Logarithmic Transform

Percentage Accurate: 40.8% → 99.1%

Time: 6.2s

Alternatives: 13

Speedup: 4.9×

• could not determine a ground truth

  1. c = -6.807688481512883
  2. x = 2.87872870495031e+205
  3. y = 1.7552308415699753e-154

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: 40.8% 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: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x + x\right)}{e^{x} + 1} \cdot y\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-75}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{expm1}\left(x\right)}} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -4.6e-8)
   (* c (log1p (* (/ (expm1 (+ x x)) (+ (exp x) 1.0)) y)))
   (if (<= y 5.5e-75)
     (* (* c y) (expm1 (* x 1.0)))
     (* c (log1p (* (/ 1.0 (/ 1.0 (expm1 x))) y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -4.6e-8) {
		tmp = c * log1p(((expm1((x + x)) / (exp(x) + 1.0)) * y));
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = c * log1p(((1.0 / (1.0 / expm1(x))) * y));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -4.6e-8) {
		tmp = c * Math.log1p(((Math.expm1((x + x)) / (Math.exp(x) + 1.0)) * y));
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = c * Math.log1p(((1.0 / (1.0 / Math.expm1(x))) * y));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -4.6e-8:
		tmp = c * math.log1p(((math.expm1((x + x)) / (math.exp(x) + 1.0)) * y))
	elif y <= 5.5e-75:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = c * math.log1p(((1.0 / (1.0 / math.expm1(x))) * y))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -4.6e-8)
		tmp = Float64(c * log1p(Float64(Float64(expm1(Float64(x + x)) / Float64(exp(x) + 1.0)) * y)));
	elseif (y <= 5.5e-75)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = Float64(c * log1p(Float64(Float64(1.0 / Float64(1.0 / expm1(x))) * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -4.6e-8], N[(c * N[Log[1 + N[(N[(N[(Exp[N[(x + x), $MachinePrecision]] - 1), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-75], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(N[(1.0 / N[(1.0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6000000000000002e-8

   1. Initial program 49.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 99.6

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. flip-- N/A

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

      9. pow-to-exp N/A

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

      10. log-E N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

   if -4.6000000000000002e-8 < y < 5.50000000000000026e-75

   1. Initial program 45.7%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 5.50000000000000026e-75 < y

   1. Initial program 20.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 97.5

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

   3. Applied rewrites 97.5%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 97.5

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. pow-neg N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow-to-exp N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lower-expm1.f64 97.4

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

   5. Applied rewrites 97.4%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(x\right)}}} \cdot y\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-75}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\frac{1}{\frac{1}{\mathsf{expm1}\left(x\right)}} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -4.6e-8)
   (* c (log1p (* (expm1 x) y)))
   (if (<= y 5.5e-75)
     (* (* c y) (expm1 (* x 1.0)))
     (* c (log1p (* (/ 1.0 (/ 1.0 (expm1 x))) y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -4.6e-8) {
		tmp = c * log1p((expm1(x) * y));
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = c * log1p(((1.0 / (1.0 / expm1(x))) * y));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -4.6e-8) {
		tmp = c * Math.log1p((Math.expm1(x) * y));
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = c * Math.log1p(((1.0 / (1.0 / Math.expm1(x))) * y));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -4.6e-8:
		tmp = c * math.log1p((math.expm1(x) * y))
	elif y <= 5.5e-75:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = c * math.log1p(((1.0 / (1.0 / math.expm1(x))) * y))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -4.6e-8)
		tmp = Float64(c * log1p(Float64(expm1(x) * y)));
	elseif (y <= 5.5e-75)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = Float64(c * log1p(Float64(Float64(1.0 / Float64(1.0 / expm1(x))) * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -4.6e-8], N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-75], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(N[(1.0 / N[(1.0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6000000000000002e-8

   1. Initial program 49.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.6%

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

   if -4.6000000000000002e-8 < y < 5.50000000000000026e-75

   1. Initial program 45.7%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if 5.50000000000000026e-75 < y

   1. Initial program 20.0%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 97.5

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

   3. Applied rewrites 97.5%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 97.5

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. pow-neg N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow-to-exp N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lower-expm1.f64 97.4

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

   5. Applied rewrites 97.4%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{1}{\mathsf{expm1}\left(x\right)}}} \cdot y\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-75}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -4.6e-8)
     t_0
     (if (<= y 5.5e-75) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -4.6e-8) {
		tmp = t_0;
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1(x) * y));
	double tmp;
	if (y <= -4.6e-8) {
		tmp = t_0;
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1(x) * y))
	tmp = 0
	if y <= -4.6e-8:
		tmp = t_0
	elif y <= 5.5e-75:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (y <= -4.6e-8)
		tmp = t_0;
	elseif (y <= 5.5e-75)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e-8], t$95$0, If[LessEqual[y, 5.5e-75], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6000000000000002e-8 or 5.50000000000000026e-75 < y

   1. Initial program 35.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.6

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

   3. Applied rewrites 98.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 98.6%

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

   if -4.6000000000000002e-8 < y < 5.50000000000000026e-75

   1. Initial program 45.7%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-292}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* c (* (expm1 x) y))))
   (if (<= t_0 -2e-292)
     t_1
     (if (<= t_0 0.0)
       (* c (log1p (* x y)))
       (if (<= t_0 2e-44) t_1 (* (log (fma (expm1 x) y 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 * (expm1(x) * y);
	double tmp;
	if (t_0 <= -2e-292) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 2e-44) {
		tmp = t_1;
	} else {
		tmp = log(fma(expm1(x), y, 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(expm1(x) * y))
	tmp = 0.0
	if (t_0 <= -2e-292)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 2e-44)
		tmp = t_1;
	else
		tmp = Float64(log(fma(expm1(x), y, 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[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-292], t$95$1, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-44], t$95$1, N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 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) < -2.0000000000000001e-292 or 0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 1.99999999999999991e-44

   1. Initial program 29.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 98.6

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 98.6

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

   6. Applied rewrites 98.6%

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

   if -2.0000000000000001e-292 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0

   1. Initial program 35.4%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 90.6

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

   3. Applied rewrites 90.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   if 1.99999999999999991e-44 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 85.7%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 96.6

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

   3. Applied rewrites 96.6%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-expm1.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-expm1.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. lower-expm1.f64 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. pow-to-exp N/A

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

   5. Applied rewrites 88.1%

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

Alternative 5: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := \mathsf{expm1}\left(x\right) \cdot y\\ t_2 := c \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-292}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 0.02:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\log t\_1 \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* (expm1 x) y)) (t_2 (* c t_1)))
   (if (<= t_0 -2e-292)
     t_2
     (if (<= t_0 0.0)
       (* c (log1p (* x y)))
       (if (<= t_0 0.02) 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 = expm1(x) * y;
	double t_2 = c * t_1;
	double tmp;
	if (t_0 <= -2e-292) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 0.02) {
		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 = Math.expm1(x) * y;
	double t_2 = c * t_1;
	double tmp;
	if (t_0 <= -2e-292) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = c * Math.log1p((x * y));
	} else if (t_0 <= 0.02) {
		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 = math.expm1(x) * y
	t_2 = c * t_1
	tmp = 0
	if t_0 <= -2e-292:
		tmp = t_2
	elif t_0 <= 0.0:
		tmp = c * math.log1p((x * y))
	elif t_0 <= 0.02:
		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(expm1(x) * y)
	t_2 = Float64(c * t_1)
	tmp = 0.0
	if (t_0 <= -2e-292)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 0.02)
		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[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(c * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-292], t$95$2, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.02], 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) < -2.0000000000000001e-292 or 0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0200000000000000004

   1. Initial program 29.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 98.0

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 98.0

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

   6. Applied rewrites 98.0%

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

   if -2.0000000000000001e-292 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0

   1. Initial program 35.4%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 90.6

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

   3. Applied rewrites 90.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 89.9%

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

   if 0.0200000000000000004 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 93.5%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 94.2

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

   4. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 94.2

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 94.2

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

   6. Applied rewrites 94.2%

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

Alternative 6: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -43:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+37}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -43.0)
     t_0
     (if (<= y 7.5e+37) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -43.0) {
		tmp = t_0;
	} else if (y <= 7.5e+37) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -43.0) {
		tmp = t_0;
	} else if (y <= 7.5e+37) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -43.0:
		tmp = t_0
	elif y <= 7.5e+37:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -43.0)
		tmp = t_0;
	elseif (y <= 7.5e+37)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -43.0], t$95$0, If[LessEqual[y, 7.5e+37], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -43 or 7.5000000000000003e37 < y

   1. Initial program 37.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.1%

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

   if -43 < y < 7.5000000000000003e37

   1. Initial program 43.0%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.5

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

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+188}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+145}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -9.5e+188)
     t_0
     (if (<= y 1.75e+145) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -9.5e+188) {
		tmp = t_0;
	} else if (y <= 1.75e+145) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -9.5e+188)
		tmp = t_0;
	elseif (y <= 1.75e+145)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e+188], t$95$0, If[LessEqual[y, 1.75e+145], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4999999999999996e188 or 1.7500000000000001e145 < y

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 5.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {\log \mathsf{E}\left(\right)}^{2} + 1\right) \]

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 51.7

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

   7. Applied rewrites 51.7%

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

   if -9.4999999999999996e188 < y < 1.7500000000000001e145

   1. Initial program 42.1%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 86.7

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

   4. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+189}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+174}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -1.3e+189) t_0 (if (<= y 3.3e+174) (* c (* (expm1 x) y)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -1.3e+189) {
		tmp = t_0;
	} else if (y <= 3.3e+174) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -1.3e+189)
		tmp = t_0;
	elseif (y <= 3.3e+174)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3e+189], t$95$0, If[LessEqual[y, 3.3e+174], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.29999999999999991e189 or 3.3000000000000001e174 < y

   1. Initial program 35.7%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 4.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + 1\right) \]

      5. associate-*r* N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {\log \mathsf{E}\left(\right)}^{2} + 1\right) \]

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 51.7

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

   7. Applied rewrites 51.7%

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

   if -1.29999999999999991e189 < y < 3.3000000000000001e174

   1. Initial program 41.7%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 92.3

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 81.1

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 81.1

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

   6. Applied rewrites 81.1%

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

Alternative 9: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -1.52 \cdot 10^{+190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+174}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* x y)))))
   (if (<= y -1.52e+190) t_0 (if (<= y 5.3e+174) (* c (* (expm1 x) y)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((x * y));
	double tmp;
	if (y <= -1.52e+190) {
		tmp = t_0;
	} else if (y <= 5.3e+174) {
		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.log((x * y));
	double tmp;
	if (y <= -1.52e+190) {
		tmp = t_0;
	} else if (y <= 5.3e+174) {
		tmp = c * (Math.expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((x * y))
	tmp = 0
	if y <= -1.52e+190:
		tmp = t_0
	elif y <= 5.3e+174:
		tmp = c * (math.expm1(x) * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(x * y)))
	tmp = 0.0
	if (y <= -1.52e+190)
		tmp = t_0;
	elseif (y <= 5.3e+174)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.52e+190], t$95$0, If[LessEqual[y, 5.3e+174], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5199999999999999e190 or 5.2999999999999998e174 < y

   1. Initial program 35.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(x \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 45.6%

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

   if -1.5199999999999999e190 < y < 5.2999999999999998e174

   1. Initial program 41.7%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 92.3

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

   3. Applied rewrites 92.3%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 81.0

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 81.0

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

   6. Applied rewrites 81.0%

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

Alternative 10: 65.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -1.52 \cdot 10^{+190}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+162}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* x y)))))
   (if (<= y -1.52e+190) t_0 (if (<= y 4.8e+162) (* (* c y) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((x * y));
	double tmp;
	if (y <= -1.52e+190) {
		tmp = t_0;
	} else if (y <= 4.8e+162) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c * log((x * y))
    if (y <= (-1.52d+190)) then
        tmp = t_0
    else if (y <= 4.8d+162) then
        tmp = (c * y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log((x * y));
	double tmp;
	if (y <= -1.52e+190) {
		tmp = t_0;
	} else if (y <= 4.8e+162) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((x * y))
	tmp = 0
	if y <= -1.52e+190:
		tmp = t_0
	elif y <= 4.8e+162:
		tmp = (c * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(x * y)))
	tmp = 0.0
	if (y <= -1.52e+190)
		tmp = t_0;
	elseif (y <= 4.8e+162)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = c * log((x * y));
	tmp = 0.0;
	if (y <= -1.52e+190)
		tmp = t_0;
	elseif (y <= 4.8e+162)
		tmp = (c * y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.52e+190], t$95$0, If[LessEqual[y, 4.8e+162], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5199999999999999e190 or 4.80000000000000018e162 < y

   1. Initial program 34.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 72.2

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(x \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 45.2%

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

   if -1.5199999999999999e190 < y < 4.80000000000000018e162

   1. Initial program 41.8%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 86.3

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot y\right) \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 68.9%

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

Alternative 11: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.45 \cdot 10^{+63}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot \left(y \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.45e+63) (* (* c y) x) (* (* c x) (* y 1.0))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.45e+63) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	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 (c <= 2.45d+63) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * (y * 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.45e+63) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 2.45e+63:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * (y * 1.0)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.45e+63)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * Float64(y * 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 2.45e+63)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * (y * 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.45e+63], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * N[(y * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.4499999999999998e63

   1. Initial program 46.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 77.9

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot y\right) \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 64.1%

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

   if 2.4499999999999998e63 < c

   1. Initial program 17.5%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot {1}^{\color{blue}{4}}\right) \]

      4. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{4}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c \cdot x\right) \cdot \color{blue}{\left(y \cdot {\log \mathsf{E}\left(\right)}^{4}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(\color{blue}{y} \cdot {\log \mathsf{E}\left(\right)}^{4}\right) \]

      7. log-E N/A

         \[\leadsto \left(c \cdot x\right) \cdot \left(y \cdot {1}^{4}\right) \]

      8. metadata-eval N/A

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

      9. lower-*.f64 60.2

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

   4. Applied rewrites 60.2%

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

Alternative 12: 63.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-75}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (* y x))))
   (if (<= y -4.6e+50) t_0 (if (<= y 5.5e-75) (* (* c y) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * (y * x);
	double tmp;
	if (y <= -4.6e+50) {
		tmp = t_0;
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = c * (y * x)
    if (y <= (-4.6d+50)) then
        tmp = t_0
    else if (y <= 5.5d-75) then
        tmp = (c * y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double t_0 = c * (y * x);
	double tmp;
	if (y <= -4.6e+50) {
		tmp = t_0;
	} else if (y <= 5.5e-75) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * (y * x)
	tmp = 0
	if y <= -4.6e+50:
		tmp = t_0
	elif y <= 5.5e-75:
		tmp = (c * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * Float64(y * x))
	tmp = 0.0
	if (y <= -4.6e+50)
		tmp = t_0;
	elseif (y <= 5.5e-75)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = c * (y * x);
	tmp = 0.0;
	if (y <= -4.6e+50)
		tmp = t_0;
	elseif (y <= 5.5e-75)
		tmp = (c * y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.6e+50], t$95$0, If[LessEqual[y, 5.5e-75], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.59999999999999994e50 or 5.50000000000000026e-75 < y

   1. Initial program 34.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.5

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

   3. Applied rewrites 98.5%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   if -4.59999999999999994e50 < y < 5.50000000000000026e-75

   1. Initial program 45.8%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 96.2

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

   4. Applied rewrites 96.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot y\right) \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 74.1%

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

Alternative 13: 61.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

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 * y) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

2. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow-to-exp N/A

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

   5. log-E N/A

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

   6. *-commutative N/A

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

   7. lower-expm1.f64 N/A

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

   8. lower-*.f64 76.7

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

4. Applied rewrites 76.7%

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

5. Taylor expanded in x around 0

   \[\leadsto \left(c \cdot y\right) \cdot x \]
6. Step-by-step derivation

7. Applied rewrites 61.8%

   \[\leadsto \left(c \cdot y\right) \cdot x \]
8. Add Preprocessing

Developer Target 1: 93.2% 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 2025119 
(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)))))