Logarithmic Transform

Percentage Accurate: 41.3% → 97.8%

Time: 5.5s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. c = 7.904992286556016e-234
  2. x = 4.48244122832868e+59
  3. y = 1.20191161246441e+256

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: 41.3% 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: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}\right)\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-220}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -2.5e-25)
   (* c (log1p (/ 1.0 (/ 2.0 (* (+ y y) (expm1 x))))))
   (if (<= y 2.8e-220) (* (* c (expm1 x)) y) (* c (log1p (* y (expm1 x)))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -2.5e-25) {
		tmp = c * log1p((1.0 / (2.0 / ((y + y) * expm1(x)))));
	} else if (y <= 2.8e-220) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = c * log1p((y * expm1(x)));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -2.5e-25) {
		tmp = c * Math.log1p((1.0 / (2.0 / ((y + y) * Math.expm1(x)))));
	} else if (y <= 2.8e-220) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = c * Math.log1p((y * Math.expm1(x)));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -2.5e-25:
		tmp = c * math.log1p((1.0 / (2.0 / ((y + y) * math.expm1(x)))))
	elif y <= 2.8e-220:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = c * math.log1p((y * math.expm1(x)))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -2.5e-25)
		tmp = Float64(c * log1p(Float64(1.0 / Float64(2.0 / Float64(Float64(y + y) * expm1(x))))));
	elseif (y <= 2.8e-220)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = Float64(c * log1p(Float64(y * expm1(x))));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -2.5e-25], N[(c * N[Log[1 + N[(1.0 / N[(2.0 / N[(N[(y + y), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-220], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(c * N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.49999999999999981e-25

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   if -2.49999999999999981e-25 < y < 2.7999999999999999e-220

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   7. 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 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 77.5

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

   10. Applied rewrites 77.5%

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

   if 2.7999999999999999e-220 < y

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-220}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y (expm1 x))))))
   (if (<= y -2.5e-25) t_0 (if (<= y 2.8e-220) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * expm1(x)));
	double tmp;
	if (y <= -2.5e-25) {
		tmp = t_0;
	} else if (y <= 2.8e-220) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((y * Math.expm1(x)));
	double tmp;
	if (y <= -2.5e-25) {
		tmp = t_0;
	} else if (y <= 2.8e-220) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((y * math.expm1(x)))
	tmp = 0
	if y <= -2.5e-25:
		tmp = t_0
	elif y <= 2.8e-220:
		tmp = (c * math.expm1(x)) * y
	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 <= -2.5e-25)
		tmp = t_0;
	elseif (y <= 2.8e-220)
		tmp = Float64(Float64(c * expm1(x)) * y);
	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, -2.5e-25], t$95$0, If[LessEqual[y, 2.8e-220], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999981e-25 or 2.7999999999999999e-220 < y

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   if -2.49999999999999981e-25 < y < 2.7999999999999999e-220

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   7. 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 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 77.5

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

   10. Applied rewrites 77.5%

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

Alternative 3: 92.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+38}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 3:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -3.9e+38)
   (* (log (fma y (expm1 x) 1.0)) c)
   (if (<= y 3.0) (* (* c (expm1 x)) y) (* c (log1p (* y x))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -3.9e+38) {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	} else if (y <= 3.0) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = c * log1p((y * x));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -3.9e+38)
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	elseif (y <= 3.0)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = Float64(c * log1p(Float64(y * x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -3.9e+38], N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 3.0], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(c * N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.90000000000000023e38

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 41.3

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.3

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.3

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

   3. Applied rewrites 51.3%

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

   if -3.90000000000000023e38 < y < 3

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   7. 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 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 77.5

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

   10. Applied rewrites 77.5%

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

   if 3 < y

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   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 66.5%

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

Alternative 4: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{-30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.4:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y x)))))
   (if (<= y -6e-30) t_0 (if (<= y 0.4) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * x));
	double tmp;
	if (y <= -6e-30) {
		tmp = t_0;
	} else if (y <= 0.4) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((y * x));
	double tmp;
	if (y <= -6e-30) {
		tmp = t_0;
	} else if (y <= 0.4) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((y * x))
	tmp = 0
	if y <= -6e-30:
		tmp = t_0
	elif y <= 0.4:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(y * x)))
	tmp = 0.0
	if (y <= -6e-30)
		tmp = t_0;
	elseif (y <= 0.4)
		tmp = Float64(Float64(y * 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 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e-30], t$95$0, If[LessEqual[y, 0.4], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.9999999999999998e-30 or 0.40000000000000002 < y

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   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 66.5%

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

   if -5.9999999999999998e-30 < y < 0.40000000000000002

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   7. 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 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 77.1

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

   10. Applied rewrites 77.1%

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

Alternative 5: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.42 \cdot 10^{+110}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-220}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.42e+110)
   (* (log (fma y x 1.0)) c)
   (if (<= y 2.8e-220) (* (* c (expm1 x)) y) (* c (* y (expm1 x))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.42e+110) {
		tmp = log(fma(y, x, 1.0)) * c;
	} else if (y <= 2.8e-220) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = c * (y * expm1(x));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.42e+110)
		tmp = Float64(log(fma(y, x, 1.0)) * c);
	elseif (y <= 2.8e-220)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = Float64(c * Float64(y * expm1(x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.42e+110], N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 2.8e-220], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4200000000000001e110

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 41.3

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.3

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.3

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

   3. Applied rewrites 51.3%

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

   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 -1.4200000000000001e110 < y < 2.7999999999999999e-220

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   7. 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 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 77.5

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

   10. Applied rewrites 77.5%

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

   if 2.7999999999999999e-220 < y

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   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 74.3

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

   6. Applied rewrites 74.3%

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

Alternative 6: 77.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5.6 \cdot 10^{+57}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 5.6e+57) (* c (* y (expm1 x))) (* (* c (expm1 x)) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 5.6e+57) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = (c * expm1(x)) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 5.6e+57) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = (c * Math.expm1(x)) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 5.6e+57:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = (c * math.expm1(x)) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 5.6e+57)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(c * expm1(x)) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5.6e+57], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.59999999999999999e57

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   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 74.3

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

   6. Applied rewrites 74.3%

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

   if 5.59999999999999999e57 < c

   1. Initial program 41.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 56.4

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

      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. lower-*.f64 56.4

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)} \]
   4. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 1}\right) \]

      2. metadata-eval N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot \color{blue}{\frac{2}{2}}\right) \]

      3. associate-/l* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}{2}}\right) \]

      4. div-flip N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\color{blue}{\frac{2}{\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot 2}}}\right) \]

      7. *-commutative N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{2 \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{2 \cdot \color{blue}{\left(y \cdot \mathsf{expm1}\left(x\right)\right)}}}\right) \]

      9. associate-*r* N/A

         \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(2 \cdot y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

      11. count-2-rev N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

      12. lower-+.f64 92.9

          \[\leadsto c \cdot \mathsf{log1p}\left(\frac{1}{\frac{2}{\color{blue}{\left(y + y\right)} \cdot \mathsf{expm1}\left(x\right)}}\right) \]

   5. Applied rewrites 92.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{2}{\left(y + y\right) \cdot \mathsf{expm1}\left(x\right)}}}\right) \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{c \cdot \left(y \cdot \left(e^{x} - 1\right)\right)} \]
   7. 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 74.3

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 77.5

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

   10. Applied rewrites 77.5%

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

Alternative 7: 74.3% 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 41.3%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 56.4

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

   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. lower-*.f64 56.4

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

   7. lift--.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-E.f64 N/A

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

   10. e-exp-1 N/A

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

   11. pow-exp N/A

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

   12. *-lft-identity N/A

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

   13. lower-expm1.f64 93.6

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

3. Applied rewrites 93.6%

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

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 74.3

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

6. Applied rewrites 74.3%

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

Alternative 8: 59.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+73}:\\ \;\;\;\;c \cdot \log 1\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -1.5e+73) (* c (log 1.0)) (* (* x y) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -1.5e+73) {
		tmp = c * log(1.0);
	} else {
		tmp = (x * y) * c;
	}
	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 <= (-1.5d+73)) then
        tmp = c * log(1.0d0)
    else
        tmp = (x * y) * c
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -1.5e+73) {
		tmp = c * Math.log(1.0);
	} else {
		tmp = (x * y) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -1.5e+73:
		tmp = c * math.log(1.0)
	else:
		tmp = (x * y) * c
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -1.5e+73)
		tmp = Float64(c * log(1.0));
	else
		tmp = Float64(Float64(x * y) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -1.5e+73)
		tmp = c * log(1.0);
	else
		tmp = (x * y) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -1.5e+73], N[(c * N[Log[1.0], $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000005e73

   1. Initial program 41.3%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 30.7%

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

   if -1.50000000000000005e73 < x

   1. Initial program 41.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-E.f64 56.3

         \[\leadsto c \cdot \left(x \cdot \left(y \cdot \log e\right)\right) \]

   4. Applied rewrites 56.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.3

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot \log e\right)\right) \cdot c \]

      6. lift-E.f64 N/A

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

      7. log-E N/A

         \[\leadsto \left(x \cdot \left(y \cdot 1\right)\right) \cdot c \]

      8. *-rgt-identity 56.3

         \[\leadsto \left(x \cdot y\right) \cdot c \]

   6. Applied rewrites 56.3%

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

Alternative 9: 56.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

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 = (x * y) * c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.3%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower-E.f64 56.3

      \[\leadsto c \cdot \left(x \cdot \left(y \cdot \log e\right)\right) \]

4. Applied rewrites 56.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 56.3

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot \log e\right)\right) \cdot c \]

   6. lift-E.f64 N/A

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

   7. log-E N/A

      \[\leadsto \left(x \cdot \left(y \cdot 1\right)\right) \cdot c \]

   8. *-rgt-identity 56.3

      \[\leadsto \left(x \cdot y\right) \cdot c \]

6. Applied rewrites 56.3%

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

Developer Target 1: 93.6% 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 2025162 
(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)))))