Logarithmic Transform

Percentage Accurate: 40.1% → 98.4%

Time: 7.1s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. c = -4.7442165451321204e-188
  2. x = 4.445490057022413e+207
  3. y = 1.2179945749979704e+188

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: 40.1% 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: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-149}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -5e-17) t_0 (if (<= y 1.85e-149) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -5e-17) {
		tmp = t_0;
	} else if (y <= 1.85e-149) {
		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((Math.expm1(x) * y));
	double tmp;
	if (y <= -5e-17) {
		tmp = t_0;
	} else if (y <= 1.85e-149) {
		tmp = (y * c) * Math.expm1(x);
	} 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 <= -5e-17:
		tmp = t_0
	elif y <= 1.85e-149:
		tmp = (y * c) * math.expm1(x)
	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 <= -5e-17)
		tmp = t_0;
	elseif (y <= 1.85e-149)
		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[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-17], t$95$0, If[LessEqual[y, 1.85e-149], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999999e-17 or 1.84999999999999995e-149 < y

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   if -4.9999999999999999e-17 < y < 1.84999999999999995e-149

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 2: 91.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+99}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+24}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -6e+99)
   (* (log (fma (expm1 x) y 1.0)) c)
   (if (<= y 8.5e+24) (* (* y c) (expm1 x)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -6e+99) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= 8.5e+24) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -6e+99)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= 8.5e+24)
		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, -6e+99], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 8.5e+24], 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 < -6.00000000000000029e99

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   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-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. *-commutative N/A

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

      11. lower-log1p.f64 N/A

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

      12. pow-to-exp N/A

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

      13. log-E N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

   5. Applied rewrites 49.7%

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

   if -6.00000000000000029e99 < y < 8.49999999999999959e24

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

   if 8.49999999999999959e24 < y

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

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

   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 66.0%

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

Alternative 3: 90.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+100}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+24}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.02e+100)
   (* (log (* (expm1 x) y)) c)
   (if (<= y 8.5e+24) (* (* y c) (expm1 x)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.02e+100) {
		tmp = log((expm1(x) * y)) * c;
	} else if (y <= 8.5e+24) {
		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 <= -1.02e+100) {
		tmp = Math.log((Math.expm1(x) * y)) * c;
	} else if (y <= 8.5e+24) {
		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 <= -1.02e+100:
		tmp = math.log((math.expm1(x) * y)) * c
	elif y <= 8.5e+24:
		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 <= -1.02e+100)
		tmp = Float64(log(Float64(expm1(x) * y)) * c);
	elseif (y <= 8.5e+24)
		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, -1.02e+100], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 8.5e+24], 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 < -1.0199999999999999e100

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

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

   4. Taylor expanded in y around inf

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

   6. Applied rewrites 19.1%

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

   if -1.0199999999999999e100 < y < 8.49999999999999959e24

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

   if 8.49999999999999959e24 < y

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

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

   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 66.0%

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

Alternative 4: 90.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -1500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+24}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -1500.0) t_0 (if (<= y 8.5e+24) (* (* 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 <= -1500.0) {
		tmp = t_0;
	} else if (y <= 8.5e+24) {
		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 <= -1500.0) {
		tmp = t_0;
	} else if (y <= 8.5e+24) {
		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 <= -1500.0:
		tmp = t_0
	elif y <= 8.5e+24:
		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 <= -1500.0)
		tmp = t_0;
	elseif (y <= 8.5e+24)
		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, -1500.0], t$95$0, If[LessEqual[y, 8.5e+24], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1500 or 8.49999999999999959e24 < y

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

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

   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 66.0%

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

   if -1500 < y < 8.49999999999999959e24

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 5: 82.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+152}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma y x 1.0)) c)))
   (if (<= y -1.15e+135) t_0 (if (<= y 2.9e+152) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -1.15e+135) {
		tmp = t_0;
	} else if (y <= 2.9e+152) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -1.15e+135)
		tmp = t_0;
	elseif (y <= 2.9e+152)
		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[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.15e+135], t$95$0, If[LessEqual[y, 2.9e+152], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1500000000000001e135 or 2.8999999999999998e152 < y

   1. Initial program 40.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \color{blue}{\left(1 + x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
   3. 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. lower-fma.f64 N/A

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

      9. lower-*.f64 38.9

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

   4. Applied rewrites 38.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 38.9

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 38.9

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

   6. Applied rewrites 38.9%

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

   if -1.1500000000000001e135 < y < 2.8999999999999998e152

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 6: 81.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(y \cdot x\right) \cdot c\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (* y x)) c)))
   (if (<= y -2.5e+135) t_0 (if (<= y 1.1e+155) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log((y * x)) * c;
	double tmp;
	if (y <= -2.5e+135) {
		tmp = t_0;
	} else if (y <= 1.1e+155) {
		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.log((y * x)) * c;
	double tmp;
	if (y <= -2.5e+135) {
		tmp = t_0;
	} else if (y <= 1.1e+155) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log((y * x)) * c
	tmp = 0
	if y <= -2.5e+135:
		tmp = t_0
	elif y <= 1.1e+155:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log(Float64(y * x)) * c)
	tmp = 0.0
	if (y <= -2.5e+135)
		tmp = t_0;
	elseif (y <= 1.1e+155)
		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[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -2.5e+135], t$95$0, If[LessEqual[y, 1.1e+155], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000015e135 or 1.1000000000000001e155 < y

   1. Initial program 40.1%

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

   2. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -2.50000000000000015e135 < y < 1.1000000000000001e155

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

Alternative 7: 65.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(y \cdot x\right) \cdot c\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{+135}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+155}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (* y x)) c)))
   (if (<= y -2.5e+135) t_0 (if (<= y 1.1e+155) (* (* y c) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log((y * x)) * c;
	double tmp;
	if (y <= -2.5e+135) {
		tmp = t_0;
	} else if (y <= 1.1e+155) {
		tmp = (y * c) * 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 = log((y * x)) * c
    if (y <= (-2.5d+135)) then
        tmp = t_0
    else if (y <= 1.1d+155) then
        tmp = (y * c) * 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 = Math.log((y * x)) * c;
	double tmp;
	if (y <= -2.5e+135) {
		tmp = t_0;
	} else if (y <= 1.1e+155) {
		tmp = (y * c) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log((y * x)) * c
	tmp = 0
	if y <= -2.5e+135:
		tmp = t_0
	elif y <= 1.1e+155:
		tmp = (y * c) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log(Float64(y * x)) * c)
	tmp = 0.0
	if (y <= -2.5e+135)
		tmp = t_0;
	elseif (y <= 1.1e+155)
		tmp = Float64(Float64(y * c) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = log((y * x)) * c;
	tmp = 0.0;
	if (y <= -2.5e+135)
		tmp = t_0;
	elseif (y <= 1.1e+155)
		tmp = (y * c) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -2.5e+135], t$95$0, If[LessEqual[y, 1.1e+155], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.50000000000000015e135 or 1.1000000000000001e155 < y

   1. Initial program 40.1%

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

   2. Taylor expanded in y around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 11.3

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

   7. Applied rewrites 11.3%

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

   if -2.50000000000000015e135 < y < 1.1000000000000001e155

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.7%

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

Alternative 8: 64.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 4.2e+61) (* (* y c) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 4.2e+61) {
		tmp = (y * c) * x;
	} else {
		tmp = (x * c) * 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 (c <= 4.2d+61) then
        tmp = (y * c) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 4.2e+61) {
		tmp = (y * c) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 4.2e+61:
		tmp = (y * c) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 4.2e+61)
		tmp = Float64(Float64(y * c) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 4.2e+61)
		tmp = (y * c) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 4.2e+61], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 4.2000000000000002e61

   1. Initial program 40.1%

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

   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 78.1

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

   4. Applied rewrites 78.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
   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 78.1

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 78.1

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

   6. Applied rewrites 78.1%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 62.7%

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

   if 4.2000000000000002e61 < c

   1. Initial program 40.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.8

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

   3. Applied rewrites 92.8%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   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 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 59.8

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

   8. Applied rewrites 59.8%

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

Alternative 9: 59.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.1%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 92.8

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

3. Applied rewrites 92.8%

   \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
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 92.8

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

5. Applied rewrites 92.8%

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

6. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 59.8

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

8. Applied rewrites 59.8%

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

Developer Target 1: 92.8% 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 2025142 
(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)))))