Logarithmic Transform

Percentage Accurate: 41.5% → 99.2%

Time: 6.9s

Alternatives: 10

Speedup: 4.9×

• could not determine a ground truth

  1. c = 5.978725999916889e+83
  2. x = 2.469197852368839e+55
  3. y = 1.0047821978166984e-113

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.5% 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: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -1 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (expm1 x) y)) c)))
   (if (<= y -1e-46) t_0 (if (<= y 5e-22) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((expm1(x) * y)) * c;
	double tmp;
	if (y <= -1e-46) {
		tmp = t_0;
	} else if (y <= 5e-22) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = Math.log1p((Math.expm1(x) * y)) * c;
	double tmp;
	if (y <= -1e-46) {
		tmp = t_0;
	} else if (y <= 5e-22) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log1p((math.expm1(x) * y)) * c
	tmp = 0
	if y <= -1e-46:
		tmp = t_0
	elif y <= 5e-22:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(expm1(x) * y)) * c)
	tmp = 0.0
	if (y <= -1e-46)
		tmp = t_0;
	elseif (y <= 5e-22)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1e-46], t$95$0, If[LessEqual[y, 5e-22], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000002e-46 or 4.99999999999999954e-22 < y

   1. Initial program 36.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

      12. lower-log1p.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 58.6%

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

      1. lift-log.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. log-E N/A

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

      8. pow-to-exp N/A

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

      9. *-commutative N/A

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

      10. pow-to-exp N/A

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

      11. log-E N/A

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

      12. *-commutative N/A

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

      13. *-rgt-identity N/A

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

      14. lower-log1p.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-expm1.f64 N/A

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

      17. lift-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -1.00000000000000002e-46 < y < 4.99999999999999954e-22

   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 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 99.5

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

   6. Applied rewrites 99.5%

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

Alternative 2: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.42:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 0.32:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.42)
   (* (log (fma (expm1 x) y 1.0)) c)
   (if (<= y 0.32) (* (* y c) (expm1 x)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -0.42) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= 0.32) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -0.42)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= 0.32)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -0.42], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.32], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.419999999999999984

   1. Initial program 49.1%

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

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

      12. lower-log1p.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 69.6%

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

   if -0.419999999999999984 < y < 0.320000000000000007

   1. Initial program 44.6%

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

   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 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 98.6

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

   6. Applied rewrites 98.6%

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

   if 0.320000000000000007 < y

   1. Initial program 17.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.2

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

   3. Applied rewrites 98.2%

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

   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 96.2%

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

Alternative 3: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+57}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 0.32:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.16e+57)
   (* (log (* (expm1 x) y)) c)
   (if (<= y 0.32) (* (* y c) (expm1 x)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.16e+57) {
		tmp = log((expm1(x) * y)) * c;
	} else if (y <= 0.32) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.16e+57) {
		tmp = Math.log((Math.expm1(x) * y)) * c;
	} else if (y <= 0.32) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = c * Math.log1p((x * y));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -1.16e+57:
		tmp = math.log((math.expm1(x) * y)) * c
	elif y <= 0.32:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = c * math.log1p((x * y))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.16e+57)
		tmp = Float64(log(Float64(expm1(x) * y)) * c);
	elseif (y <= 0.32)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.16e+57], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.32], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.16000000000000003e57

   1. Initial program 48.1%

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

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. *-commutative N/A

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

      9. log-E N/A

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

      10. pow-to-exp N/A

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

      11. *-commutative N/A

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

      12. lower-log1p.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 65.6

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

   8. Applied rewrites 65.6%

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

   if -1.16000000000000003e57 < y < 0.320000000000000007

   1. Initial program 45.3%

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

   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 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 95.5

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 95.5

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

   6. Applied rewrites 95.5%

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

   if 0.320000000000000007 < y

   1. Initial program 17.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.2

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

   3. Applied rewrites 98.2%

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

   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 96.2%

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

Alternative 4: 89.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -130:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.32:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -130.0) t_0 (if (<= y 0.32) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -130.0) {
		tmp = t_0;
	} else if (y <= 0.32) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -130.0) {
		tmp = t_0;
	} else if (y <= 0.32) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -130.0:
		tmp = t_0
	elif y <= 0.32:
		tmp = (y * c) * math.expm1(x)
	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 <= -130.0)
		tmp = t_0;
	elseif (y <= 0.32)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -130.0], t$95$0, If[LessEqual[y, 0.32], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -130 or 0.320000000000000007 < y

   1. Initial program 37.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.1

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

   3. Applied rewrites 99.1%

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

   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 76.2%

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

   if -130 < y < 0.320000000000000007

   1. Initial program 44.6%

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

   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 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 98.6

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 98.6

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

   6. Applied rewrites 98.6%

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

Alternative 5: 79.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+191}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+73}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -9.5e+191) t_0 (if (<= y 1.15e+73) (* (* y c) (expm1 x)) 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+191) {
		tmp = t_0;
	} else if (y <= 1.15e+73) {
		tmp = (y * c) * expm1(x);
	} 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+191)
		tmp = t_0;
	elseif (y <= 1.15e+73)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e+191], t$95$0, If[LessEqual[y, 1.15e+73], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4999999999999998e191 or 1.15e73 < y

   1. Initial program 29.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 7.7%

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

   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 51.3

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

   7. Applied rewrites 51.3%

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

   if -9.4999999999999998e191 < y < 1.15e73

   1. Initial program 44.5%

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

   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%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 86.7

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 86.7

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

   6. Applied rewrites 86.7%

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

Alternative 6: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 5e-22) (* (* y c) (expm1 x)) (* (* (expm1 x) y) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 5e-22) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = (expm1(x) * y) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 5e-22) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = (Math.expm1(x) * y) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 5e-22:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = (math.expm1(x) * y) * c
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 5e-22)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(Float64(expm1(x) * y) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 5e-22], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999954e-22

   1. Initial program 46.1%

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

   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 79.6

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

   4. Applied rewrites 79.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 79.6

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 79.6

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

   6. Applied rewrites 79.6%

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

   if 4.99999999999999954e-22 < y

   1. Initial program 19.8%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.4

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

   6. Applied rewrites 66.4%

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

Alternative 7: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.32:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 0.32) (* (* y c) (expm1 x)) (* c (* x y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 0.32) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 0.32) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 0.32:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = c * (x * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 0.32)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * Float64(x * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 0.32], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.320000000000000007

   1. Initial program 46.0%

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

   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 79.7

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

   4. Applied rewrites 79.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 79.7

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 79.7

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

   6. Applied rewrites 79.7%

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

   if 0.320000000000000007 < y

   1. Initial program 17.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 64.1

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

   4. Applied rewrites 64.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.1%

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

Alternative 8: 62.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 8.6 \cdot 10^{+122}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot \left(y \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 8.6e+122) (* (* y c) x) (* (* c x) (* y 1.0))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 8.6e+122) {
		tmp = (y * c) * 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 <= 8.6d+122) then
        tmp = (y * c) * 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 <= 8.6e+122) {
		tmp = (y * c) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 8.6e+122:
		tmp = (y * c) * x
	else:
		tmp = (c * x) * (y * 1.0)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 8.6e+122)
		tmp = Float64(Float64(y * c) * 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 <= 8.6e+122)
		tmp = (y * c) * x;
	else
		tmp = (c * x) * (y * 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 8.6e+122], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * N[(y * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 8.59999999999999943e122

   1. Initial program 46.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(c \cdot \left(x \cdot \left(-3 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \left(2 \cdot \left({y}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(c \cdot \left(-1 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right)} \]

   4. Applied rewrites 51.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3 \cdot \left(y \cdot y\right), 1, \mathsf{fma}\left(2 \cdot \left(\left(y \cdot y\right) \cdot y\right), 1, y \cdot 1\right)\right) \cdot x\right) \cdot c, 0.16666666666666666, \left(0.5 \cdot c\right) \cdot \mathsf{fma}\left(y, 1, -\left(y \cdot y\right) \cdot 1\right)\right), x, \left(c \cdot y\right) \cdot 1\right) \cdot x} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.0

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

   7. Applied rewrites 63.0%

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

   if 8.59999999999999943e122 < c

   1. Initial program 15.8%

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

   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. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 58.4

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

   4. Applied rewrites 58.4%

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

Alternative 9: 62.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 5e-22) (* (* y c) x) (* c (* x y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 5e-22) {
		tmp = (y * c) * x;
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-22) then
        tmp = (y * c) * x
    else
        tmp = c * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 5e-22) {
		tmp = (y * c) * x;
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 5e-22:
		tmp = (y * c) * x
	else:
		tmp = c * (x * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 5e-22)
		tmp = Float64(Float64(y * c) * x);
	else
		tmp = Float64(c * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (y <= 5e-22)
		tmp = (y * c) * x;
	else
		tmp = c * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 5e-22], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999954e-22

   1. Initial program 46.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(c \cdot \left(x \cdot \left(-3 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \left(2 \cdot \left({y}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(c \cdot \left(-1 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right)} \]

   4. Applied rewrites 53.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3 \cdot \left(y \cdot y\right), 1, \mathsf{fma}\left(2 \cdot \left(\left(y \cdot y\right) \cdot y\right), 1, y \cdot 1\right)\right) \cdot x\right) \cdot c, 0.16666666666666666, \left(0.5 \cdot c\right) \cdot \mathsf{fma}\left(y, 1, -\left(y \cdot y\right) \cdot 1\right)\right), x, \left(c \cdot y\right) \cdot 1\right) \cdot x} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.9

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

   7. Applied rewrites 61.9%

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

   if 4.99999999999999954e-22 < y

   1. Initial program 19.8%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 66.4

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

   4. Applied rewrites 66.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.9%

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

Alternative 10: 60.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

double code(double c, double x, double y) {
	return (y * c) * 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 = (y * c) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(c \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right) + x \cdot \left(\frac{1}{6} \cdot \left(c \cdot \left(x \cdot \left(-3 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \left(2 \cdot \left({y}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{3}\right)\right)\right)\right) + \frac{1}{2} \cdot \left(c \cdot \left(-1 \cdot \left({y}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right) + y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)\right)} \]

4. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(-3 \cdot \left(y \cdot y\right), 1, \mathsf{fma}\left(2 \cdot \left(\left(y \cdot y\right) \cdot y\right), 1, y \cdot 1\right)\right) \cdot x\right) \cdot c, 0.16666666666666666, \left(0.5 \cdot c\right) \cdot \mathsf{fma}\left(y, 1, -\left(y \cdot y\right) \cdot 1\right)\right), x, \left(c \cdot y\right) \cdot 1\right) \cdot x} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 60.8

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

7. Applied rewrites 60.8%

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

Developer Target 1: 93.4% 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 2025110 
(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)))))