Logarithmic Transform

Percentage Accurate: 41.8% → 99.3%

Time: 4.6s

Alternatives: 10

Speedup: 4.9×

• could not determine a ground truth

  1. c = 2.4043972052377044e+248
  2. x = 4.3104973207310754e+80
  3. y = 3.9668402211218013e-26

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: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{if}\;y \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* (expm1 x) y)) c)))
   (if (<= y -1e-19) t_0 (if (<= y 2e-33) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((expm1(x) * y)) * c;
	double tmp;
	if (y <= -1e-19) {
		tmp = t_0;
	} else if (y <= 2e-33) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = Math.log1p((Math.expm1(x) * y)) * c;
	double tmp;
	if (y <= -1e-19) {
		tmp = t_0;
	} else if (y <= 2e-33) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log1p((math.expm1(x) * y)) * c
	tmp = 0
	if y <= -1e-19:
		tmp = t_0
	elif y <= 2e-33:
		tmp = (c * y) * math.expm1((x * 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(expm1(x) * y)) * c)
	tmp = 0.0
	if (y <= -1e-19)
		tmp = t_0;
	elseif (y <= 2e-33)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1e-19], t$95$0, If[LessEqual[y, 2e-33], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999998e-20 or 2.0000000000000001e-33 < y

   1. Initial program 37.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. 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 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-expm1.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. log-E N/A

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

      14. pow-to-exp N/A

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

      15. lower-log1p.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 59.9%

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

      1. lift-log.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower-log1p.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-expm1.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -9.9999999999999998e-20 < y < 2.0000000000000001e-33

   1. Initial program 45.5%

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

   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 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 2: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ t_1 := \mathsf{expm1}\left(x\right) \cdot y\\ t_2 := c \cdot t\_1\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-300}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\log t\_1 \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (- (pow E x) 1.0) y)) (t_1 (* (expm1 x) y)) (t_2 (* c t_1)))
   (if (<= t_0 -4e-300)
     t_2
     (if (<= t_0 0.0)
       (* c (log1p (* x y)))
       (if (<= t_0 0.005) t_2 (* (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 t_2 = c * t_1;
	double tmp;
	if (t_0 <= -4e-300) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 0.005) {
		tmp = t_2;
	} 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 t_2 = c * t_1;
	double tmp;
	if (t_0 <= -4e-300) {
		tmp = t_2;
	} else if (t_0 <= 0.0) {
		tmp = c * Math.log1p((x * y));
	} else if (t_0 <= 0.005) {
		tmp = t_2;
	} 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
	t_2 = c * t_1
	tmp = 0
	if t_0 <= -4e-300:
		tmp = t_2
	elif t_0 <= 0.0:
		tmp = c * math.log1p((x * y))
	elif t_0 <= 0.005:
		tmp = t_2
	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)
	t_2 = Float64(c * t_1)
	tmp = 0.0
	if (t_0 <= -4e-300)
		tmp = t_2;
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 0.005)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(c * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-300], t$95$2, If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.005], t$95$2, N[(N[Log[t$95$1], $MachinePrecision] * c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -4.0000000000000001e-300 or 0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0050000000000000001

   1. Initial program 28.8%

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

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

      2. lower-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 98.2

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 98.2

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

   6. Applied rewrites 98.2%

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

   if -4.0000000000000001e-300 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 0.0

   1. Initial program 36.8%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 91.1

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

   3. Applied rewrites 91.1%

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

   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 90.8%

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

   if 0.0050000000000000001 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 28.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 1.5

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

   4. Applied rewrites 1.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 1.5

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 1.5

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

   6. Applied rewrites 1.5%

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

Alternative 3: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -46000000000:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(x\right), y, 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 0.41:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -46000000000.0)
   (* (log (fma (expm1 x) y 1.0)) c)
   (if (<= y 0.41) (* (* c y) (expm1 (* x 1.0))) (* c (log1p (* x y))))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -46000000000.0) {
		tmp = log(fma(expm1(x), y, 1.0)) * c;
	} else if (y <= 0.41) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = c * log1p((x * y));
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -46000000000.0)
		tmp = Float64(log(fma(expm1(x), y, 1.0)) * c);
	elseif (y <= 0.41)
		tmp = Float64(Float64(c * y) * expm1(Float64(x * 1.0)));
	else
		tmp = Float64(c * log1p(Float64(x * y)));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -46000000000.0], N[(N[Log[N[(N[(Exp[x] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.41], N[(N[(c * y), $MachinePrecision] * N[(Exp[N[(x * 1.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e10

   1. Initial program 50.4%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.6

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

   3. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-expm1.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. *-commutative N/A

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

      13. log-E N/A

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

      14. pow-to-exp N/A

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

      15. lower-log1p.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 70.5%

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

   if -4.6e10 < y < 0.409999999999999976

   1. Initial program 44.9%

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

   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 98.3

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

   4. Applied rewrites 98.3%

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

   if 0.409999999999999976 < y

   1. Initial program 16.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 98.4

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

   3. Applied rewrites 98.4%

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

   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 96.9%

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

Alternative 4: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -58:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.41:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -58.0) t_0 (if (<= y 0.41) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -58.0) {
		tmp = t_0;
	} else if (y <= 0.41) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -58.0) {
		tmp = t_0;
	} else if (y <= 0.41) {
		tmp = (c * y) * Math.expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -58.0:
		tmp = t_0
	elif y <= 0.41:
		tmp = (c * y) * math.expm1((x * 1.0))
	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 <= -58.0)
		tmp = t_0;
	elseif (y <= 0.41)
		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[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -58.0], t$95$0, If[LessEqual[y, 0.41], 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 < -58 or 0.409999999999999976 < y

   1. Initial program 37.3%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 99.1

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

   3. Applied rewrites 99.1%

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

   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 76.0%

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

   if -58 < y < 0.409999999999999976

   1. Initial program 44.7%

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

   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 98.7

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

   4. Applied rewrites 98.7%

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

Alternative 5: 80.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;\left(c \cdot y\right) \cdot \mathsf{expm1}\left(x \cdot 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -3.8e+79)
     t_0
     (if (<= y 3.2e+196) (* (* c y) (expm1 (* x 1.0))) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -3.8e+79) {
		tmp = t_0;
	} else if (y <= 3.2e+196) {
		tmp = (c * y) * expm1((x * 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -3.8e+79)
		tmp = t_0;
	elseif (y <= 3.2e+196)
		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[(c * N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e+79], t$95$0, If[LessEqual[y, 3.2e+196], 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 < -3.8000000000000002e79 or 3.19999999999999993e196 < y

   1. Initial program 41.2%

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

      1. lift-E.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 41.2

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

   3. Applied rewrites 41.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. pow-to-exp N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 44.1

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

   6. Applied rewrites 44.1%

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

   if -3.8000000000000002e79 < y < 3.19999999999999993e196

   1. Initial program 42.0%

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

   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 90.7

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

   4. Applied rewrites 90.7%

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

Alternative 6: 77.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(\mathsf{fma}\left(y, x, 1\right)\right)\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (fma y x 1.0)))))
   (if (<= y -1.1e+158) t_0 (if (<= y 3.2e+196) (* c (* (expm1 x) y)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log(fma(y, x, 1.0));
	double tmp;
	if (y <= -1.1e+158) {
		tmp = t_0;
	} else if (y <= 3.2e+196) {
		tmp = c * (expm1(x) * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(c * log(fma(y, x, 1.0)))
	tmp = 0.0
	if (y <= -1.1e+158)
		tmp = t_0;
	elseif (y <= 3.2e+196)
		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 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1e+158], t$95$0, If[LessEqual[y, 3.2e+196], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1000000000000001e158 or 3.19999999999999993e196 < y

   1. Initial program 38.0%

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

      1. lift-E.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-*.f64 38.0

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

   3. Applied rewrites 38.0%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-commutative N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. log-E N/A

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

      11. pow-to-exp N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 50.7

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

   6. Applied rewrites 50.7%

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

   if -1.1000000000000001e158 < y < 3.19999999999999993e196

   1. Initial program 42.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 92.7

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

   3. Applied rewrites 92.7%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 82.2

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 82.2

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

   6. Applied rewrites 82.2%

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

Alternative 7: 74.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-58}:\\ \;\;\;\;c \cdot \left(\mathsf{expm1}\left(x\right) \cdot y\right)\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-77}:\\ \;\;\;\;c \cdot \log \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= x -2.7e-58)
   (* c (* (expm1 x) y))
   (if (<= x -4.5e-77) (* c (log (* x y))) (* (* c y) x))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.7e-58) {
		tmp = c * (expm1(x) * y);
	} else if (x <= -4.5e-77) {
		tmp = c * log((x * y));
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -2.7e-58) {
		tmp = c * (Math.expm1(x) * y);
	} else if (x <= -4.5e-77) {
		tmp = c * Math.log((x * y));
	} else {
		tmp = (c * y) * x;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -2.7e-58:
		tmp = c * (math.expm1(x) * y)
	elif x <= -4.5e-77:
		tmp = c * math.log((x * y))
	else:
		tmp = (c * y) * x
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2.7e-58)
		tmp = Float64(c * Float64(expm1(x) * y));
	elseif (x <= -4.5e-77)
		tmp = Float64(c * log(Float64(x * y)));
	else
		tmp = Float64(Float64(c * y) * x);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2.7e-58], N[(c * N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-77], N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6999999999999999e-58

   1. Initial program 49.5%

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

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

   3. Applied rewrites 99.5%

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

   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

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

      2. lower-expm1.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 67.7

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

      6. lift-*.f64 N/A

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

      7. *-rgt-identity 67.7

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

   6. Applied rewrites 67.7%

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

   if -2.6999999999999999e-58 < x < -4.5000000000000001e-77

   1. Initial program 23.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 25.0

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

   4. Applied rewrites 25.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 25.0%

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

   if -4.5000000000000001e-77 < x

   1. Initial program 37.8%

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

   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 81.0

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

   4. Applied rewrites 81.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 80.4%

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

Alternative 8: 65.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \log \left(x \cdot y\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+196}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log (* x y)))))
   (if (<= y -3.6e+158) t_0 (if (<= y 3.2e+196) (* (* c y) x) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log((x * y));
	double tmp;
	if (y <= -3.6e+158) {
		tmp = t_0;
	} else if (y <= 3.2e+196) {
		tmp = (c * y) * 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((x * y))
    if (y <= (-3.6d+158)) then
        tmp = t_0
    else if (y <= 3.2d+196) then
        tmp = (c * y) * 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((x * y));
	double tmp;
	if (y <= -3.6e+158) {
		tmp = t_0;
	} else if (y <= 3.2e+196) {
		tmp = (c * y) * x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log((x * y))
	tmp = 0
	if y <= -3.6e+158:
		tmp = t_0
	elif y <= 3.2e+196:
		tmp = (c * y) * x
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log(Float64(x * y)))
	tmp = 0.0
	if (y <= -3.6e+158)
		tmp = t_0;
	elseif (y <= 3.2e+196)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	t_0 = c * log((x * y));
	tmp = 0.0;
	if (y <= -3.6e+158)
		tmp = t_0;
	elseif (y <= 3.2e+196)
		tmp = (c * y) * x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.6e+158], t$95$0, If[LessEqual[y, 3.2e+196], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999988e158 or 3.19999999999999993e196 < y

   1. Initial program 37.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-to-exp N/A

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

      4. log-E N/A

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

      5. *-commutative N/A

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

      6. lower-expm1.f64 N/A

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

      7. lower-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 44.4%

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

   if -3.59999999999999988e158 < y < 3.19999999999999993e196

   1. Initial program 42.5%

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

   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 86.8

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

   4. Applied rewrites 86.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 68.7%

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

Alternative 9: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.9 \cdot 10^{+212}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(c \cdot x\right) \cdot \left(y \cdot 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.9e+212) (* (* c y) x) (* (* c x) (* y 1.0))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.9e+212) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	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) :: tmp
    if (c <= 1.9d+212) then
        tmp = (c * y) * x
    else
        tmp = (c * x) * (y * 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.9e+212) {
		tmp = (c * y) * x;
	} else {
		tmp = (c * x) * (y * 1.0);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.9e+212:
		tmp = (c * y) * x
	else:
		tmp = (c * x) * (y * 1.0)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.9e+212)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(c * x) * Float64(y * 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1.9e+212)
		tmp = (c * y) * x;
	else
		tmp = (c * x) * (y * 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.9e+212], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(c * x), $MachinePrecision] * N[(y * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.89999999999999994e212

   1. Initial program 44.0%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.3%

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

   if 1.89999999999999994e212 < c

   1. Initial program 15.0%

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

   2. Taylor expanded in x around 0

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

      1. associate-*r* N/A

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

      2. log-E N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 59.8

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

   4. Applied rewrites 59.8%

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

Alternative 10: 60.9% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

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 76.3

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

4. Applied rewrites 76.3%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 60.9%

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

Developer Target 1: 93.7% 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 2025122 
(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)))))