Logarithmic Transform

Percentage Accurate: 41.4% → 99.3%

Time: 5.8s

Alternatives: 10

Speedup: 4.9×

• could not determine a ground truth

  1. c = 4.921781290497682e-304
  2. x = 1.8697278773818675e+108
  3. y = 1.6652808415827174e-148

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: 41.4% 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: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{if}\;y \leq -5 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-15}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 x) y)))))
   (if (<= y -5e-18) t_0 (if (<= y 2.8e-15) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1(x) * y));
	double tmp;
	if (y <= -5e-18) {
		tmp = t_0;
	} else if (y <= 2.8e-15) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1(x) * y));
	double tmp;
	if (y <= -5e-18) {
		tmp = t_0;
	} else if (y <= 2.8e-15) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1(x) * y))
	tmp = 0
	if y <= -5e-18:
		tmp = t_0
	elif y <= 2.8e-15:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(x) * y)))
	tmp = 0.0
	if (y <= -5e-18)
		tmp = t_0;
	elseif (y <= 2.8e-15)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-18], t$95$0, If[LessEqual[y, 2.8e-15], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.00000000000000036e-18 or 2.80000000000000014e-15 < y

   1. Initial program 37.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. rem-exp-log N/A

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      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

      1. *-rgt-identity 99.1

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

   6. Applied rewrites 99.1%

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

   if -5.00000000000000036e-18 < y < 2.80000000000000014e-15

   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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. mult-flip N/A

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

      7. unpow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. *-rgt-identity 99.4

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

   7. Applied rewrites 99.4%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.000195:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), \left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\right) \cdot -0.5\right), \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.000195)
   (* c (log1p (* (expm1 x) y)))
   (if (<= y 8e+68)
     (*
      (fma
       y
       (fma
        y
        (fma
         (* (pow (expm1 x) 3.0) c)
         0.3333333333333333
         (* (* (* (pow (expm1 x) 4.0) y) c) -0.25))
        (* (* (* (expm1 x) (expm1 x)) c) -0.5))
       (* (expm1 x) c))
      y)
     (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -0.000195) {
		tmp = c * log1p((expm1(x) * y));
	} else if (y <= 8e+68) {
		tmp = fma(y, fma(y, fma((pow(expm1(x), 3.0) * c), 0.3333333333333333, (((pow(expm1(x), 4.0) * y) * c) * -0.25)), (((expm1(x) * expm1(x)) * c) * -0.5)), (expm1(x) * c)) * y;
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -0.000195)
		tmp = Float64(c * log1p(Float64(expm1(x) * y)));
	elseif (y <= 8e+68)
		tmp = Float64(fma(y, fma(y, fma(Float64((expm1(x) ^ 3.0) * c), 0.3333333333333333, Float64(Float64(Float64((expm1(x) ^ 4.0) * y) * c) * -0.25)), Float64(Float64(Float64(expm1(x) * expm1(x)) * c) * -0.5)), Float64(expm1(x) * c)) * y);
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -0.000195], N[(c * N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8e+68], N[(N[(y * N[(y * N[(N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 3.0], $MachinePrecision] * c), $MachinePrecision] * 0.3333333333333333 + N[(N[(N[(N[Power[N[(Exp[x] - 1), $MachinePrecision], 4.0], $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(Exp[x] - 1), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.94999999999999996e-4

   1. Initial program 49.3%

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

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      15. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

      1. *-rgt-identity 99.6

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

   6. Applied rewrites 99.6%

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

   if -1.94999999999999996e-4 < y < 7.99999999999999962e68

   1. Initial program 43.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. rem-exp-log N/A

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      15. lower-*.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + y \cdot \left(\frac{-1}{4} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{4}\right)\right) + \frac{1}{3} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left({\left(\mathsf{expm1}\left(x\right)\right)}^{3} \cdot c, 0.3333333333333333, \left(\left({\left(\mathsf{expm1}\left(x\right)\right)}^{4} \cdot y\right) \cdot c\right) \cdot -0.25\right), \left(\left(\mathsf{expm1}\left(x\right) \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\right) \cdot -0.5\right), \mathsf{expm1}\left(x\right) \cdot c\right) \cdot y} \]

   if 7.99999999999999962e68 < y

   1. Initial program 13.5%

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

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      15. lower-*.f64 98.3

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.2%

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

Alternative 3: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+51}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 1.96:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -3.5e+51)
   (* (log (fma (expm1 x) y 1.0)) c)
   (if (<= y 1.96) (* (* (expm1 x) c) y) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -3.5e+51) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= 1.96) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -3.5e+51)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= 1.96)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -3.5e+51], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.96], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e51

   1. Initial program 49.0%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 73.7%

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

   4. Taylor expanded in x around 0

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

      1. *-rgt-identity 73.7

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

   6. Applied rewrites 73.7%

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

   if -3.5e51 < y < 1.96

   1. Initial program 44.7%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

   5. Taylor expanded in y around 0

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 96.1

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 96.1

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

   7. Applied rewrites 96.1%

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

   if 1.96 < y

   1. Initial program 17.2%

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

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      15. lower-*.f64 98.4

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 97.1%

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

Alternative 4: 91.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-305}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-13}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)))
   (if (<= t_0 -5e-305)
     (* (* (expm1 x) c) y)
     (if (<= t_0 0.0)
       (* c (log1p (* x y)))
       (if (<= t_0 4e-13)
         (* (* y c) (expm1 x))
         (* (log (* (expm1 x) y)) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = (expm1(x) * c) * y;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 4e-13) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = log((expm1(x) * y)) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = (Math.pow(Math.E, x) - 1.0) * y;
	double tmp;
	if (t_0 <= -5e-305) {
		tmp = (Math.expm1(x) * c) * y;
	} else if (t_0 <= 0.0) {
		tmp = c * Math.log1p((x * y));
	} else if (t_0 <= 4e-13) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = Math.log((Math.expm1(x) * y)) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = (math.pow(math.e, x) - 1.0) * y
	tmp = 0
	if t_0 <= -5e-305:
		tmp = (math.expm1(x) * c) * y
	elif t_0 <= 0.0:
		tmp = c * math.log1p((x * y))
	elif t_0 <= 4e-13:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = math.log((math.expm1(x) * y)) * c
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	tmp = 0.0
	if (t_0 <= -5e-305)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 4e-13)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(log(Float64(expm1(x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-305], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e-13], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -4.99999999999999985e-305

   1. Initial program 29.3%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

   5. Taylor expanded in y around 0

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 97.1

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 97.1

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

   7. Applied rewrites 97.1%

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

   if -4.99999999999999985e-305 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -0.0

   1. Initial program 35.9%

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

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      15. lower-*.f64 91.1

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

   3. Applied rewrites 91.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 91.0%

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

   if -0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 4.0000000000000001e-13

   1. Initial program 30.4%

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

   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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. mult-flip N/A

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

      7. unpow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-rgt-identity 99.8

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

   7. Applied rewrites 99.8%

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

   if 4.0000000000000001e-13 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 94.4%

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

   4. Taylor expanded in y around inf

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 89.9

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 89.9

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

   6. Applied rewrites 89.9%

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

Alternative 5: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -3.55 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.75:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -3.55e+22) t_0 (if (<= y 2.75) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -3.55e+22) {
		tmp = t_0;
	} else if (y <= 2.75) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -3.55e+22) {
		tmp = t_0;
	} else if (y <= 2.75) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -3.55e+22:
		tmp = t_0
	elif y <= 2.75:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -3.55e+22)
		tmp = t_0;
	elseif (y <= 2.75)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.55e+22], t$95$0, If[LessEqual[y, 2.75], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.5500000000000001e22 or 2.75 < y

   1. Initial program 36.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. rem-exp-log N/A

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

      12. log-pow-rev N/A

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

      13. lower-expm1.f64 N/A

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

      14. log-E N/A

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

      15. lower-*.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 77.0%

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

   if -3.5500000000000001e22 < y < 2.75

   1. Initial program 44.5%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. mult-flip N/A

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

      7. unpow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in x around 0

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

      1. *-rgt-identity 97.5

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

   7. Applied rewrites 97.5%

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

Alternative 6: 79.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+239}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -3.3e+53) t_0 (if (<= y 2.3e+239) (* (* (expm1 x) c) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -3.3e+53) {
		tmp = t_0;
	} else if (y <= 2.3e+239) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -3.3e+53)
		tmp = t_0;
	elseif (y <= 2.3e+239)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.3e+53], t$95$0, If[LessEqual[y, 2.3e+239], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3000000000000002e53 or 2.3000000000000002e239 < y

   1. Initial program 44.4%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. log-E N/A

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

      4. metadata-eval N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. log-E N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. unpow1 N/A

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

      11. metadata-eval N/A

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

      12. pow-pow N/A

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

      13. inv-pow N/A

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

      14. unpow-prod-down N/A

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

      15. mult-flip N/A

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

      16. inv-pow N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. unpow1 N/A

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

      20. lower-fma.f64 42.7

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

   4. Applied rewrites 42.7%

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

   if -3.3000000000000002e53 < y < 2.3000000000000002e239

   1. Initial program 40.5%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

   5. Taylor expanded in y around 0

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 90.6

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 90.6

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

   7. Applied rewrites 90.6%

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

Alternative 7: 77.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 5e-26) (* (* y c) (expm1 x)) (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 5e-26) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 5e-26) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 5e-26:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 5e-26)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5e-26], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.00000000000000019e-26

   1. Initial program 48.9%

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

   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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. mult-flip N/A

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

      7. unpow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in x around 0

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

      1. *-rgt-identity 77.1

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

   7. Applied rewrites 77.1%

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

   if 5.00000000000000019e-26 < c

   1. Initial program 21.2%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

   5. Taylor expanded in y around 0

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

      1. lower-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 78.6

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 78.6

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

   7. Applied rewrites 78.6%

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

Alternative 8: 76.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 75.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

5. Taylor expanded in y around 0

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

   1. lower-expm1.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 76.4

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

   6. lift-*.f64 N/A

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

   7. *-rgt-identity 76.4

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

7. Applied rewrites 76.4%

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

Alternative 9: 62.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 5.5e-87) (* (* y c) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 5.5e-87) {
		tmp = (y * c) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 5.5d-87) then
        tmp = (y * c) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 5.5e-87) {
		tmp = (y * c) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 5.5e-87:
		tmp = (y * c) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 5.5e-87)
		tmp = Float64(Float64(y * c) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 5.5e-87)
		tmp = (y * c) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5.5e-87], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.5000000000000004e-87

   1. Initial program 49.2%

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

   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. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. pow-pow N/A

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

      5. inv-pow N/A

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

      6. mult-flip N/A

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

      7. unpow-prod-down N/A

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

      8. metadata-eval N/A

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

      9. inv-pow N/A

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

      10. pow-pow N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-*.f64 N/A

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.5%

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

   if 5.5000000000000004e-87 < c

   1. Initial program 24.9%

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

   2. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.1

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

   7. Applied rewrites 58.1%

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

Alternative 10: 59.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.4%

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

2. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 75.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\right) \cdot y\right) \cdot c, -0.5, \mathsf{expm1}\left(x \cdot 1\right) \cdot c\right) \cdot y} \]

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 59.0

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

7. Applied rewrites 59.0%

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

Developer Target 1: 93.5% 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 2025120 
(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)))))