Logarithmic Transform

Percentage Accurate: 41.2% → 99.1%

Time: 5.1s

Alternatives: 9

Speedup: 4.9×

• could not determine a ground truth

  1. c = -1.0255777523998591e-97
  2. x = 4.623773640851071e+28
  3. y = 9.513243043582193e-32

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -215:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x \cdot 3\right) \cdot y}{\left(e^{x} - -1\right) + e^{x + x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -215.0)
   (* (log1p (* (expm1 x) y)) c)
   (if (<= y 5.3e-11)
     (* (* y c) (expm1 x))
     (*
      c
      (log1p
       (/ (* (expm1 (* x 3.0)) y) (+ (- (exp x) -1.0) (exp (+ x x)))))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -215.0) {
		tmp = log1p((expm1(x) * y)) * c;
	} else if (y <= 5.3e-11) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * log1p(((expm1((x * 3.0)) * y) / ((exp(x) - -1.0) + exp((x + x)))));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= -215.0) {
		tmp = Math.log1p((Math.expm1(x) * y)) * c;
	} else if (y <= 5.3e-11) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = c * Math.log1p(((Math.expm1((x * 3.0)) * y) / ((Math.exp(x) - -1.0) + Math.exp((x + x)))));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= -215.0:
		tmp = math.log1p((math.expm1(x) * y)) * c
	elif y <= 5.3e-11:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = c * math.log1p(((math.expm1((x * 3.0)) * y) / ((math.exp(x) - -1.0) + math.exp((x + x)))))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -215.0)
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	elseif (y <= 5.3e-11)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * log1p(Float64(Float64(expm1(Float64(x * 3.0)) * y) / Float64(Float64(exp(x) - -1.0) + exp(Float64(x + x))))));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -215.0], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 5.3e-11], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(N[(N[(Exp[N[(x * 3.0), $MachinePrecision]] - 1), $MachinePrecision] * y), $MachinePrecision] / N[(N[(N[Exp[x], $MachinePrecision] - -1.0), $MachinePrecision] + N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -215

   1. Initial program 41.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 94.1

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

   5. Applied rewrites 51.3%

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

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-*.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if -215 < y < 5.2999999999999998e-11

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 76.5

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

   6. Applied rewrites 76.5%

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

   if 5.2999999999999998e-11 < y

   1. Initial program 41.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. lift-E.f64 N/A

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

      8. pow-to-exp N/A

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

      9. lift-pow.f64 N/A

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

      10. flip3-- N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

      11. associate-*l/ N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left({\left({e}^{x}\right)}^{3} - {1}^{3}\right) \cdot y}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}}\right) \]

      12. lower-/.f64 N/A

          \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\left({\left({e}^{x}\right)}^{3} - {1}^{3}\right) \cdot y}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}}\right) \]

   5. Applied rewrites 93.9%

      \[\leadsto c \cdot \mathsf{log1p}\left(\color{blue}{\frac{\mathsf{expm1}\left(x \cdot 3\right) \cdot y}{\left(e^{x} - -1\right) + e^{x + x}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -215:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-11}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (expm1 x) y)) c)))
   (if (<= y -215.0) t_0 (if (<= y 5.3e-11) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((expm1(x) * y)) * c;
	double tmp;
	if (y <= -215.0) {
		tmp = t_0;
	} else if (y <= 5.3e-11) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = Math.log1p((Math.expm1(x) * y)) * c;
	double tmp;
	if (y <= -215.0) {
		tmp = t_0;
	} else if (y <= 5.3e-11) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log1p((math.expm1(x) * y)) * c
	tmp = 0
	if y <= -215.0:
		tmp = t_0
	elif y <= 5.3e-11:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(expm1(x) * y)) * c)
	tmp = 0.0
	if (y <= -215.0)
		tmp = t_0;
	elseif (y <= 5.3e-11)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -215.0], t$95$0, If[LessEqual[y, 5.3e-11], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -215 or 5.2999999999999998e-11 < y

   1. Initial program 41.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 94.1

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

   5. Applied rewrites 51.3%

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

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-*.f64 94.1

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

   7. Applied rewrites 94.1%

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

   if -215 < y < 5.2999999999999998e-11

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 76.5

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

   6. Applied rewrites 76.5%

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

Alternative 3: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -960000000:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -960000000.0)
   (* (log (fma (expm1 x) y 1.0)) c)
   (if (<= y 2.3e+56) (* (* y c) (expm1 x)) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -960000000.0) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= 2.3e+56) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -960000000.0)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= 2.3e+56)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -960000000.0], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 2.3e+56], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.6e8

   1. Initial program 41.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 94.1

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

   5. Applied rewrites 51.3%

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

   if -9.6e8 < y < 2.30000000000000015e56

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 76.5

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

   6. Applied rewrites 76.5%

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

   if 2.30000000000000015e56 < y

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 66.4

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity 66.4

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

   6. Applied rewrites 66.4%

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

Alternative 4: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -215:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -215.0) t_0 (if (<= y 2.3e+56) (* (* 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 <= -215.0) {
		tmp = t_0;
	} else if (y <= 2.3e+56) {
		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 <= -215.0) {
		tmp = t_0;
	} else if (y <= 2.3e+56) {
		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 <= -215.0:
		tmp = t_0
	elif y <= 2.3e+56:
		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 <= -215.0)
		tmp = t_0;
	elseif (y <= 2.3e+56)
		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, -215.0], t$95$0, If[LessEqual[y, 2.3e+56], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -215 or 2.30000000000000015e56 < y

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 66.4

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity 66.4

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

   6. Applied rewrites 66.4%

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

   if -215 < y < 2.30000000000000015e56

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 76.5

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

   6. Applied rewrites 76.5%

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

Alternative 5: 82.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(x, y, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 10^{+204}:\\ \;\;\;\;\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma x y 1.0)) c)))
   (if (<= y -1.1e+153)
     t_0
     (if (<= y 2.3e+56)
       (* (* (expm1 x) c) y)
       (if (<= y 1e+204) (* (* y (expm1 x)) c) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(x, y, 1.0)) * c;
	double tmp;
	if (y <= -1.1e+153) {
		tmp = t_0;
	} else if (y <= 2.3e+56) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 1e+204) {
		tmp = (y * expm1(x)) * c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(x, y, 1.0)) * c)
	tmp = 0.0
	if (y <= -1.1e+153)
		tmp = t_0;
	elseif (y <= 2.3e+56)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 1e+204)
		tmp = Float64(Float64(y * expm1(x)) * c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(x * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.1e+153], t$95$0, If[LessEqual[y, 2.3e+56], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1e+204], N[(N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e153 or 9.99999999999999989e203 < y

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-E.f64 39.9

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

   4. Applied rewrites 39.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.9

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 39.9

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

      8. lift-*.f64 N/A

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

      9. lift-log.f64 N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-rgt-identity 39.9

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

   6. Applied rewrites 39.9%

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

   if -1.1e153 < y < 2.30000000000000015e56

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.2

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-lft-identity N/A

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

      14. lower-expm1.f64 76.9

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

   6. Applied rewrites 76.9%

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

   if 2.30000000000000015e56 < y < 9.99999999999999989e203

   1. Initial program 41.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 94.1

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 74.4

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

   8. Applied rewrites 74.4%

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

Alternative 6: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 7 \cdot 10^{+144}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 7e+144) (* (* y c) (expm1 x)) (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 7e+144) {
		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 <= 7e+144) {
		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 <= 7e+144:
		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 <= 7e+144)
		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, 7e+144], 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 < 6.9999999999999996e144

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 76.5

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

   6. Applied rewrites 76.5%

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

   if 6.9999999999999996e144 < c

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.2

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-lft-identity N/A

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

      14. lower-expm1.f64 76.9

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

   6. Applied rewrites 76.9%

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

Alternative 7: 77.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{+56}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 2.3e+56) (* (* y c) (expm1 x)) (* c (* x y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 2.3e+56) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 2.3e+56) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 2.3e+56:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = c * (x * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 2.3e+56)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * Float64(x * y));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 2.3e+56], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.30000000000000015e56

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 46.0

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lift-E.f64 N/A

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

      11. log-E N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 76.5

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

   6. Applied rewrites 76.5%

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

   if 2.30000000000000015e56 < y

   1. Initial program 41.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.5

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.5

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 94.1

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

   3. Applied rewrites 94.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 56.1

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

   6. Applied rewrites 56.1%

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

Alternative 8: 62.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 9 \cdot 10^{+151}:\\ \;\;\;\;x \cdot \left(c \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 9e+151) (* x (* c y)) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 9e+151) {
		tmp = x * (c * y);
	} 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 <= 9d+151) then
        tmp = x * (c * y)
    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 <= 9e+151) {
		tmp = x * (c * y);
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 9e+151:
		tmp = x * (c * y)
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 9e+151)
		tmp = Float64(x * Float64(c * y));
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 9e+151)
		tmp = x * (c * y);
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 9e+151], N[(x * N[(c * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 8.9999999999999997e151

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.2

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-lft-identity N/A

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

      14. lower-expm1.f64 76.9

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 58.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 61.0

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

   11. Applied rewrites 61.0%

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

   if 8.9999999999999997e151 < c

   1. Initial program 41.2%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 46.2

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

      8. lift--.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow-to-exp N/A

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

      11. lift-E.f64 N/A

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

      12. log-E N/A

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

      13. *-lft-identity N/A

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

      14. lower-expm1.f64 76.9

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

   6. Applied rewrites 76.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 58.6%

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

Alternative 9: 61.0% accurate, 4.9× speedup

Math

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

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(x * Float64(c * y))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

2. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-E.f64 46.2

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

4. Applied rewrites 46.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 46.2

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

   8. lift--.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-to-exp N/A

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

   11. lift-E.f64 N/A

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

   12. log-E N/A

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

   13. *-lft-identity N/A

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

   14. lower-expm1.f64 76.9

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

6. Applied rewrites 76.9%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 58.6%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 61.0

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

11. Applied rewrites 61.0%

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

Developer Target 1: 94.1% 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 2025149 
(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)))))