Logarithmic Transform

Percentage Accurate: 42.0% → 99.1%

Time: 5.6s

Alternatives: 8

Speedup: 19.8×

• could not determine a ground truth

  1. c = 4.361628018635704e+122
  2. x = 3.123659494396257e+171
  3. y = 1.2206059819180713e+99

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: 42.0% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -1 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (expm1 x) y)) c)))
   (if (<= y -1e-76) t_0 (if (<= y 5e+17) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((expm1(x) * y)) * c;
	double tmp;
	if (y <= -1e-76) {
		tmp = t_0;
	} else if (y <= 5e+17) {
		tmp = (c * y) * expm1((x * 1.0));
	} 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 <= -1e-76) {
		tmp = t_0;
	} else if (y <= 5e+17) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} 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 <= -1e-76:
		tmp = t_0
	elif y <= 5e+17:
		tmp = (c * y) * math.expm1((x * 1.0))
	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 <= -1e-76)
		tmp = t_0;
	elseif (y <= 5e+17)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	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, -1e-76], t$95$0, If[LessEqual[y, 5e+17], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999927e-77 or 5e17 < y

   1. Initial program 37.1%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.0%

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

   if -9.99999999999999927e-77 < y < 5e17

   1. Initial program 46.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 99.1

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

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{e}^{x} - 1 \leq -5 \cdot 10^{-13}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= (- (pow E x) 1.0) -5e-13)
   (* (* (expm1 x) y) c)
   (* (log1p (* x y)) c)))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[c_, x_, y_] := If[LessEqual[N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision], -5e-13], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) < -4.9999999999999999e-13

   1. Initial program 54.5%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 66.8

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

   5. Applied rewrites 66.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 66.8

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

      4. pow-to-exp 66.8

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

      5. log-E 66.8

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

      6. *-commutative 66.8

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

      7. *-rgt-identity 66.8

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 66.8

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

      11. lift-*.f64 N/A

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

      12. *-rgt-identity 66.8

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

   7. Applied rewrites 66.8%

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

   if -4.9999999999999999e-13 < (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64))

   1. Initial program 36.2%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{x} \cdot y\right) \cdot c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -9.6)
   (* (log1p (* x y)) c)
   (if (<= y 5e+17)
     (* (* c y) (expm1 (* x 1.0)))
     (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -9.6) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 5e+17) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -9.6)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 5e+17)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -9.6], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 5e+17], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.59999999999999964

   1. Initial program 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.5%

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

   if -9.59999999999999964 < y < 5e17

   1. Initial program 44.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 5e17 < y

   1. Initial program 15.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(x \cdot \left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)} \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \log \left(1 + \left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \]

   5. Applied rewrites 49.9%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, 1, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{c \cdot \log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot x, 1, \frac{1}{2}\right), x, 1\right) \cdot x\right) \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot x, 1, \frac{1}{2}\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]

      3. lower-*.f64 49.9

         \[\leadsto \color{blue}{\log \left(1 + \left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, 1, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]

   7. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.6:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -9.6)
   (* (log1p (* x y)) c)
   (if (<= y 5e+17)
     (* (* c y) (expm1 (* x 1.0)))
     (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -9.6) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 5e+17) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -9.6)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 5e+17)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -9.6], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 5e+17], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.59999999999999964

   1. Initial program 51.1%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.5%

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

   if -9.59999999999999964 < y < 5e17

   1. Initial program 44.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 5e17 < y

   1. Initial program 15.9%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot x\right)\right)} \cdot y\right) \cdot c \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(1 + \frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right) \cdot y\right) \cdot c \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(1 + \frac{1}{2} \cdot x\right) \cdot \color{blue}{x}\right) \cdot y\right) \cdot c \]

      3. +-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(\frac{1}{2} \cdot x + 1\right) \cdot x\right) \cdot y\right) \cdot c \]

      4. lower-fma.f64 97.8

         \[\leadsto \mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c \]

   7. Applied rewrites 97.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -9.6:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* x y)) c)))
   (if (<= y -9.6) t_0 (if (<= y 5e+17) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((x * y)) * c;
	double tmp;
	if (y <= -9.6) {
		tmp = t_0;
	} else if (y <= 5e+17) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = Math.log1p((x * y)) * c;
	double tmp;
	if (y <= -9.6) {
		tmp = t_0;
	} else if (y <= 5e+17) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log1p((x * y)) * c
	tmp = 0
	if y <= -9.6:
		tmp = t_0
	elif y <= 5e+17:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(x * y)) * c)
	tmp = 0.0
	if (y <= -9.6)
		tmp = t_0;
	elseif (y <= 5e+17)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -9.6], t$95$0, If[LessEqual[y, 5e+17], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.59999999999999964 or 5e17 < y

   1. Initial program 38.5%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.0%

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

   if -9.59999999999999964 < y < 5e17

   1. Initial program 44.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-146}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -2.3e-146) (* (* (expm1 x) y) c) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.3e-146) {
		tmp = (expm1(x) * y) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.3e-146) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -2.3e-146:
		tmp = (math.expm1(x) * y) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2.3e-146)
		tmp = Float64(Float64(expm1(x) * y) * c);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2.3e-146], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e-146

   1. Initial program 46.4%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 68.4

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

      4. pow-to-exp 68.4

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

      5. log-E 68.4

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

      6. *-commutative 68.4

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

      7. *-rgt-identity 68.4

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 68.4

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

      11. lift-*.f64 N/A

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

      12. *-rgt-identity 68.4

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

   7. Applied rewrites 68.4%

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

   if -2.3000000000000001e-146 < x

   1. Initial program 38.3%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 81.9

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

   7. Applied rewrites 81.9%

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

Alternative 7: 63.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 8.2 \cdot 10^{+109}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 8.2e+109) (* (* c y) x) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 8.2e+109) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * 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 <= 8.2d+109) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 8.2e+109) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 8.2e+109:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 8.2e+109)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 8.2e+109)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 8.2e+109], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 8.19999999999999939e109

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

      1. lift-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

   if 8.19999999999999939e109 < c

   1. Initial program 16.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 60.6

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

   5. Applied rewrites 60.6%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 60.6

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

   7. Applied rewrites 60.6%

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

Alternative 8: 62.3% accurate, 19.8× speedup

Math

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

FPCore

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

C

double code(double c, double x, double y) {
	return (c * y) * 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 = (c * y) * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.0%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 55.0%

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

6. Taylor expanded in x around 0

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

   1. lift-*.f64 62.3

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

8. Applied rewrites 62.3%

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

Developer Target 1: 93.4% accurate, 1.0× 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 2025088 
(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)))))