Logarithmic Transform

Percentage Accurate: 41.7% → 98.9%

Time: 5.2s

Alternatives: 8

Speedup: 4.9×

• could not determine a ground truth

  1. c = 2.7404260655980755e+208
  2. x = 2.02633924821503e+224
  3. y = 2.873542373521297e-287

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: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;y \leq -1 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-107}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y (expm1 x))))))
   (if (<= y -1e-26) t_0 (if (<= y 4.5e-107) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * expm1(x)));
	double tmp;
	if (y <= -1e-26) {
		tmp = t_0;
	} else if (y <= 4.5e-107) {
		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((y * Math.expm1(x)));
	double tmp;
	if (y <= -1e-26) {
		tmp = t_0;
	} else if (y <= 4.5e-107) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((y * math.expm1(x)))
	tmp = 0
	if y <= -1e-26:
		tmp = t_0
	elif y <= 4.5e-107:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(y * expm1(x))))
	tmp = 0.0
	if (y <= -1e-26)
		tmp = t_0;
	elseif (y <= 4.5e-107)
		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[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1e-26], t$95$0, If[LessEqual[y, 4.5e-107], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1e-26 or 4.50000000000000016e-107 < y

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   if -1e-26 < y < 4.50000000000000016e-107

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.2

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

   8. Applied rewrites 76.2%

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

Alternative 2: 91.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{+49}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 1.62:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.18e+49)
   (* (log (fma y (expm1 x) 1.0)) c)
   (if (<= y 1.62) (* (* y c) (expm1 x)) (* c (log1p (* y x))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.18e+49) {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	} else if (y <= 1.62) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * log1p((y * x));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.18e+49)
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	elseif (y <= 1.62)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * log1p(Float64(y * x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.18e+49], N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.62], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1800000000000001e49

   1. Initial program 41.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.4

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

   3. Applied rewrites 51.4%

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

   if -1.1800000000000001e49 < y < 1.6200000000000001

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if 1.6200000000000001 < y

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 66.1%

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

Alternative 3: 89.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot x\right)\\ \mathbf{if}\;y \leq -11000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.62:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* y x)))))
   (if (<= y -11000.0) t_0 (if (<= y 1.62) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * x));
	double tmp;
	if (y <= -11000.0) {
		tmp = t_0;
	} else if (y <= 1.62) {
		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((y * x));
	double tmp;
	if (y <= -11000.0) {
		tmp = t_0;
	} else if (y <= 1.62) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((y * x))
	tmp = 0
	if y <= -11000.0:
		tmp = t_0
	elif y <= 1.62:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(y * x)))
	tmp = 0.0
	if (y <= -11000.0)
		tmp = t_0;
	elseif (y <= 1.62)
		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[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -11000.0], t$95$0, If[LessEqual[y, 1.62], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -11000 or 1.6200000000000001 < y

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 66.1%

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

   if -11000 < y < 1.6200000000000001

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.2

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

   8. Applied rewrites 76.2%

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

Alternative 4: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{+168}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-107}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -6.3e+168)
   (* (log (fma y x 1.0)) c)
   (if (<= y 4.5e-107) (* (* y c) (expm1 x)) (* c (* y (expm1 x))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -6.3e+168) {
		tmp = log(fma(y, x, 1.0)) * c;
	} else if (y <= 4.5e-107) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * (y * expm1(x));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -6.3e+168)
		tmp = Float64(log(fma(y, x, 1.0)) * c);
	elseif (y <= 4.5e-107)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * Float64(y * expm1(x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -6.3e+168], N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 4.5e-107], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.2999999999999997e168

   1. Initial program 41.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.7

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.4

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

   3. Applied rewrites 51.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 39.4%

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

   if -6.2999999999999997e168 < y < 4.50000000000000016e-107

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if 4.50000000000000016e-107 < y

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

Alternative 5: 77.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-107}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y 4.5e-107) (* (* y c) (expm1 x)) (* c (* y (expm1 x)))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= 4.5e-107) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = c * (y * expm1(x));
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (y <= 4.5e-107) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = c * (y * Math.expm1(x));
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if y <= 4.5e-107:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = c * (y * math.expm1(x))
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= 4.5e-107)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(c * Float64(y * expm1(x)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, 4.5e-107], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.50000000000000016e-107

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-expm1.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if 4.50000000000000016e-107 < y

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

Alternative 6: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-145}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(c \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -5.8e-145) (* c (* y (expm1 x))) (* y (* c x))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -5.8e-145) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = y * (c * x);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -5.8e-145) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = y * (c * x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -5.8e-145:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = y * (c * x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -5.8e-145)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(y * Float64(c * x));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -5.8e-145], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(c * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.79999999999999968e-145

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if -5.79999999999999968e-145 < x

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.0

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 59.0

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

   9. Applied rewrites 59.0%

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

Alternative 7: 59.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5 \cdot 10^{-28}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(c \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 5e-28) (* c (* x y)) (* y (* c x))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 5e-28) {
		tmp = c * (x * y);
	} else {
		tmp = y * (c * x);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 5d-28) then
        tmp = c * (x * y)
    else
        tmp = y * (c * x)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 5e-28) {
		tmp = c * (x * y);
	} else {
		tmp = y * (c * x);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 5e-28:
		tmp = c * (x * y)
	else:
		tmp = y * (c * x)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 5e-28)
		tmp = Float64(c * Float64(x * y));
	else
		tmp = Float64(y * Float64(c * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 5e-28)
		tmp = c * (x * y);
	else
		tmp = y * (c * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5e-28], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(y * N[(c * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.0000000000000002e-28

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   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 56.0

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

   6. Applied rewrites 56.0%

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

   if 5.0000000000000002e-28 < c

   1. Initial program 41.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.4

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

      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 56.4

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

      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. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.6

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

   3. Applied rewrites 93.6%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-expm1.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

      8. lower-expm1.f64 76.0

         \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

   6. Applied rewrites 76.0%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

   7. Taylor expanded in x around 0

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

      1. lower-*.f64 59.0

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

   9. Applied rewrites 59.0%

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

Alternative 8: 58.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lower-log1p.f64 56.4

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

   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 56.4

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

   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. lift-E.f64 N/A

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

   10. e-exp-1 N/A

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

   11. pow-exp N/A

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

   12. *-lft-identity N/A

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

   13. lower-expm1.f64 93.6

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

3. Applied rewrites 93.6%

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

4. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-expm1.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto y \cdot \mathsf{fma}\left(\frac{-1}{2}, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \left(e^{x} - 1\right)\right) \]

   8. lower-expm1.f64 76.0

      \[\leadsto y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right) \]

6. Applied rewrites 76.0%

   \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(-0.5, c \cdot \left(y \cdot {\left(\mathsf{expm1}\left(x\right)\right)}^{2}\right), c \cdot \mathsf{expm1}\left(x\right)\right)} \]

7. Taylor expanded in x around 0

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

   1. lower-*.f64 59.0

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

9. Applied rewrites 59.0%

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

Developer Target 1: 93.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025159 
(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)))))