Logarithmic Transform

Percentage Accurate: 41.0% → 99.1%

Time: 6.1s

Alternatives: 8

Speedup: 19.8×

• could not determine a ground truth

  1. c = -1.0352161513244161e-47
  2. x = 1.2929160872686249e+76
  3. y = 1.0675825680400917e+293

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: 41.0% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -4e-16)
   (* (log1p (* (expm1 x) y)) c)
   (if (<= y 1.0)
     (* (expm1 x) (* y c))
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -4e-16) {
		tmp = log1p((expm1(x) * y)) * c;
	} else if (y <= 1.0) {
		tmp = expm1(x) * (y * c);
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -4e-16)
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	elseif (y <= 1.0)
		tmp = Float64(expm1(x) * Float64(y * c));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -4e-16], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(Exp[x] - 1), $MachinePrecision] * N[(y * c), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999999e-16

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

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

   if -3.9999999999999999e-16 < y < 1

   1. Initial program 32.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. log-E N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if 1 < y

   1. Initial program 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)} \cdot y\right) \cdot c \]

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.32e-14)
   (* (log1p (* x y)) c)
   (if (<= y 1.0)
     (* (expm1 x) (* y c))
     (*
      (log1p
       (*
        (*
         (fma
          (fma (fma 0.041666666666666664 x 0.16666666666666666) x 0.5)
          x
          1.0)
         x)
        y))
      c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.32e-14) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 1.0) {
		tmp = expm1(x) * (y * c);
	} else {
		tmp = log1p(((fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.32e-14)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 1.0)
		tmp = Float64(expm1(x) * Float64(y * c));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(fma(0.041666666666666664, x, 0.16666666666666666), x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.32e-14], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(Exp[x] - 1), $MachinePrecision] * N[(y * c), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(N[(0.041666666666666664 * x + 0.16666666666666666), $MachinePrecision] * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32e-14

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.3%

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

   if -1.32e-14 < y < 1

   1. Initial program 32.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. log-E N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if 1 < y

   1. Initial program 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot x\right)\right)\right)\right)} \cdot y\right) \cdot c \]

   7. Applied rewrites 99.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \cdot c \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x, 0.16666666666666666\right), x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.32e-14)
   (* (log1p (* x y)) c)
   (if (<= y 1.0)
     (* (expm1 x) (* y c))
     (* (log1p (* (* (fma (fma 0.16666666666666666 x 0.5) x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.32e-14) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 1.0) {
		tmp = expm1(x) * (y * c);
	} else {
		tmp = log1p(((fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.32e-14)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 1.0)
		tmp = Float64(expm1(x) * Float64(y * c));
	else
		tmp = Float64(log1p(Float64(Float64(fma(fma(0.16666666666666666, x, 0.5), x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.32e-14], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(Exp[x] - 1), $MachinePrecision] * N[(y * c), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32e-14

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.3%

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

   if -1.32e-14 < y < 1

   1. Initial program 32.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. log-E N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if 1 < y

   1. Initial program 30.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(x \cdot \left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right)\right)} \cdot y\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto c \cdot \log \left(1 + \left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \]

      2. lower-*.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \left(\left(\log \mathsf{E}\left(\right) + x \cdot \left(\frac{1}{6} \cdot \left(x \cdot {\log \mathsf{E}\left(\right)}^{3}\right) + \frac{1}{2} \cdot {\log \mathsf{E}\left(\right)}^{2}\right)\right) \cdot \color{blue}{x}\right) \cdot y\right) \]

   5. Applied rewrites 55.7%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, 1, 0.5\right), x, 1\right) \cdot x\right)} \cdot y\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 55.7

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

   7. Applied rewrites 99.4%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -1.32e-14)
   (* (log1p (* x y)) c)
   (if (<= y 1.0)
     (* (expm1 x) (* y c))
     (* (log1p (* (* (fma 0.5 x 1.0) x) y)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -1.32e-14) {
		tmp = log1p((x * y)) * c;
	} else if (y <= 1.0) {
		tmp = expm1(x) * (y * c);
	} else {
		tmp = log1p(((fma(0.5, x, 1.0) * x) * y)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -1.32e-14)
		tmp = Float64(log1p(Float64(x * y)) * c);
	elseif (y <= 1.0)
		tmp = Float64(expm1(x) * Float64(y * c));
	else
		tmp = Float64(log1p(Float64(Float64(fma(0.5, x, 1.0) * x) * y)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -1.32e-14], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(Exp[x] - 1), $MachinePrecision] * N[(y * c), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.32e-14

   1. Initial program 48.6%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 62.3%

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

   if -1.32e-14 < y < 1

   1. Initial program 32.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. log-E N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 98.0%

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

   if 1 < y

   1. Initial program 30.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 98.9

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

   7. Applied rewrites 98.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right) \cdot y\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -1.32e-14) (not (<= y 1.0)))
   (* (log1p (* x y)) c)
   (* (expm1 x) (* y c))))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.32e-14) || !(y <= 1.0)) {
		tmp = log1p((x * y)) * c;
	} else {
		tmp = expm1(x) * (y * c);
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -1.32e-14) || !(y <= 1.0)) {
		tmp = Math.log1p((x * y)) * c;
	} else {
		tmp = Math.expm1(x) * (y * c);
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -1.32e-14) or not (y <= 1.0):
		tmp = math.log1p((x * y)) * c
	else:
		tmp = math.expm1(x) * (y * c)
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -1.32e-14) || !(y <= 1.0))
		tmp = Float64(log1p(Float64(x * y)) * c);
	else
		tmp = Float64(expm1(x) * Float64(y * c));
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -1.32e-14], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(Exp[x] - 1), $MachinePrecision] * N[(y * c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.32e-14 or 1 < y

   1. Initial program 41.3%

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

      1. lift-*.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 76.6%

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

   if -1.32e-14 < y < 1

   1. Initial program 32.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow-to-exp N/A

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

      5. log-E N/A

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

      6. *-commutative N/A

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

      7. lower-expm1.f64 N/A

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

      8. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-expm1.f64 N/A

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

      5. *-commutative N/A

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

      6. pow-to-exp N/A

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

      7. log-E N/A

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

      8. *-commutative N/A

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

      9. *-rgt-identity N/A

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

      10. lower-expm1.f64 N/A

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

      11. *-rgt-identity N/A

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

      12. lift-expm1.f64 N/A

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

      13. *-commutative N/A

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

   7. Applied rewrites 98.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.32 \cdot 10^{-14} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\mathsf{log1p}\left(x \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right) \cdot \left(y \cdot c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow-to-exp N/A

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

   5. log-E N/A

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

   6. *-commutative N/A

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

   7. lower-expm1.f64 N/A

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

   8. lower-*.f64 77.7

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

5. Applied rewrites 77.7%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-expm1.f64 N/A

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

   5. *-commutative N/A

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

   6. pow-to-exp N/A

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

   7. log-E N/A

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

   8. *-commutative N/A

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

   9. *-rgt-identity N/A

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

   10. lower-expm1.f64 N/A

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

   11. *-rgt-identity N/A

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

   12. lift-expm1.f64 N/A

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

   13. *-commutative N/A

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

7. Applied rewrites 77.7%

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

8. Final simplification 77.7%

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

Alternative 7: 61.4% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. pow-to-exp N/A

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

   5. log-E N/A

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

   6. *-commutative N/A

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

   7. lower-expm1.f64 N/A

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

   8. lower-*.f64 77.7

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

5. Applied rewrites 77.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 61.3%

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

Alternative 8: 58.2% accurate, 19.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.0%

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

3. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. log-E N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 57.8

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

5. Applied rewrites 57.8%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 57.8

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

7. Applied rewrites 57.8%

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

Developer Target 1: 93.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025085 
(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)))))