Logarithmic Transform

Percentage Accurate: 41.7% → 98.5%

Time: 7.2s

Alternatives: 8

Speedup: 19.8×

• could not determine a ground truth

  1. c = -8.691280899783748e+239
  2. x = 1.9801541604395484e+247
  3. y = 1.8773447618992845e+106

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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.52 \cdot 10^{-153}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (expm1 x) y)) c)))
   (if (<= y -1.85e-5)
     t_0
     (if (<= y 1.52e-153) (* (* 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 <= -1.85e-5) {
		tmp = t_0;
	} else if (y <= 1.52e-153) {
		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 <= -1.85e-5) {
		tmp = t_0;
	} else if (y <= 1.52e-153) {
		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 <= -1.85e-5:
		tmp = t_0
	elif y <= 1.52e-153:
		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 <= -1.85e-5)
		tmp = t_0;
	elseif (y <= 1.52e-153)
		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, -1.85e-5], t$95$0, If[LessEqual[y, 1.52e-153], 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 < -1.84999999999999991e-5 or 1.52000000000000004e-153 < y

   1. Initial program 37.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 97.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 97.7%

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

   if -1.84999999999999991e-5 < y < 1.52000000000000004e-153

   1. Initial program 47.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 2: 89.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1400:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1400.0)
   (* (log1p (* x y)) c)
   (if (<= y 9.5e-27)
     (* (* 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 <= -1400.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 9.5e-27) {
		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 <= -1400.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 9.5e-27)
		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, -1400.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 9.5e-27], 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 < -1400

   1. Initial program 50.4%

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 63.3

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

      4. *-rgt-identity 63.3

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

      5. *-commutative 63.3

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

      6. log-E 63.3

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

      7. pow-to-exp 63.3

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

   7. Applied rewrites 63.3%

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

   if -1400 < y < 9.50000000000000037e-27

   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 99.0

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

   5. Applied rewrites 99.0%

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

   if 9.50000000000000037e-27 < y

   1. Initial program 19.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.5%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      3. lower-expm1.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(\color{blue}{x} \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      5. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      6. log-E N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      7. pow-to-exp N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right)\right) \cdot y\right) \cdot c \]

      8. *-commutative N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \cdot c \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(\left(1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \cdot c \]

      10. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\right) \cdot x\right) \cdot y\right) \cdot c \]

      11. *-commutative N/A

          \[\leadsto \mathsf{log1p}\left(\left(\left(\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x + 1\right) \cdot x\right) \cdot y\right) \cdot c \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{log1p}\left(\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot x, x, 1\right) \cdot x\right) \cdot y\right) \cdot c \]

      13. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(\left(\mathsf{fma}\left(\frac{1}{6} \cdot x + \frac{1}{2}, x, 1\right) \cdot x\right) \cdot y\right) \cdot c \]

      14. lower-fma.f64 93.3

          \[\leadsto \mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c \]

   7. Applied rewrites 93.3%

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

Alternative 3: 89.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1400:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(y \cdot x, 0.5, y\right) \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1400.0)
   (* (log1p (* x y)) c)
   (if (<= y 9.5e-27)
     (* (* c y) (expm1 (* x 1.0)))
     (* (log1p (* (fma (* y x) 0.5 y) x)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1400.0) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 9.5e-27) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = log1p((fma((y * x), 0.5, y) * x)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1400.0)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 9.5e-27)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = Float64(log1p(Float64(fma(Float64(y * x), 0.5, y) * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1400.0], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 9.5e-27], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(y * x), $MachinePrecision] * 0.5 + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1400

   1. Initial program 50.4%

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 63.3

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

      4. *-rgt-identity 63.3

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

      5. *-commutative 63.3

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

      6. log-E 63.3

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

      7. pow-to-exp 63.3

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

   7. Applied rewrites 63.3%

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

   if -1400 < y < 9.50000000000000037e-27

   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 99.0

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

   5. Applied rewrites 99.0%

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

   if 9.50000000000000037e-27 < y

   1. Initial program 19.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.5%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 93.2

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

   7. Applied rewrites 93.2%

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

Alternative 4: 89.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -1400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* x y)) c)))
   (if (<= y -1400.0)
     t_0
     (if (<= y 9.5e-27) (* (* 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 <= -1400.0) {
		tmp = t_0;
	} else if (y <= 9.5e-27) {
		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 <= -1400.0) {
		tmp = t_0;
	} else if (y <= 9.5e-27) {
		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 <= -1400.0:
		tmp = t_0
	elif y <= 9.5e-27:
		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 <= -1400.0)
		tmp = t_0;
	elseif (y <= 9.5e-27)
		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, -1400.0], t$95$0, If[LessEqual[y, 9.5e-27], 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 < -1400 or 9.50000000000000037e-27 < y

   1. Initial program 38.1%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.2%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 75.1

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

      4. *-rgt-identity 75.1

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

      5. *-commutative 75.1

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

      6. log-E 75.1

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

      7. pow-to-exp 75.1

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

   7. Applied rewrites 75.1%

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

   if -1400 < y < 9.50000000000000037e-27

   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 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 5: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-5}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -3.2e-5) (* (* (expm1 x) y) c) (* (log1p (* x y)) c)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -3.2e-5) {
		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 (x <= -3.2e-5) {
		tmp = (Math.expm1(x) * y) * c;
	} else {
		tmp = Math.log1p((x * y)) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -3.2e-5:
		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 (x <= -3.2e-5)
		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[x, -3.2e-5], 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 x < -3.19999999999999986e-5

   1. Initial program 53.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.8%

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

   5. Taylor expanded in y around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. lower-expm1.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lift-expm1.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-expm1.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 67.2

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

      15. lift-*.f64 N/A

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

      16. *-rgt-identity 67.2

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

   7. Applied rewrites 67.2%

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

   if -3.19999999999999986e-5 < x

   1. Initial program 36.4%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 90.9%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 89.7

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

      4. *-rgt-identity 89.7

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

      5. *-commutative 89.7

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

      6. log-E 89.7

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

      7. pow-to-exp 89.7

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

   7. Applied rewrites 89.7%

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

Alternative 6: 74.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.7%

   \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
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 93.7%

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

5. Taylor expanded in y around 0

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

   1. lift-expm1.f64 N/A

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

   2. *-rgt-identity N/A

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

   3. lower-expm1.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. *-commutative N/A

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

   6. log-E N/A

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

   7. pow-to-exp N/A

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

   8. lower-expm1.f64 N/A

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

   9. *-rgt-identity N/A

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

   10. lift-expm1.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-expm1.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 74.2

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

   15. lift-*.f64 N/A

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

   16. *-rgt-identity 74.2

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

7. Applied rewrites 74.2%

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

Alternative 7: 58.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.55 \cdot 10^{+91}:\\ \;\;\;\;\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.55e+91) (* (* y x) c) (* (* c x) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.55e+91) {
		tmp = (y * x) * c;
	} 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 <= 1.55d+91) then
        tmp = (y * x) * c
    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 <= 1.55e+91) {
		tmp = (y * x) * c;
	} else {
		tmp = (c * x) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.55e+91:
		tmp = (y * x) * c
	else:
		tmp = (c * x) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.55e+91)
		tmp = Float64(Float64(y * x) * c);
	else
		tmp = Float64(Float64(c * x) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1.55e+91)
		tmp = (y * x) * c;
	else
		tmp = (c * x) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.55e+91], N[(N[(y * x), $MachinePrecision] * c), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.54999999999999999e91

   1. Initial program 46.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 95.6%

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

   5. Taylor expanded in x around 0

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

      1. lift-expm1.f64 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 58.5

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

   7. Applied rewrites 58.5%

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

   if 1.54999999999999999e91 < c

   1. Initial program 17.9%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 59.1

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

   5. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
   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 59.1

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

   7. Applied rewrites 59.1%

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

Alternative 8: 59.1% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log \mathsf{E}\left(\right)\right)\right)} \]
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 59.1

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

5. Applied rewrites 59.1%

   \[\leadsto \color{blue}{\left(c \cdot x\right) \cdot \left(y \cdot 1\right)} \]
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 59.1

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

7. Applied rewrites 59.1%

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

Developer Target 1: 93.7% 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 2025089 
(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)))))