Logarithmic Transform

Percentage Accurate: 41.4% → 99.3%

Time: 6.1s

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. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.4

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -0.000195)
   (* c (log1p (* (expm1 x) y)))
   (if (<= y 8e+68)
     (*
      (fma
       y
       (fma
        (* (* (expm1 x) (expm1 x)) c)
        -0.5
        (*
         (fma
          (* (pow (expm1 x) 3.0) c)
          0.3333333333333333
          (* (* (* (pow (expm1 x) 4.0) y) c) -0.25))
         y))
       (* (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(((expm1(x) * expm1(x)) * c), -0.5, (fma((pow(expm1(x), 3.0) * c), 0.3333333333333333, (((pow(expm1(x), 4.0) * y) * c) * -0.25)) * y)), (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(Float64(Float64(expm1(x) * expm1(x)) * c), -0.5, Float64(fma(Float64((expm1(x) ^ 3.0) * c), 0.3333333333333333, Float64(Float64(Float64((expm1(x) ^ 4.0) * y) * c) * -0.25)) * y)), 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[(N[(N[(N[(Exp[x] - 1), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * -0.5 + N[(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] * y), $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. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around 0

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

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

   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. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.3

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

   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. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.4

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

   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. 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 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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 99.8

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

   6. Applied rewrites 99.8%

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

   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-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 95.6

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

   3. Applied rewrites 95.6%

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

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\log \left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c} \]
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. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 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. 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 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 97.5

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 97.5

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

   6. Applied rewrites 97.5%

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

Alternative 6: 79.8% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma x y 1.0)) c)))
   (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 = log(fma(x, y, 1.0)) * c;
	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(log(fma(x, y, 1.0)) * c)
	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[(N[Log[N[(x * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $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. 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 75.5%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 42.7%

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

   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: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(y \cdot x\right) \cdot c\\ \mathbf{if}\;y \leq -5 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+244}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (* y x)) c)))
   (if (<= y -5e+229) t_0 (if (<= y 1.4e+244) (* (* (expm1 x) c) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log((y * x)) * c;
	double tmp;
	if (y <= -5e+229) {
		tmp = t_0;
	} else if (y <= 1.4e+244) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = Math.log((y * x)) * c;
	double tmp;
	if (y <= -5e+229) {
		tmp = t_0;
	} else if (y <= 1.4e+244) {
		tmp = (Math.expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log((y * x)) * c
	tmp = 0
	if y <= -5e+229:
		tmp = t_0
	elif y <= 1.4e+244:
		tmp = (math.expm1(x) * c) * y
	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 <= -5e+229)
		tmp = t_0;
	elseif (y <= 1.4e+244)
		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[(N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -5e+229], t$95$0, If[LessEqual[y, 1.4e+244], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.0000000000000005e229 or 1.39999999999999995e244 < y

   1. Initial program 35.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 85.2%

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

   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 55.9

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

   9. Applied rewrites 55.9%

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

   if -5.0000000000000005e229 < y < 1.39999999999999995e244

   1. Initial program 41.8%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot \left(y \cdot {\left({\mathsf{E}\left(\right)}^{x} - 1\right)}^{2}\right)\right) + c \cdot \left({\mathsf{E}\left(\right)}^{x} - 1\right)\right)} \]
   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 81.5%

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

   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 81.9

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 81.9

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

   7. Applied rewrites 81.9%

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

Alternative 8: 67.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(y \cdot x\right) \cdot c\\ t_1 := c \cdot \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+230}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -0.02:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+244}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (* y x)) c)) (t_1 (* c (* y x))))
   (if (<= y -4e+230)
     t_0
     (if (<= y -0.02)
       t_1
       (if (<= y 4.5e-14) (* (* c y) x) (if (<= y 1.5e+244) t_1 t_0))))))

C

double code(double c, double x, double y) {
	double t_0 = log((y * x)) * c;
	double t_1 = c * (y * x);
	double tmp;
	if (y <= -4e+230) {
		tmp = t_0;
	} else if (y <= -0.02) {
		tmp = t_1;
	} else if (y <= 4.5e-14) {
		tmp = (c * y) * x;
	} else if (y <= 1.5e+244) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = log((y * x)) * c
    t_1 = c * (y * x)
    if (y <= (-4d+230)) then
        tmp = t_0
    else if (y <= (-0.02d0)) then
        tmp = t_1
    else if (y <= 4.5d-14) then
        tmp = (c * y) * x
    else if (y <= 1.5d+244) then
        tmp = t_1
    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 t_1 = c * (y * x);
	double tmp;
	if (y <= -4e+230) {
		tmp = t_0;
	} else if (y <= -0.02) {
		tmp = t_1;
	} else if (y <= 4.5e-14) {
		tmp = (c * y) * x;
	} else if (y <= 1.5e+244) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log((y * x)) * c
	t_1 = c * (y * x)
	tmp = 0
	if y <= -4e+230:
		tmp = t_0
	elif y <= -0.02:
		tmp = t_1
	elif y <= 4.5e-14:
		tmp = (c * y) * x
	elif y <= 1.5e+244:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log(Float64(y * x)) * c)
	t_1 = Float64(c * Float64(y * x))
	tmp = 0.0
	if (y <= -4e+230)
		tmp = t_0;
	elseif (y <= -0.02)
		tmp = t_1;
	elseif (y <= 4.5e-14)
		tmp = Float64(Float64(c * y) * x);
	elseif (y <= 1.5e+244)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = log((y * x)) * c;
	t_1 = c * (y * x);
	tmp = 0.0;
	if (y <= -4e+230)
		tmp = t_0;
	elseif (y <= -0.02)
		tmp = t_1;
	elseif (y <= 4.5e-14)
		tmp = (c * y) * x;
	elseif (y <= 1.5e+244)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+230], t$95$0, If[LessEqual[y, -0.02], t$95$1, If[LessEqual[y, 4.5e-14], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.5e+244], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0000000000000004e230 or 1.4999999999999999e244 < y

   1. Initial program 35.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   6. Applied rewrites 85.3%

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

   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 55.9

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

   9. Applied rewrites 55.9%

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

   if -4.0000000000000004e230 < y < -0.0200000000000000004 or 4.4999999999999998e-14 < y < 1.4999999999999999e244

   1. Initial program 37.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.4

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

   6. Applied rewrites 58.4%

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

   if -0.0200000000000000004 < y < 4.4999999999999998e-14

   1. Initial program 44.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 74.4%

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

Alternative 9: 62.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Python

def code(c, x, y):
	tmp = 0
	if c <= 5.5e-87:
		tmp = (c * y) * 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(c * y) * 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 = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5.5e-87], N[(N[(c * y), $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. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.5%

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

   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)))))