Logarithmic Transform

Percentage Accurate: 41.2% → 98.7%

Time: 5.5s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. c = -1.4041941951932862e+20
  2. x = 3.7047291531860333e+241
  3. y = 6.321854381713738e-126

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: 41.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-144}:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -2e-15) t_0 (if (<= y 9e-144) (* y (* (expm1 x) c)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -2e-15) {
		tmp = t_0;
	} else if (y <= 9e-144) {
		tmp = y * (expm1(x) * c);
	} 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 <= -2e-15) {
		tmp = t_0;
	} else if (y <= 9e-144) {
		tmp = y * (Math.expm1(x) * c);
	} 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 <= -2e-15:
		tmp = t_0
	elif y <= 9e-144:
		tmp = y * (math.expm1(x) * c)
	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 <= -2e-15)
		tmp = t_0;
	elseif (y <= 9e-144)
		tmp = Float64(y * Float64(expm1(x) * c));
	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, -2e-15], t$95$0, If[LessEqual[y, 9e-144], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0000000000000002e-15 or 8.9999999999999996e-144 < y

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 93.6

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

   5. Applied rewrites 93.6%

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

   if -2.0000000000000002e-15 < y < 8.9999999999999996e-144

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

Alternative 2: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+93}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -3.8e+93)
   (* (log (fma (expm1 x) y 1.0)) c)
   (if (<= y 1.1) (* y (* (expm1 x) c)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -3.8e+93) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= 1.1) {
		tmp = y * (expm1(x) * c);
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -3.8e+93)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= 1.1)
		tmp = Float64(y * Float64(expm1(x) * c));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -3.8e+93], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.1], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.7999999999999998e93

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. *-commutative N/A

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

      11. lower-log1p.f64 N/A

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

      12. pow-to-exp N/A

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

      13. log-E N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

   5. Applied rewrites 51.5%

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

   if -3.7999999999999998e93 < y < 1.1000000000000001

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 1.1000000000000001 < y

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

      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 66.1

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

      4. *-rgt-identity 66.1

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

      5. *-commutative 66.1

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

      6. log-E 66.1

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

      7. pow-to-exp 66.1

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

   6. Applied rewrites 66.1%

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

Alternative 3: 90.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+93}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -4.1e+93)
   (* (log (* (expm1 x) y)) c)
   (if (<= y 1.1) (* y (* (expm1 x) c)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -4.1e+93) {
		tmp = log((expm1(x) * y)) * c;
	} else if (y <= 1.1) {
		tmp = y * (expm1(x) * c);
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -4.1e+93) {
		tmp = Math.log((Math.expm1(x) * y)) * c;
	} else if (y <= 1.1) {
		tmp = y * (Math.expm1(x) * c);
	} else {
		tmp = c * Math.log1p((x * y));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -4.1e+93:
		tmp = math.log((math.expm1(x) * y)) * c
	elif y <= 1.1:
		tmp = y * (math.expm1(x) * c)
	else:
		tmp = c * math.log1p((x * y))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -4.1e+93)
		tmp = Float64(log(Float64(expm1(x) * y)) * c);
	elseif (y <= 1.1)
		tmp = Float64(y * Float64(expm1(x) * c));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -4.1e+93], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.1], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1000000000000001e93

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around inf

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

   6. Applied rewrites 21.1%

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

   if -4.1000000000000001e93 < y < 1.1000000000000001

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 1.1000000000000001 < y

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

      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 66.1

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

      4. *-rgt-identity 66.1

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

      5. *-commutative 66.1

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

      6. log-E 66.1

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

      7. pow-to-exp 66.1

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

   6. Applied rewrites 66.1%

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

Alternative 4: 89.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -9.0) t_0 (if (<= y 1.1) (* y (* (expm1 x) c)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -9.0) {
		tmp = t_0;
	} else if (y <= 1.1) {
		tmp = y * (expm1(x) * c);
	} 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 <= -9.0) {
		tmp = t_0;
	} else if (y <= 1.1) {
		tmp = y * (Math.expm1(x) * c);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -9.0:
		tmp = t_0
	elif y <= 1.1:
		tmp = y * (math.expm1(x) * c)
	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 <= -9.0)
		tmp = t_0;
	elseif (y <= 1.1)
		tmp = Float64(y * Float64(expm1(x) * c));
	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, -9.0], t$95$0, If[LessEqual[y, 1.1], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9 or 1.1000000000000001 < y

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

      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 66.1

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

      4. *-rgt-identity 66.1

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

      5. *-commutative 66.1

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

      6. log-E 66.1

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

      7. pow-to-exp 66.1

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

   6. Applied rewrites 66.1%

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

   if -9 < y < 1.1000000000000001

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

Alternative 5: 81.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+124}:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -4e+137) t_0 (if (<= y 4.2e+124) (* y (* (expm1 x) c)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -4e+137) {
		tmp = t_0;
	} else if (y <= 4.2e+124) {
		tmp = y * (expm1(x) * c);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -4e+137)
		tmp = t_0;
	elseif (y <= 4.2e+124)
		tmp = Float64(y * Float64(expm1(x) * c));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+137], t$95$0, If[LessEqual[y, 4.2e+124], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000001e137 or 4.20000000000000023e124 < y

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 30.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

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

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

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

      7. metadata-eval N/A

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

      8. associate-*r* N/A

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

      9. *-rgt-identity N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 39.9

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

   7. Applied rewrites 39.9%

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

   if -4.0000000000000001e137 < y < 4.20000000000000023e124

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

Alternative 6: 79.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\log \left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;y \leq 1.05:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -2e+138)
   (* (log (* y x)) c)
   (if (<= y 1.05) (* y (* (expm1 x) c)) (* c (* y x)))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -2e+138) {
		tmp = log((y * x)) * c;
	} else if (y <= 1.05) {
		tmp = y * (expm1(x) * c);
	} else {
		tmp = c * (y * x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -2e+138) {
		tmp = Math.log((y * x)) * c;
	} else if (y <= 1.05) {
		tmp = y * (Math.expm1(x) * c);
	} else {
		tmp = c * (y * x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -2e+138:
		tmp = math.log((y * x)) * c
	elif y <= 1.05:
		tmp = y * (math.expm1(x) * c)
	else:
		tmp = c * (y * x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -2e+138)
		tmp = Float64(log(Float64(y * x)) * c);
	elseif (y <= 1.05)
		tmp = Float64(y * Float64(expm1(x) * c));
	else
		tmp = Float64(c * Float64(y * x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -2e+138], N[(N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.05], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000001e138

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around inf

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

   6. Applied rewrites 21.1%

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

   7. Taylor expanded in x around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 12.4

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

   9. Applied rewrites 12.4%

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

   if -2.0000000000000001e138 < y < 1.05000000000000004

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 1.05000000000000004 < y

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 55.4%

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

Alternative 7: 65.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(y \cdot x\right) \cdot c\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{+213}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (* y x)) c)))
   (if (<= y -1.9e+138) t_0 (if (<= y 1.18e+213) (* (* c y) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log((y * x)) * c;
	double tmp;
	if (y <= -1.9e+138) {
		tmp = t_0;
	} else if (y <= 1.18e+213) {
		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 = log((y * x)) * c
    if (y <= (-1.9d+138)) then
        tmp = t_0
    else if (y <= 1.18d+213) 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 = Math.log((y * x)) * c;
	double tmp;
	if (y <= -1.9e+138) {
		tmp = t_0;
	} else if (y <= 1.18e+213) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log((y * x)) * c
	tmp = 0
	if y <= -1.9e+138:
		tmp = t_0
	elif y <= 1.18e+213:
		tmp = (c * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log(Float64(y * x)) * c)
	tmp = 0.0
	if (y <= -1.9e+138)
		tmp = t_0;
	elseif (y <= 1.18e+213)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = log((y * x)) * c;
	tmp = 0.0;
	if (y <= -1.9e+138)
		tmp = t_0;
	elseif (y <= 1.18e+213)
		tmp = (c * y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.9e+138], t$95$0, If[LessEqual[y, 1.18e+213], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.90000000000000006e138 or 1.17999999999999999e213 < y

   1. Initial program 41.2%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around inf

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

   6. Applied rewrites 21.1%

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

   7. Taylor expanded in x around 0

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

      1. sum-log N/A

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

      2. lower-log.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 12.4

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

   9. Applied rewrites 12.4%

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

   if -1.90000000000000006e138 < y < 1.17999999999999999e213

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

   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 61.2

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

      4. *-rgt-identity 61.2

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

      5. *-commutative 61.2

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

      6. log-E 61.2

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

      7. pow-to-exp 61.2

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

   7. Applied rewrites 61.2%

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

Alternative 8: 62.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 6 \cdot 10^{+142}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 6e+142) (* (* c y) x) (* y (* x c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 6e+142) {
		tmp = (c * y) * x;
	} else {
		tmp = y * (x * c);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 6d+142) then
        tmp = (c * y) * x
    else
        tmp = y * (x * c)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 6e+142) {
		tmp = (c * y) * x;
	} else {
		tmp = y * (x * c);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 6e+142:
		tmp = (c * y) * x
	else:
		tmp = y * (x * c)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 6e+142)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(y * Float64(x * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 6e+142)
		tmp = (c * y) * x;
	else
		tmp = y * (x * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 6e+142], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(x * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.99999999999999949e142

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

   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 61.2

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

      4. *-rgt-identity 61.2

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

      5. *-commutative 61.2

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

      6. log-E 61.2

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

      7. pow-to-exp 61.2

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

   7. Applied rewrites 61.2%

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

   if 5.99999999999999949e142 < c

   1. Initial program 41.2%

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

   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 76.3

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

   4. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.3

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.3

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

   6. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.5

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

   8. Applied rewrites 76.5%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 58.5

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

   11. Applied rewrites 58.5%

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

Alternative 9: 58.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

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 76.3

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

4. Applied rewrites 76.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 76.3

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

   4. lift-*.f64 N/A

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

   5. *-rgt-identity 76.3

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

6. Applied rewrites 76.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-expm1.f64 76.5

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

8. Applied rewrites 76.5%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 58.5

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

11. Applied rewrites 58.5%

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

Developer Target 1: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025127 
(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)))))