Logarithmic Transform

Percentage Accurate: 41.1% → 98.9%

Time: 6.9s

Alternatives: 10

Speedup: 4.9×

• could not determine a ground truth

  1. c = 6.153978883099629e-157
  2. x = 2.6130473392076588e+163
  3. y = 5.593762564336087e-159

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: 41.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(x \cdot 1\right) \cdot y\right)\\ \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* (expm1 (* x 1.0)) y)))))
   (if (<= y -1.4) t_0 (if (<= y 3.4e-84) (* (* (expm1 x) c) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((expm1((x * 1.0)) * y));
	double tmp;
	if (y <= -1.4) {
		tmp = t_0;
	} else if (y <= 3.4e-84) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((Math.expm1((x * 1.0)) * y));
	double tmp;
	if (y <= -1.4) {
		tmp = t_0;
	} else if (y <= 3.4e-84) {
		tmp = (Math.expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((math.expm1((x * 1.0)) * y))
	tmp = 0
	if y <= -1.4:
		tmp = t_0
	elif y <= 3.4e-84:
		tmp = (math.expm1(x) * c) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(expm1(Float64(x * 1.0)) * y)))
	tmp = 0.0
	if (y <= -1.4)
		tmp = t_0;
	elseif (y <= 3.4e-84)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4], t$95$0, If[LessEqual[y, 3.4e-84], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999 or 3.40000000000000021e-84 < y

   1. Initial program 36.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.6

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

   3. Applied rewrites 98.6%

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

   if -1.3999999999999999 < y < 3.40000000000000021e-84

   1. Initial program 45.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 88.6

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

   3. Applied rewrites 88.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 99.5%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 99.2

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

   9. Applied rewrites 99.2%

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

Alternative 2: 91.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := \mathsf{expm1}\left(x\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-281}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-32}:\\ \;\;\;\;c \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\log t\_1 \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* (expm1 x) y)))
   (if (<= t_0 -5e-281)
     (* (* (expm1 x) c) y)
     (if (<= t_0 0.0)
       (* c (log1p (* x y)))
       (if (<= t_0 5e-32) (* c t_1) (* (log t_1) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double t_1 = expm1(x) * y;
	double tmp;
	if (t_0 <= -5e-281) {
		tmp = (expm1(x) * c) * y;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 5e-32) {
		tmp = c * t_1;
	} else {
		tmp = log(t_1) * c;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = (Math.pow(Math.E, x) - 1.0) * y;
	double t_1 = Math.expm1(x) * y;
	double tmp;
	if (t_0 <= -5e-281) {
		tmp = (Math.expm1(x) * c) * y;
	} else if (t_0 <= 0.0) {
		tmp = c * Math.log1p((x * y));
	} else if (t_0 <= 5e-32) {
		tmp = c * t_1;
	} else {
		tmp = Math.log(t_1) * c;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = (math.pow(math.e, x) - 1.0) * y
	t_1 = math.expm1(x) * y
	tmp = 0
	if t_0 <= -5e-281:
		tmp = (math.expm1(x) * c) * y
	elif t_0 <= 0.0:
		tmp = c * math.log1p((x * y))
	elif t_0 <= 5e-32:
		tmp = c * t_1
	else:
		tmp = math.log(t_1) * c
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	t_1 = Float64(expm1(x) * y)
	tmp = 0.0
	if (t_0 <= -5e-281)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 5e-32)
		tmp = Float64(c * t_1);
	else
		tmp = Float64(log(t_1) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-281], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-32], N[(c * t$95$1), $MachinePrecision], N[(N[Log[t$95$1], $MachinePrecision] * c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -4.9999999999999998e-281

   1. Initial program 28.6%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 97.5

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

   9. Applied rewrites 97.5%

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

   if -4.9999999999999998e-281 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -0.0

   1. Initial program 35.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 91.0

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

   3. Applied rewrites 91.0%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 90.1%

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

   if -0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 5e-32

   1. Initial program 31.0%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 99.9

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

   3. Applied rewrites 99.9%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 99.9

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 99.9

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

   6. Applied rewrites 99.9%

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

   if 5e-32 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 88.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 96.3

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in y around inf

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

   6. Applied rewrites 86.4%

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

Alternative 3: 89.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -250:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-12}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -250.0) t_0 (if (<= y 8.2e-12) (* (* (expm1 x) c) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -250.0) {
		tmp = t_0;
	} else if (y <= 8.2e-12) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -250.0) {
		tmp = t_0;
	} else if (y <= 8.2e-12) {
		tmp = (Math.expm1(x) * c) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -250.0:
		tmp = t_0
	elif y <= 8.2e-12:
		tmp = (math.expm1(x) * c) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -250.0)
		tmp = t_0;
	elseif (y <= 8.2e-12)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -250.0], t$95$0, If[LessEqual[y, 8.2e-12], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -250 or 8.19999999999999979e-12 < y

   1. Initial program 37.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 98.9

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 75.8%

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

   if -250 < y < 8.19999999999999979e-12

   1. Initial program 43.7%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 89.6

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

Alternative 4: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -4 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+217}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma y x 1.0)) c)))
   (if (<= y -4e+113)
     t_0
     (if (<= y 3.4e-84)
       (* (* (expm1 x) c) y)
       (if (<= y 1.9e+217) (* c (* (expm1 x) y)) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -4e+113) {
		tmp = t_0;
	} else if (y <= 3.4e-84) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 1.9e+217) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -4e+113)
		tmp = t_0;
	elseif (y <= 3.4e-84)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 1.9e+217)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -4e+113], t$95$0, If[LessEqual[y, 3.4e-84], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.9e+217], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e113 or 1.90000000000000001e217 < y

   1. Initial program 42.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 45.6

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

   4. Applied rewrites 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 45.6

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 45.6

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

   6. Applied rewrites 45.6%

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

   if -4e113 < y < 3.40000000000000021e-84

   1. Initial program 46.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 90.5

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

   3. Applied rewrites 90.5%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 91.6%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 91.3

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

   9. Applied rewrites 91.3%

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

   if 3.40000000000000021e-84 < y < 1.90000000000000001e217

   1. Initial program 24.4%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 83.1

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 83.1

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

   6. Applied rewrites 83.1%

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

Alternative 5: 81.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -4 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+137}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma y x 1.0)) c)))
   (if (<= y -4e+113)
     t_0
     (if (<= y 1.7e+137) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -4e+113) {
		tmp = t_0;
	} else if (y <= 1.7e+137) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -4e+113)
		tmp = t_0;
	elseif (y <= 1.7e+137)
		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[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -4e+113], t$95$0, If[LessEqual[y, 1.7e+137], 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 < -4e113 or 1.69999999999999993e137 < y

   1. Initial program 36.9%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 46.2

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

   4. Applied rewrites 46.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 46.2

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 46.2

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

   6. Applied rewrites 46.2%

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

   if -4e113 < y < 1.69999999999999993e137

   1. Initial program 42.3%

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

   2. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 90.0

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

   4. Applied rewrites 90.0%

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

Alternative 6: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+217}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* y x)))))
   (if (<= y -7.5e+141)
     t_0
     (if (<= y 3.4e-84)
       (* (* (expm1 x) c) y)
       (if (<= y 2.05e+217) (* c (* (expm1 x) y)) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((y * x));
	double tmp;
	if (y <= -7.5e+141) {
		tmp = t_0;
	} else if (y <= 3.4e-84) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 2.05e+217) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log((y * x));
	double tmp;
	if (y <= -7.5e+141) {
		tmp = t_0;
	} else if (y <= 3.4e-84) {
		tmp = (Math.expm1(x) * c) * y;
	} else if (y <= 2.05e+217) {
		tmp = c * (Math.expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((y * x))
	tmp = 0
	if y <= -7.5e+141:
		tmp = t_0
	elif y <= 3.4e-84:
		tmp = (math.expm1(x) * c) * y
	elif y <= 2.05e+217:
		tmp = c * (math.expm1(x) * y)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(y * x)))
	tmp = 0.0
	if (y <= -7.5e+141)
		tmp = t_0;
	elseif (y <= 3.4e-84)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 2.05e+217)
		tmp = Float64(c * Float64(expm1(x) * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+141], t$95$0, If[LessEqual[y, 3.4e-84], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.05e+217], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999937e141 or 2.0500000000000001e217 < y

   1. Initial program 40.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 47.8

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 42.2

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

   7. Applied rewrites 42.2%

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

   if -7.49999999999999937e141 < y < 3.40000000000000021e-84

   1. Initial program 46.6%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 90.9

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

   3. Applied rewrites 90.9%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 89.5%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 89.3

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

   9. Applied rewrites 89.3%

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

   if 3.40000000000000021e-84 < y < 2.0500000000000001e217

   1. Initial program 24.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 97.5

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

   3. Applied rewrites 97.5%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lower-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. log-E N/A

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

      9. pow-to-exp N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-expm1.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 83.1

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

      17. lift-*.f64 N/A

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

      18. *-rgt-identity 83.1

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

   6. Applied rewrites 83.1%

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

Alternative 7: 80.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 10^{+17}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+217}:\\ \;\;\;\;\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* y x)))))
   (if (<= y -7.5e+141)
     t_0
     (if (<= y 1e+17)
       (* (* (expm1 x) c) y)
       (if (<= y 2.05e+217) (* (* y x) c) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((y * x));
	double tmp;
	if (y <= -7.5e+141) {
		tmp = t_0;
	} else if (y <= 1e+17) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 2.05e+217) {
		tmp = (y * x) * c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log((y * x));
	double tmp;
	if (y <= -7.5e+141) {
		tmp = t_0;
	} else if (y <= 1e+17) {
		tmp = (Math.expm1(x) * c) * y;
	} else if (y <= 2.05e+217) {
		tmp = (y * x) * c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((y * x))
	tmp = 0
	if y <= -7.5e+141:
		tmp = t_0
	elif y <= 1e+17:
		tmp = (math.expm1(x) * c) * y
	elif y <= 2.05e+217:
		tmp = (y * x) * c
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(y * x)))
	tmp = 0.0
	if (y <= -7.5e+141)
		tmp = t_0;
	elseif (y <= 1e+17)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 2.05e+217)
		tmp = Float64(Float64(y * x) * c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+141], t$95$0, If[LessEqual[y, 1e+17], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.05e+217], N[(N[(y * x), $MachinePrecision] * c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.49999999999999937e141 or 2.0500000000000001e217 < y

   1. Initial program 40.7%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 47.8

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 42.2

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

   7. Applied rewrites 42.2%

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

   if -7.49999999999999937e141 < y < 1e17

   1. Initial program 44.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 91.7

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

   3. Applied rewrites 91.7%

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

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(c \cdot \left(e^{x} - 1\right) + y \cdot \left(\frac{-1}{2} \cdot \left(c \cdot {\left(e^{x} - 1\right)}^{2}\right) + \frac{1}{3} \cdot \left(c \cdot \left(y \cdot {\left(e^{x} - 1\right)}^{3}\right)\right)\right)\right)} \]

   6. Applied rewrites 90.6%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 90.2

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

   9. Applied rewrites 90.2%

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

   if 1e17 < y < 2.0500000000000001e217

   1. Initial program 18.6%

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

   2. Taylor expanded in x around 0

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

      1. log-E N/A

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

      2. metadata-eval N/A

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

      3. log-E N/A

         \[\leadsto c \cdot \left(x \cdot \left(y \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \]

      4. associate-*r* N/A

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

      5. log-E N/A

         \[\leadsto c \cdot \left(\left(x \cdot y\right) \cdot {1}^{2}\right) \]

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 73.3

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

   4. Applied rewrites 73.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 73.3

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.3

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

   6. Applied rewrites 73.3%

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

Alternative 8: 64.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(y \cdot x\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+216}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* y x)))))
   (if (<= y -8e+125) t_0 (if (<= y 2.3e+216) (* (* y c) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((y * x));
	double tmp;
	if (y <= -8e+125) {
		tmp = t_0;
	} else if (y <= 2.3e+216) {
		tmp = (y * c) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = c * log((y * x))
    if (y <= (-8d+125)) then
        tmp = t_0
    else if (y <= 2.3d+216) then
        tmp = (y * c) * x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log((y * x));
	double tmp;
	if (y <= -8e+125) {
		tmp = t_0;
	} else if (y <= 2.3e+216) {
		tmp = (y * c) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((y * x))
	tmp = 0
	if y <= -8e+125:
		tmp = t_0
	elif y <= 2.3e+216:
		tmp = (y * c) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(y * x)))
	tmp = 0.0
	if (y <= -8e+125)
		tmp = t_0;
	elseif (y <= 2.3e+216)
		tmp = Float64(Float64(y * c) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = c * log((y * x));
	tmp = 0.0;
	if (y <= -8e+125)
		tmp = t_0;
	elseif (y <= 2.3e+216)
		tmp = (y * c) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(y * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e+125], t$95$0, If[LessEqual[y, 2.3e+216], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.9999999999999994e125 or 2.29999999999999996e216 < y

   1. Initial program 41.1%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. log-E N/A

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

      3. metadata-eval N/A

         \[\leadsto c \cdot \log \left(x \cdot \left(y \cdot {1}^{2}\right) + 1\right) \]

      4. log-E N/A

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

      5. associate-*r* N/A

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

      6. log-E N/A

         \[\leadsto c \cdot \log \left(\left(x \cdot y\right) \cdot {1}^{2} + 1\right) \]

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 46.5

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

   4. Applied rewrites 46.5%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 40.4

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

   7. Applied rewrites 40.4%

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

   if -7.9999999999999994e125 < y < 2.29999999999999996e216

   1. Initial program 41.2%

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

   2. Taylor expanded in x around 0

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

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

   4. Applied rewrites 64.4%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.7

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

   7. Applied rewrites 69.7%

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

Alternative 9: 63.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 5e-26) (* (* y c) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 5e-26) {
		tmp = (y * c) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 5d-26) then
        tmp = (y * c) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 5e-26) {
		tmp = (y * c) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 5e-26:
		tmp = (y * c) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 5e-26)
		tmp = Float64(Float64(y * c) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 5e-26)
		tmp = (y * c) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 5e-26], N[(N[(y * c), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.00000000000000019e-26

   1. Initial program 48.2%

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

   2. Taylor expanded in x around 0

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

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.1

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

   7. Applied rewrites 64.1%

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

   if 5.00000000000000019e-26 < c

   1. Initial program 22.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-E.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-log1p.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow-to-exp N/A

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

      12. log-E N/A

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

      13. *-commutative N/A

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

      14. lower-expm1.f64 N/A

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

      15. lower-*.f64 88.7

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

   3. Applied rewrites 88.7%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 88.7

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

      3. lower-expm1.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. log-E N/A

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

      7. pow-to-exp N/A

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

      8. flip-- N/A

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

      9. pow-to-exp N/A

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

      10. log-E N/A

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

      11. *-commutative N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.0%

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

Alternative 10: 61.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

2. Taylor expanded in x around 0

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

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

4. Applied rewrites 54.9%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 61.9

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

7. Applied rewrites 61.9%

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

Developer Target 1: 93.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025117 
(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)))))