e-308)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(y * x)));
	elseif (t_0 <= 2e-29)
		tmp = t_1;
	else
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-308], t$95$1, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-29], t$95$1, N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -1.9999999999999998e-308 or -0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 1.99999999999999989e-29

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot \left(e^{x} - 1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \left(y \cdot \color{blue}{\left(e^{x} - 1\right)}\right) \]

      3. lower-expm1.f64 74.5

         \[\leadsto c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -0.0

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{x}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 66.7%

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{x}\right) \]

   if 1.99999999999999989e-29 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 42.2

         \[\leadsto \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \cdot c} \]

   3. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := y \cdot \mathsf{expm1}\left(x\right)\\ t_2 := c \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 0.004:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\log t\_1 \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* y (expm1 x))) (t_2 (* c t_1)))
   (if (<= t_0 -2e-308)
     t_2
     (if (<= t_0 0.0)
       (* c (log1p (* y x)))
       (if (<= t_0 0.004) t_2 (* (log t_1) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double t_1 = y * expm1(x);
	double t_2 = c * t_1;
	double tmp;
	if (t_0 <= -2e-308) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((y * x));
	} else if (t_0 <= 0.004) {
		tmp = t_2;
	} else {
		tmp = log(t_1) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = (Math.pow(Math.E, x) - 1.0) * y;
	double t_1 = y * Math.expm1(x);
	double t_2 = c * t_1;
	double tmp;
	if (t_0 <= -2e-308) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = c * Math.log1p((y * x));
	} else if (t_0 <= 0.004) {
		tmp = t_2;
	} else {
		tmp = Math.log(t_1) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = (math.pow(math.e, x) - 1.0) * y
	t_1 = y * math.expm1(x)
	t_2 = c * t_1
	tmp = 0
	if t_0 <= -2e-308:
		tmp = t_2
	elif t_0 <= 0.0:
		tmp = c * math.log1p((y * x))
	elif t_0 <= 0.004:
		tmp = t_2
	else:
		tmp = math.log(t_1) * c
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	t_1 = Float64(y * expm1(x))
	t_2 = Float64(c * t_1)
	tmp = 0.0
	if (t_0 <= -2e-308)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(y * x)));
	elseif (t_0 <= 0.004)
		tmp = t_2;
	else
		tmp = Float64(log(t_1) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(c * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-308], t$95$2, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.004], t$95$2, N[(N[Log[t$95$1], $MachinePrecision] * c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -1.9999999999999998e-308 or -0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0040000000000000001

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot \left(e^{x} - 1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \left(y \cdot \color{blue}{\left(e^{x} - 1\right)}\right) \]

      3. lower-expm1.f64 74.5

         \[\leadsto c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   if -1.9999999999999998e-308 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -0.0

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{x}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 66.7%

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{x}\right) \]

   if 0.0040000000000000001 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in y around inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-expm1.f64 N/A

         \[\leadsto c \cdot \left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto c \cdot \left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \]

      5. lower-log.f64 N/A

         \[\leadsto c \cdot \left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right) \]

      6. lower-/.f64 5.1

         \[\leadsto c \cdot \left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right) \]

   6. Applied rewrites 5.1%

      \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot c} \]

      3. lower-*.f64 5.1

         \[\leadsto \color{blue}{\left(\log \left(\mathsf{expm1}\left(x\right)\right) + -1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot c} \]

   8. Applied rewrites 20.1%

      \[\leadsto \color{blue}{\log \left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{e}^{x} - 1 \leq -0.002:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (- (pow E x) 1.0) -0.002)
   (* c (* y (expm1 x)))
   (* c (log1p (* y x)))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((pow(((double) M_E), x) - 1.0) <= -0.002) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = c * log1p((y * x));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((Math.pow(Math.E, x) - 1.0) <= -0.002) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = c * Math.log1p((y * x));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (math.pow(math.e, x) - 1.0) <= -0.002:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = c * math.log1p((y * x))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (Float64((exp(1) ^ x) - 1.0) <= -0.002)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(c * log1p(Float64(y * x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision], -0.002], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) < -2e-3

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot \left(e^{x} - 1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \left(y \cdot \color{blue}{\left(e^{x} - 1\right)}\right) \]

      3. lower-expm1.f64 74.5

         \[\leadsto c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   if -2e-3 < (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64))

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{x}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 66.7%

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -8 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{+95}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma y x 1.0)) c)))
   (if (<= y -8e+115) t_0 (if (<= y 2.45e+95) (* c (* y (expm1 x))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -8e+115) {
		tmp = t_0;
	} else if (y <= 2.45e+95) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -8e+115)
		tmp = t_0;
	elseif (y <= 2.45e+95)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -8e+115], t$95$0, If[LessEqual[y, 2.45e+95], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.0000000000000001e115 or 2.4499999999999999e95 < y

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 42.2

         \[\leadsto \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \cdot c} \]

   3. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c} \]

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 40.1%

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

   if -8.0000000000000001e115 < y < 2.4499999999999999e95

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(y \cdot \left(e^{x} - 1\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \left(y \cdot \color{blue}{\left(e^{x} - 1\right)}\right) \]

      3. lower-expm1.f64 74.5

         \[\leadsto c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (* y (expm1 x))))

C

double code(double c, double x, double y) {
	return c * (y * expm1(x));
}

Java

public static double code(double c, double x, double y) {
	return c * (y * Math.expm1(x));
}

Python

def code(c, x, y):
	return c * (y * math.expm1(x))

Julia

function code(c, x, y)
	return Float64(c * Float64(y * expm1(x)))
end

Wolfram

code[c_, x_, y_] := N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.2%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
2. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

   3. lower-log1p.f64 56.6

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

   5. *-commutative N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

   6. *-rgt-identity N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

   8. *-rgt-identity 56.6

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

   10. sub-negate-rev N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

   11. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

   12. sub-negate N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

   13. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   15. lift-pow.f64 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   16. lift-E.f64 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   17. e-exp-1 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   18. pow-exp N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   19. *-lft-identity N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   20. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

   21. lower-expm1.f64 93.7

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

3. Applied rewrites 93.7%

   \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

4. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

      \[\leadsto c \cdot \color{blue}{\left(y \cdot \left(e^{x} - 1\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto c \cdot \left(y \cdot \color{blue}{\left(e^{x} - 1\right)}\right) \]

   3. lower-expm1.f64 74.5

      \[\leadsto c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right) \]

6. Applied rewrites 74.5%

   \[\leadsto \color{blue}{c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
7. Add Preprocessing

Alternative 8: 59.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+120}:\\ \;\;\;\;c \cdot \log 1\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -4.7e+120) (* c (log 1.0)) (* c (* x y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.7e+120) {
		tmp = c * log(1.0);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.7d+120)) then
        tmp = c * log(1.0d0)
    else
        tmp = c * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.7e+120) {
		tmp = c * Math.log(1.0);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -4.7e+120:
		tmp = c * math.log(1.0)
	else:
		tmp = c * (x * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -4.7e+120)
		tmp = Float64(c * log(1.0));
	else
		tmp = Float64(c * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -4.7e+120)
		tmp = c * log(1.0);
	else
		tmp = c * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -4.7e+120], N[(c * N[Log[1.0], $MachinePrecision]), $MachinePrecision], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999993e120

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 31.3%

      \[\leadsto c \cdot \log \color{blue}{1} \]

   if -4.69999999999999993e120 < x

   1. Initial program 42.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   2. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

      3. lower-log1p.f64 56.6

         \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

      5. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

      8. *-rgt-identity 56.6

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

      9. lift--.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

      10. sub-negate-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

      11. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

      12. sub-negate N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

      13. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

      14. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      15. lift-pow.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      16. lift-E.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      17. e-exp-1 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      18. pow-exp N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      19. *-lft-identity N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      20. remove-double-neg N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

      21. lower-expm1.f64 93.7

          \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

   3. Applied rewrites 93.7%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 57.0

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

   6. Applied rewrites 57.0%

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

Alternative 9: 57.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.2%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
2. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto c \cdot \color{blue}{\log \left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto c \cdot \log \color{blue}{\left(1 + \left({e}^{x} - 1\right) \cdot y\right)} \]

   3. lower-log1p.f64 56.6

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\left({e}^{x} - 1\right) \cdot y\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left({e}^{x} - 1\right) \cdot y}\right) \]

   5. *-commutative N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y \cdot \left({e}^{x} - 1\right)}\right) \]

   6. *-rgt-identity N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right)} \cdot \left({e}^{x} - 1\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot 1\right) \cdot \left({e}^{x} - 1\right)}\right) \]

   8. *-rgt-identity 56.6

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{y} \cdot \left({e}^{x} - 1\right)\right) \]

   9. lift--.f64 N/A

      \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left({e}^{x} - 1\right)}\right) \]

   10. sub-negate-rev N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\mathsf{neg}\left(\left(1 - {e}^{x}\right)\right)\right)}\right) \]

   11. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\mathsf{neg}\left(\left(1 - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right)}\right)\right)\right)\right) \]

   12. sub-negate N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - 1\right)}\right) \]

   13. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({e}^{x}\right)\right)\right)\right) - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)}\right)\right) \]

   14. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   15. lift-pow.f64 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{{e}^{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   16. lift-E.f64 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\mathsf{E}\left(\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   17. e-exp-1 N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left({\color{blue}{\left(e^{1}\right)}}^{x} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   18. pow-exp N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(\color{blue}{e^{1 \cdot x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   19. *-lft-identity N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{\color{blue}{x}} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

   20. remove-double-neg N/A

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \left(e^{x} - \color{blue}{1}\right)\right) \]

   21. lower-expm1.f64 93.7

       \[\leadsto c \cdot \mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(x\right)}\right) \]

3. Applied rewrites 93.7%

   \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]

4. Taylor expanded in x around 0

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

   1. lower-*.f64 57.0

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

6. Applied rewrites 57.0%

   \[\leadsto c \cdot \color{blue}{\left(x \cdot y\right)} \]
7. Add Preprocessing

Developer Target 1: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \end{array} \]

FPCore

(FPCore (c x y) :precision binary64 (* c (log1p (* (expm1 x) y))))

C

double code(double c, double x, double y) {
	return c * log1p((expm1(x) * y));
}

Java

public static double code(double c, double x, double y) {
	return c * Math.log1p((Math.expm1(x) * y));
}

Python

def code(c, x, y):
	return c * math.log1p((math.expm1(x) * y))

Julia

function code(c, x, y)
	return Float64(c * log1p(Float64(expm1(x) * y)))
end

Wolfram

code[c_, x_, y_] := N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025151 
(FPCore (c x y)
  :name "Logarithmic Transform"
  :precision binary64

  :alt
  (* c (log1p (* (expm1 x) y)))

  (* c (log (+ 1.0 (* (- (pow E x) 1.0) y)))))