Logarithmic Transform

Percentage Accurate: 41.6% → 99.4%

Time: 5.9s

Alternatives: 7

Speedup: 4.9×

• could not determine a ground truth

  1. c = -1.0324420636159316e+77
  2. x = 9.794637172244746e+175
  3. y = 2.4845683072954954e+240

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: 41.6% 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: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right) \cdot c\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log1p (* y (expm1 x))) c)))
   (if (<= y -6.6e-14) t_0 (if (<= y 1.8e-22) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log1p((y * expm1(x))) * c;
	double tmp;
	if (y <= -6.6e-14) {
		tmp = t_0;
	} else if (y <= 1.8e-22) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = Math.log1p((y * Math.expm1(x))) * c;
	double tmp;
	if (y <= -6.6e-14) {
		tmp = t_0;
	} else if (y <= 1.8e-22) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = math.log1p((y * math.expm1(x))) * c
	tmp = 0
	if y <= -6.6e-14:
		tmp = t_0
	elif y <= 1.8e-22:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(log1p(Float64(y * expm1(x))) * c)
	tmp = 0.0
	if (y <= -6.6e-14)
		tmp = t_0;
	elseif (y <= 1.8e-22)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -6.6e-14], t$95$0, If[LessEqual[y, 1.8e-22], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5999999999999996e-14 or 1.7999999999999999e-22 < y

   1. Initial program 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 50.8

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

   5. Applied rewrites 50.8%

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(x\right)}, y, 1\right)\right) \cdot c \]
   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. *-rgt-identity N/A

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

      5. *-commutative N/A

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

      6. pow-exp N/A

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

      7. e-exp-1 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-log1p.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

      16. lift-expm1.f64 93.1

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

   7. Applied rewrites 93.1%

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

   if -6.5999999999999996e-14 < y < 1.7999999999999999e-22

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

Alternative 2: 89.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -21000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.1:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* c (log1p (* x y)))))
   (if (<= y -21000.0) t_0 (if (<= y 1.1) (* (* y c) (expm1 x)) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -21000.0) {
		tmp = t_0;
	} else if (y <= 1.1) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -21000.0) {
		tmp = t_0;
	} else if (y <= 1.1) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -21000.0:
		tmp = t_0
	elif y <= 1.1:
		tmp = (y * c) * math.expm1(x)
	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 <= -21000.0)
		tmp = t_0;
	elseif (y <= 1.1)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -21000.0], t$95$0, If[LessEqual[y, 1.1], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -21000 or 1.1000000000000001 < y

   1. Initial program 41.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. e-exp-1 N/A

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

      12. pow-exp N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-expm1.f64 N/A

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

      16. log-E N/A

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

      17. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative 65.5

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

      2. pow-exp 65.5

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

      3. e-exp-1 65.5

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

   6. Applied rewrites 65.5%

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

   if -21000 < y < 1.1000000000000001

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

Alternative 3: 82.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(x, y, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.23:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+182}:\\ \;\;\;\;c \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (let* ((t_0 (* (log (fma x y 1.0)) c)))
   (if (<= y -7.8e+134)
     t_0
     (if (<= y 0.23)
       (* y (* (expm1 x) c))
       (if (<= y 3.5e+182) (* c (* y x)) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(x, y, 1.0)) * c;
	double tmp;
	if (y <= -7.8e+134) {
		tmp = t_0;
	} else if (y <= 0.23) {
		tmp = y * (expm1(x) * c);
	} else if (y <= 3.5e+182) {
		tmp = c * (y * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(x, y, 1.0)) * c)
	tmp = 0.0
	if (y <= -7.8e+134)
		tmp = t_0;
	elseif (y <= 0.23)
		tmp = Float64(y * Float64(expm1(x) * c));
	elseif (y <= 3.5e+182)
		tmp = Float64(c * Float64(y * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(x * y + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -7.8e+134], t$95$0, If[LessEqual[y, 0.23], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+182], N[(c * N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.79999999999999967e134 or 3.50000000000000023e182 < y

   1. Initial program 41.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   3. Applied rewrites 50.8%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative 39.1

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

      2. pow-exp 39.1

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

      3. e-exp-1 39.1

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

   6. Applied rewrites 39.1%

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

   if -7.79999999999999967e134 < y < 0.23000000000000001

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. pow-exp N/A

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

      8. e-exp-1 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

      16. lift-expm1.f64 77.3

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

   8. Applied rewrites 77.3%

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

   if 0.23000000000000001 < y < 3.50000000000000023e182

   1. Initial program 41.6%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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. e-exp-1 N/A

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

      12. pow-exp N/A

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

      13. *-commutative N/A

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

      14. log-E N/A

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

      15. lower-expm1.f64 N/A

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

      16. log-E N/A

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

      17. lower-*.f64 93.1

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

   3. Applied rewrites 93.1%

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

   4. Taylor expanded in x around 0

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

      1. lift-*.f64 N/A

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

      2. lift-expm1.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lower-expm1.f64 N/A

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

      6. lower-expm1.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. lift-expm1.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. pow-exp N/A

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

      13. e-exp-1 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 56.0

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

   6. Applied rewrites 56.0%

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

Alternative 4: 78.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 0.00055:\\ \;\;\;\;\left(y \cdot c\right) \cdot \mathsf{expm1}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 0.00055) (* (* y c) (expm1 x)) (* y (* (expm1 x) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 0.00055) {
		tmp = (y * c) * expm1(x);
	} else {
		tmp = y * (expm1(x) * c);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 0.00055) {
		tmp = (y * c) * Math.expm1(x);
	} else {
		tmp = y * (Math.expm1(x) * c);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 0.00055:
		tmp = (y * c) * math.expm1(x)
	else:
		tmp = y * (math.expm1(x) * c)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 0.00055)
		tmp = Float64(Float64(y * c) * expm1(x));
	else
		tmp = Float64(y * Float64(expm1(x) * c));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 0.00055], N[(N[(y * c), $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 5.50000000000000033e-4

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

   if 5.50000000000000033e-4 < c

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. pow-exp N/A

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

      8. e-exp-1 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

      16. lift-expm1.f64 77.3

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

   8. Applied rewrites 77.3%

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

Alternative 5: 69.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 4 \cdot 10^{-107}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\mathsf{expm1}\left(x\right) \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 4e-107) (* (* c y) x) (* y (* (expm1 x) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 4e-107) {
		tmp = (c * y) * x;
	} else {
		tmp = y * (expm1(x) * c);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 4e-107) {
		tmp = (c * y) * x;
	} else {
		tmp = y * (Math.expm1(x) * c);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 4e-107:
		tmp = (c * y) * x
	else:
		tmp = y * (math.expm1(x) * c)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 4e-107)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(y * Float64(expm1(x) * c));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 4e-107], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 4e-107

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative 61.6

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

      2. pow-exp 61.6

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

      3. e-exp-1 61.6

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

   7. Applied rewrites 61.6%

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

   if 4e-107 < c

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. pow-exp N/A

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

      8. e-exp-1 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

      16. lift-expm1.f64 77.3

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

   8. Applied rewrites 77.3%

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

Alternative 6: 63.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 1.7 \cdot 10^{-5}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 1.7e-5) (* (* c y) x) (* y (* x c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.7e-5) {
		tmp = (c * y) * x;
	} else {
		tmp = y * (x * c);
	}
	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 <= 1.7d-5) then
        tmp = (c * y) * x
    else
        tmp = y * (x * c)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 1.7e-5) {
		tmp = (c * y) * x;
	} else {
		tmp = y * (x * c);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 1.7e-5:
		tmp = (c * y) * x
	else:
		tmp = y * (x * c)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 1.7e-5)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(y * Float64(x * c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 1.7e-5)
		tmp = (c * y) * x;
	else
		tmp = y * (x * c);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 1.7e-5], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(y * N[(x * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 1.7e-5

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative 61.6

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

      2. pow-exp 61.6

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

      3. e-exp-1 61.6

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

   7. Applied rewrites 61.6%

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

   if 1.7e-5 < c

   1. Initial program 41.6%

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

   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. e-exp-1 N/A

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

      5. pow-exp 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. log-E N/A

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

      8. lower-expm1.f64 N/A

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

      9. log-E N/A

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

      10. lower-*.f64 76.9

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

   4. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.9

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 76.9

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

   6. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-expm1.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. pow-exp N/A

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

      8. e-exp-1 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-commutative N/A

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

      15. *-rgt-identity N/A

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

      16. lift-expm1.f64 77.3

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 59.5

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

   11. Applied rewrites 59.5%

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

Alternative 7: 59.5% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

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 * (x * c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.6%

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

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. e-exp-1 N/A

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

   5. pow-exp 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. log-E N/A

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

   8. lower-expm1.f64 N/A

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

   9. log-E N/A

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

   10. lower-*.f64 76.9

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

4. Applied rewrites 76.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 76.9

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

   4. lift-*.f64 N/A

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

   5. *-rgt-identity 76.9

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

6. Applied rewrites 76.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-expm1.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-rgt-identity N/A

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

   6. *-commutative N/A

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

   7. pow-exp N/A

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

   8. e-exp-1 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. e-exp-1 N/A

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

   13. pow-exp N/A

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

   14. *-commutative N/A

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

   15. *-rgt-identity N/A

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

   16. lift-expm1.f64 77.3

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

8. Applied rewrites 77.3%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 59.5

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

11. Applied rewrites 59.5%

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

Developer Target 1: 93.1% 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 2025134 
(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)))))