Logarithmic Transform

Percentage Accurate: 41.3% → 99.4%

Time: 6.6s

Alternatives: 12

Speedup: 4.9×

• could not determine a ground truth

  1. c = -5.045335304834244e-275
  2. x = 4.107266058576246e+147
  3. y = 5.981173788629503e+238

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\frac{y \cdot 1}{\frac{1}{\mathsf{expm1}\left(x\right)}}\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-104}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (/ (* y 1.0) (/ 1.0 (expm1 x)))))))
   (if (<= y -9.5e-16)
     t_0
     (if (<= y 2.05e-104) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p(((y * 1.0) / (1.0 / expm1(x))));
	double tmp;
	if (y <= -9.5e-16) {
		tmp = t_0;
	} else if (y <= 2.05e-104) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p(((y * 1.0) / (1.0 / Math.expm1(x))));
	double tmp;
	if (y <= -9.5e-16) {
		tmp = t_0;
	} else if (y <= 2.05e-104) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p(((y * 1.0) / (1.0 / math.expm1(x))))
	tmp = 0
	if y <= -9.5e-16:
		tmp = t_0
	elif y <= 2.05e-104:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(Float64(y * 1.0) / Float64(1.0 / expm1(x)))))
	tmp = 0.0
	if (y <= -9.5e-16)
		tmp = t_0;
	elseif (y <= 2.05e-104)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(y * 1.0), $MachinePrecision] / N[(1.0 / N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e-16], t$95$0, If[LessEqual[y, 2.05e-104], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5000000000000005e-16 or 2.04999999999999992e-104 < y

   1. Initial program 36.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.5

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

   3. Applied rewrites 98.5%

      \[\leadsto c \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)} \]
   4. 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 98.5

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. unpow1 N/A

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

      9. metadata-eval N/A

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

      10. pow-neg N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. pow-to-exp N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. *-rgt-identity N/A

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

      18. lower-expm1.f64 98.5

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

   5. Applied rewrites 98.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-expm1.f64 N/A

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

      10. lift-/.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if -9.5000000000000005e-16 < y < 2.04999999999999992e-104

   1. Initial program 46.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-24}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -9.5e-16)
     t_0
     (if (<= y 9.5e-24) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -9.5e-16) {
		tmp = t_0;
	} else if (y <= 9.5e-24) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1(x) * y));
	double tmp;
	if (y <= -9.5e-16) {
		tmp = t_0;
	} else if (y <= 9.5e-24) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1(x) * y))
	tmp = 0
	if y <= -9.5e-16:
		tmp = t_0
	elif y <= 9.5e-24:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (y <= -9.5e-16)
		tmp = t_0;
	elseif (y <= 9.5e-24)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e-16], t$95$0, If[LessEqual[y, 9.5e-24], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5000000000000005e-16 or 9.50000000000000029e-24 < y

   1. Initial program 37.5%

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

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

   6. Applied rewrites 99.1%

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

   if -9.5000000000000005e-16 < y < 9.50000000000000029e-24

   1. Initial program 44.4%

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

   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 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 3: 91.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+221}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -1.35e+221)
     (* (log (fma (expm1 x) y 1.0)) c)
     (if (<= y -40000.0)
       t_0
       (if (<= y 1.02e+22) (* (* c y) (expm1 (* x 1.0))) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -1.35e+221) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= -40000.0) {
		tmp = t_0;
	} else if (y <= 1.02e+22) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -1.35e+221)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= -40000.0)
		tmp = t_0;
	elseif (y <= 1.02e+22)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.35e+221], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, -40000.0], t$95$0, If[LessEqual[y, 1.02e+22], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35e221

   1. Initial program 50.9%

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

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

      2. lift-log1p.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-expm1.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-expm1.f64 N/A

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

      8. *-rgt-identity N/A

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

      9. lower-expm1.f64 N/A

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

      10. *-rgt-identity N/A

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

      11. *-commutative N/A

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

      12. log-E N/A

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

      13. pow-to-exp N/A

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

   5. Applied rewrites 88.7%

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

   if -1.35e221 < y < -4e4 or 1.02e22 < y

   1. Initial program 35.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 78.7

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

      4. *-rgt-identity 78.7

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

      5. *-commutative 78.7

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

      6. log-E 78.7

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

      7. pow-to-exp 78.7

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

   6. Applied rewrites 78.7%

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

   if -4e4 < y < 1.02e22

   1. Initial program 43.6%

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

   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%

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

Alternative 4: 91.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -9.5 \cdot 10^{+227}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -9.5e+227)
     (* (log (* (expm1 x) y)) c)
     (if (<= y -40000.0)
       t_0
       (if (<= y 1.02e+22) (* (* c y) (expm1 (* x 1.0))) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -9.5e+227) {
		tmp = log((expm1(x) * y)) * c;
	} else if (y <= -40000.0) {
		tmp = t_0;
	} else if (y <= 1.02e+22) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -9.5e+227) {
		tmp = Math.log((Math.expm1(x) * y)) * c;
	} else if (y <= -40000.0) {
		tmp = t_0;
	} else if (y <= 1.02e+22) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -9.5e+227:
		tmp = math.log((math.expm1(x) * y)) * c
	elif y <= -40000.0:
		tmp = t_0
	elif y <= 1.02e+22:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -9.5e+227)
		tmp = Float64(log(Float64(expm1(x) * y)) * c);
	elseif (y <= -40000.0)
		tmp = t_0;
	elseif (y <= 1.02e+22)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.5e+227], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, -40000.0], t$95$0, If[LessEqual[y, 1.02e+22], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000005e227

   1. Initial program 50.6%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 85.8

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

   4. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.8

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 85.8

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

   6. Applied rewrites 85.8%

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

   if -9.5000000000000005e227 < y < -4e4 or 1.02e22 < y

   1. Initial program 35.5%

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

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 78.3

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

      4. *-rgt-identity 78.3

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

      5. *-commutative 78.3

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

      6. log-E 78.3

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

      7. pow-to-exp 78.3

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

   6. Applied rewrites 78.3%

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

   if -4e4 < y < 1.02e22

   1. Initial program 43.6%

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

   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%

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

Alternative 5: 89.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{+22}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -40000.0)
     t_0
     (if (<= y 1.02e+22) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -40000.0) {
		tmp = t_0;
	} else if (y <= 1.02e+22) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -40000.0) {
		tmp = t_0;
	} else if (y <= 1.02e+22) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -40000.0:
		tmp = t_0
	elif y <= 1.02e+22:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -40000.0)
		tmp = t_0;
	elseif (y <= 1.02e+22)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -40000.0], t$95$0, If[LessEqual[y, 1.02e+22], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4e4 or 1.02e22 < y

   1. Initial program 37.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 74.8

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

      4. *-rgt-identity 74.8

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

      5. *-commutative 74.8

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

      6. log-E 74.8

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

      7. pow-to-exp 74.8

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

   6. Applied rewrites 74.8%

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

   if -4e4 < y < 1.02e22

   1. Initial program 43.6%

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

   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%

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

Alternative 6: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(x, y, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{+189}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma x y 1.0)) c)))
   (if (<= y -5.8e+98)
     t_0
     (if (<= y 5.1e+189) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(x, y, 1.0)) * c;
	double tmp;
	if (y <= -5.8e+98) {
		tmp = t_0;
	} else if (y <= 5.1e+189) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(x, y, 1.0)) * c)
	tmp = 0.0
	if (y <= -5.8e+98)
		tmp = t_0;
	elseif (y <= 5.1e+189)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(x * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -5.8e+98], t$95$0, If[LessEqual[y, 5.1e+189], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8000000000000002e98 or 5.10000000000000049e189 < y

   1. Initial program 39.7%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(1 + x \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 45.6%

      \[\leadsto c \cdot \log \left(1 + x \cdot y\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.6

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 45.6

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

   9. Applied rewrites 45.6%

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

   if -5.8000000000000002e98 < y < 5.10000000000000049e189

   1. Initial program 41.8%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 90.4

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

   4. Applied rewrites 90.4%

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

Alternative 7: 76.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(x, y, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+235}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma x y 1.0)) c)))
   (if (<= y -1.45e+134) t_0 (if (<= y 2.9e+235) (* c (* (expm1 x) y)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(x, y, 1.0)) * c;
	double tmp;
	if (y <= -1.45e+134) {
		tmp = t_0;
	} else if (y <= 2.9e+235) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(x, y, 1.0)) * c)
	tmp = 0.0
	if (y <= -1.45e+134)
		tmp = t_0;
	elseif (y <= 2.9e+235)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(x * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.45e+134], t$95$0, If[LessEqual[y, 2.9e+235], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.45000000000000006e134 or 2.90000000000000021e235 < y

   1. Initial program 41.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 45.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(1 + x \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 46.8%

      \[\leadsto c \cdot \log \left(1 + x \cdot y\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.8

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 46.8

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

   9. Applied rewrites 46.8%

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

   if -1.45000000000000006e134 < y < 2.90000000000000021e235

   1. Initial program 41.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. 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 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 82.6

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 82.6

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

   6. Applied rewrites 82.6%

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

Alternative 8: 76.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -1.52 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+235}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* x y)))))
   (if (<= y -1.52e+208) t_0 (if (<= y 2.9e+235) (* c (* (expm1 x) y)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((x * y));
	double tmp;
	if (y <= -1.52e+208) {
		tmp = t_0;
	} else if (y <= 2.9e+235) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log((x * y));
	double tmp;
	if (y <= -1.52e+208) {
		tmp = t_0;
	} else if (y <= 2.9e+235) {
		tmp = c * (Math.expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((x * y))
	tmp = 0
	if y <= -1.52e+208:
		tmp = t_0
	elif y <= 2.9e+235:
		tmp = c * (math.expm1(x) * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(x * y)))
	tmp = 0.0
	if (y <= -1.52e+208)
		tmp = t_0;
	elseif (y <= 2.9e+235)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.52e+208], t$95$0, If[LessEqual[y, 2.9e+235], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.51999999999999995e208 or 2.90000000000000021e235 < y

   1. Initial program 38.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 80.8

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

   4. Applied rewrites 80.8%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 49.5

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

      4. *-rgt-identity 49.5

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

      5. *-commutative 49.5

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

      6. log-E 49.5

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

      7. pow-to-exp 49.5

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

   7. Applied rewrites 49.5%

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

   if -1.51999999999999995e208 < y < 2.90000000000000021e235

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 93.2

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

   3. Applied rewrites 93.2%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 79.9

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 79.9

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

   6. Applied rewrites 79.9%

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

Alternative 9: 64.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+190}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* x y)))))
   (if (<= y -6.6e+98) t_0 (if (<= y 1.1e+190) (* (* c y) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((x * y));
	double tmp;
	if (y <= -6.6e+98) {
		tmp = t_0;
	} else if (y <= 1.1e+190) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * log((x * y))
    if (y <= (-6.6d+98)) then
        tmp = t_0
    else if (y <= 1.1d+190) then
        tmp = (c * y) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log((x * y));
	double tmp;
	if (y <= -6.6e+98) {
		tmp = t_0;
	} else if (y <= 1.1e+190) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((x * y))
	tmp = 0
	if y <= -6.6e+98:
		tmp = t_0
	elif y <= 1.1e+190:
		tmp = (c * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(x * y)))
	tmp = 0.0
	if (y <= -6.6e+98)
		tmp = t_0;
	elseif (y <= 1.1e+190)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = c * log((x * y));
	tmp = 0.0;
	if (y <= -6.6e+98)
		tmp = t_0;
	elseif (y <= 1.1e+190)
		tmp = (c * y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.6e+98], t$95$0, If[LessEqual[y, 1.1e+190], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.60000000000000056e98 or 1.1e190 < y

   1. Initial program 39.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 68.0

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

   4. Applied rewrites 68.0%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 37.8

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

      4. *-rgt-identity 37.8

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

      5. *-commutative 37.8

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

      6. log-E 37.8

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

      7. pow-to-exp 37.8

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

   7. Applied rewrites 37.8%

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

   if -6.60000000000000056e98 < y < 1.1e190

   1. Initial program 41.8%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 90.3

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

   4. Applied rewrites 90.3%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 71.6

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

      4. *-rgt-identity 71.6

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

      5. *-commutative 71.6

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

      6. log-E 71.6

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

      7. pow-to-exp 71.6

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

   7. Applied rewrites 71.6%

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

Alternative 10: 62.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 3.8 \cdot 10^{+118}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot \left(y \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 3.8e+118) (* (* c y) x) (* (* c x) (* y 1.0))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 3.8e+118) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 3.8d+118) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * (y * 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 3.8e+118) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 3.8e+118:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * (y * 1.0)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 3.8e+118)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * Float64(y * 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 3.8e+118)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * (y * 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 3.8e+118], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * N[(y * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 3.80000000000000016e118

   1. Initial program 45.9%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 77.3

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

   4. Applied rewrites 77.3%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 63.1

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

      4. *-rgt-identity 63.1

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

      5. *-commutative 63.1

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

      6. log-E 63.1

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

      7. pow-to-exp 63.1

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

   7. Applied rewrites 63.1%

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

   if 3.80000000000000016e118 < c

   1. Initial program 16.9%

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

   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 61.0

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

   4. Applied rewrites 61.0%

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

Alternative 11: 62.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+83}:\\ \;\;\;\;c \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -2e+83) (* c (* y x)) (* (* c y) x)))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -2e+83) {
		tmp = c * (y * x);
	} else {
		tmp = (c * y) * x;
	}
	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 <= (-2d+83)) then
        tmp = c * (y * x)
    else
        tmp = (c * y) * x
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -2e+83) {
		tmp = c * (y * x);
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -2e+83:
		tmp = c * (y * x)
	else:
		tmp = (c * y) * x
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -2e+83)
		tmp = Float64(c * Float64(y * x));
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (y <= -2e+83)
		tmp = c * (y * x);
	else
		tmp = (c * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -2e+83], N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000006e83

   1. Initial program 49.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 34.1

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

   6. Applied rewrites 34.1%

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

   if -2.00000000000000006e83 < y

   1. Initial program 39.6%

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

   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 86.8

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 68.9

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

      4. *-rgt-identity 68.9

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

      5. *-commutative 68.9

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

      6. log-E 68.9

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

      7. pow-to-exp 68.9

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

   7. Applied rewrites 68.9%

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

Alternative 12: 61.4% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.3%

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

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 76.2

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

4. Applied rewrites 76.2%

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

5. Taylor expanded in x around 0

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

   1. lift-expm1.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lower-expm1.f64 61.4

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

   4. *-rgt-identity 61.4

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

   5. *-commutative 61.4

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

   6. log-E 61.4

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

   7. pow-to-exp 61.4

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

7. Applied rewrites 61.4%

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

Developer Target 1: 93.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025114 
(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)))))