Logarithmic Transform

Percentage Accurate: 41.7% → 88.8%

Time: 21.4s

Alternatives: 9

Speedup: 5.0×

• unsoundness detected in the egraph

• could not determine a ground truth

  1. c = -3.137197119719614e-254
  2. x = 7.437649481379717e+205
  3. y = 2.4814580720731816e+153

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.7% accurate, 1.0× speedup

Math

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

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: 88.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+46}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \mathbf{elif}\;y \leq 0.03:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+228}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right), 1\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= y -8.2e+46)
  (* (log (fma y (expm1 x) 1.0)) c)
  (if (<= y 0.03)
    (* (* (expm1 x) c) y)
    (if (<= y 2.7e+228)
      (* c (* x y))
      (*
       (log
        (fma
         y
         (*
          x
          (+
           1.0
           (*
            x
            (+
             0.5
             (*
              x
              (+ 0.16666666666666666 (* 0.041666666666666664 x)))))))
         1.0))
       c)))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -8.2e+46) {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	} else if (y <= 0.03) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 2.7e+228) {
		tmp = c * (x * y);
	} else {
		tmp = log(fma(y, (x * (1.0 + (x * (0.5 + (x * (0.16666666666666666 + (0.041666666666666664 * x))))))), 1.0)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -8.2e+46)
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	elseif (y <= 0.03)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 2.7e+228)
		tmp = Float64(c * Float64(x * y));
	else
		tmp = Float64(log(fma(y, Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(0.16666666666666666 + Float64(0.041666666666666664 * x))))))), 1.0)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -8.2e+46], N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 0.03], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.7e+228], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(y * N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * N[(0.16666666666666666 + N[(0.041666666666666664 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.2e46

   1. Initial program 41.7%

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

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

      3. lower-*.f64 41.7%

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

   3. Applied rewrites 51.7%

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

   if -8.2e46 < y < 0.029999999999999999

   1. Initial program 41.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.7%

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.7%

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

   6. Applied rewrites 76.7%

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

   if 0.029999999999999999 < y < 2.7000000000000002e228

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

   if 2.7000000000000002e228 < y

   1. Initial program 41.7%

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

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

      3. lower-*.f64 41.7%

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

   3. Applied rewrites 51.7%

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

   4. Taylor expanded in x around 0

      \[\leadsto \log \left(\mathsf{fma}\left(y, \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}, 1\right)\right) \cdot c \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)}, 1\right)\right) \cdot c \]

      2. lower-+.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \left(1 + \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right), 1\right)\right) \cdot c \]

      3. lower-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \left(1 + x \cdot \color{blue}{\left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)}\right), 1\right)\right) \cdot c \]

      4. lower-+.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \left(1 + x \cdot \left(\frac{1}{2} + \color{blue}{x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}\right)\right), 1\right)\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{24} \cdot x\right)}\right)\right), 1\right)\right) \cdot c \]

      6. lower-+.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{24} \cdot x}\right)\right)\right), 1\right)\right) \cdot c \]

      7. lower-*.f64 36.7%

         \[\leadsto \log \left(\mathsf{fma}\left(y, x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot \color{blue}{x}\right)\right)\right), 1\right)\right) \cdot c \]

   6. Applied rewrites 36.7%

      \[\leadsto \log \left(\mathsf{fma}\left(y, \color{blue}{x \cdot \left(1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot x\right)\right)\right)}, 1\right)\right) \cdot c \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.03:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+228}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* (log (fma y (expm1 x) 1.0)) c)))
  (if (<= y -8.2e+46)
    t_0
    (if (<= y 0.03)
      (* (* (expm1 x) c) y)
      (if (<= y 3e+228) (* c (* x y)) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, expm1(x), 1.0)) * c;
	double tmp;
	if (y <= -8.2e+46) {
		tmp = t_0;
	} else if (y <= 0.03) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 3e+228) {
		tmp = c * (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, expm1(x), 1.0)) * c)
	tmp = 0.0
	if (y <= -8.2e+46)
		tmp = t_0;
	elseif (y <= 0.03)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 3e+228)
		tmp = Float64(c * Float64(x * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -8.2e+46], t$95$0, If[LessEqual[y, 0.03], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 3e+228], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2e46 or 3.0000000000000001e228 < y

   1. Initial program 41.7%

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

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

      3. lower-*.f64 41.7%

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

   3. Applied rewrites 51.7%

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

   if -8.2e46 < y < 0.029999999999999999

   1. Initial program 41.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.7%

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.7%

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

   6. Applied rewrites 76.7%

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

   if 0.029999999999999999 < y < 3.0000000000000001e228

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

Alternative 3: 82.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 0.03:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+228}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* (log (fma y x 1.0)) c)))
  (if (<= y -4.8e+131)
    t_0
    (if (<= y 0.03)
      (* (* (expm1 x) c) y)
      (if (<= y 3e+228) (* c (* x y)) t_0)))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -4.8e+131) {
		tmp = t_0;
	} else if (y <= 0.03) {
		tmp = (expm1(x) * c) * y;
	} else if (y <= 3e+228) {
		tmp = c * (x * y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -4.8e+131)
		tmp = t_0;
	elseif (y <= 0.03)
		tmp = Float64(Float64(expm1(x) * c) * y);
	elseif (y <= 3e+228)
		tmp = Float64(c * Float64(x * y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -4.8e+131], t$95$0, If[LessEqual[y, 0.03], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 3e+228], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.7999999999999999e131 or 3.0000000000000001e228 < y

   1. Initial program 41.7%

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

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

      3. lower-*.f64 41.7%

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

   3. Applied rewrites 51.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 40.2%

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

   6. Applied rewrites 40.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. remove-double-neg N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 40.2%

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

   8. Applied rewrites 40.2%

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

   if -4.7999999999999999e131 < y < 0.029999999999999999

   1. Initial program 41.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.7%

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.7%

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

   6. Applied rewrites 76.7%

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

   if 0.029999999999999999 < y < 3.0000000000000001e228

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

Alternative 4: 79.6% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq 200000000000:\\ \;\;\;\;\left|c\right| \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot \left|c\right|\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1.0 c)
 (if (<= (fabs c) 200000000000.0)
   (* (fabs c) (* y (expm1 x)))
   (* (* (expm1 x) (fabs c)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 200000000000.0) {
		tmp = fabs(c) * (y * expm1(x));
	} else {
		tmp = (expm1(x) * fabs(c)) * y;
	}
	return copysign(1.0, c) * tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (Math.abs(c) <= 200000000000.0) {
		tmp = Math.abs(c) * (y * Math.expm1(x));
	} else {
		tmp = (Math.expm1(x) * Math.abs(c)) * y;
	}
	return Math.copySign(1.0, c) * tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if math.fabs(c) <= 200000000000.0:
		tmp = math.fabs(c) * (y * math.expm1(x))
	else:
		tmp = (math.expm1(x) * math.fabs(c)) * y
	return math.copysign(1.0, c) * tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (abs(c) <= 200000000000.0)
		tmp = Float64(abs(c) * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(expm1(x) * abs(c)) * y);
	end
	return Float64(copysign(1.0, c) * tmp)
end

Wolfram

code[c_, x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 200000000000.0], N[(N[Abs[c], $MachinePrecision] * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * N[Abs[c], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 2e11

   1. Initial program 41.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.7%

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

   4. Applied rewrites 73.7%

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

   if 2e11 < c

   1. Initial program 41.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.7%

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

   4. Applied rewrites 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 76.7%

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

   6. Applied rewrites 76.7%

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

Alternative 5: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-27}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= x -2e-27) (* c (* y (expm1 x))) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -2e-27) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -2e-27) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -2e-27:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -2e-27)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -2e-27], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-27

   1. Initial program 41.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.7%

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

   4. Applied rewrites 73.7%

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

   if -2.0000000000000001e-27 < x

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.0%

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

   6. Applied rewrites 59.0%

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

Alternative 6: 64.6% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq 10^{-18}:\\ \;\;\;\;\left(y \cdot \left|c\right|\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left|c\right|\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1.0 c)
 (if (<= (fabs c) 1e-18) (* (* y (fabs c)) x) (* (* x (fabs c)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 1e-18) {
		tmp = (y * fabs(c)) * x;
	} else {
		tmp = (x * fabs(c)) * y;
	}
	return copysign(1.0, c) * tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (Math.abs(c) <= 1e-18) {
		tmp = (y * Math.abs(c)) * x;
	} else {
		tmp = (x * Math.abs(c)) * y;
	}
	return Math.copySign(1.0, c) * tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if math.fabs(c) <= 1e-18:
		tmp = (y * math.fabs(c)) * x
	else:
		tmp = (x * math.fabs(c)) * y
	return math.copysign(1.0, c) * tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (abs(c) <= 1e-18)
		tmp = Float64(Float64(y * abs(c)) * x);
	else
		tmp = Float64(Float64(x * abs(c)) * y);
	end
	return Float64(copysign(1.0, c) * tmp)
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (abs(c) <= 1e-18)
		tmp = (y * abs(c)) * x;
	else
		tmp = (x * abs(c)) * y;
	end
	tmp_2 = (sign(c) * abs(1.0)) * tmp;
end

Wolfram

code[c_, x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 1e-18], N[(N[(y * N[Abs[c], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Abs[c], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.0000000000000001e-18

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 61.6%

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

   6. Applied rewrites 61.6%

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

   if 1.0000000000000001e-18 < c

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.0%

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

   6. Applied rewrites 59.0%

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

Alternative 7: 61.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+71}:\\ \;\;\;\;0 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= x -7e+71) (* 0.0 (* x y)) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -7e+71) {
		tmp = 0.0 * (x * y);
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

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 (x <= (-7d+71)) then
        tmp = 0.0d0 * (x * y)
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -7e+71) {
		tmp = 0.0 * (x * y);
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -7e+71:
		tmp = 0.0 * (x * y)
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -7e+71)
		tmp = Float64(0.0 * Float64(x * y));
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -7e+71)
		tmp = 0.0 * (x * y);
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -7e+71], N[(0.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999998e71

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto 0 \cdot \left(\color{blue}{x} \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 30.6%

      \[\leadsto 0 \cdot \left(\color{blue}{x} \cdot y\right) \]

   if -6.9999999999999998e71 < x

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 59.0%

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

   6. Applied rewrites 59.0%

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

Alternative 8: 58.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+71}:\\ \;\;\;\;0 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(x \cdot y\right)\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= x -7e+71) (* 0.0 (* x y)) (* c (* x y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -7e+71) {
		tmp = 0.0 * (x * y);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Fortran

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 (x <= (-7d+71)) then
        tmp = 0.0d0 * (x * y)
    else
        tmp = c * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -7e+71) {
		tmp = 0.0 * (x * y);
	} else {
		tmp = c * (x * y);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -7e+71:
		tmp = 0.0 * (x * y)
	else:
		tmp = c * (x * y)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -7e+71)
		tmp = Float64(0.0 * Float64(x * y));
	else
		tmp = Float64(c * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (x <= -7e+71)
		tmp = 0.0 * (x * y);
	else
		tmp = c * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -7e+71], N[(0.0 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(c * N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.9999999999999998e71

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto 0 \cdot \left(\color{blue}{x} \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 30.6%

      \[\leadsto 0 \cdot \left(\color{blue}{x} \cdot y\right) \]

   if -6.9999999999999998e71 < x

   1. Initial program 41.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.0%

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

   4. Applied rewrites 56.0%

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

Alternative 9: 56.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.7%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 56.0%

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

4. Applied rewrites 56.0%

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

Developer Target 1: 93.4% accurate, 1.4× speedup

Math

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

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 2025322 
(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)))))