Logarithmic Transform

Percentage Accurate: 41.7% → 99.2%

Time: 31.5s

Alternatives: 7

Speedup: 19.8×

• Could not converge on a ground truth

  1. c = -5.587094410358279e+154
  2. x = 8.841219160768542e+114
  3. y = 3.730294375662143e-107

Specification

Math

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

FPCore

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

Initial Program: 41.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-6} \lor \neg \left(y \leq 4.1 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -9.8e-6) (not (<= y 4.1e-14)))
   (* (log1p (* (expm1 x) y)) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -9.8e-6) || !(y <= 4.1e-14)) {
		tmp = log1p((expm1(x) * y)) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -9.8e-6) || !(y <= 4.1e-14)) {
		tmp = Math.log1p((Math.expm1(x) * y)) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -9.8e-6) or not (y <= 4.1e-14):
		tmp = math.log1p((math.expm1(x) * y)) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -9.8e-6) || !(y <= 4.1e-14))
		tmp = Float64(log1p(Float64(expm1(x) * y)) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -9.8e-6], N[Not[LessEqual[y, 4.1e-14]], $MachinePrecision]], N[(N[Log[1 + N[(N[(Exp[x] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.79999999999999934e-6 or 4.1000000000000002e-14 < y

   1. Initial program 37.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 37.2

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 37.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 37.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. *-rgt-identity 99.7

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

   6. Applied rewrites 99.7%

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

   if -9.79999999999999934e-6 < y < 4.1000000000000002e-14

   1. Initial program 47.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.9

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 72.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 90.8

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

   4. Applied rewrites 90.8%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{-6} \lor \neg \left(y \leq 4.1 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(x\right) \cdot y\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, y\right) \cdot x\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -17.0)
   (* (log1p (* y x)) c)
   (if (<= y 1.0)
     (* (* (expm1 x) c) y)
     (* (log1p (* (fma (* y (fma 0.16666666666666666 x 0.5)) x y) x)) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = log1p((y * x)) * c;
	} else if (y <= 1.0) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((fma((y * fma(0.16666666666666666, x, 0.5)), x, y) * x)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(log1p(Float64(y * x)) * c);
	elseif (y <= 1.0)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(fma(Float64(y * fma(0.16666666666666666, x, 0.5)), x, y) * x)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -17.0], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(N[(N[(y * N[(0.16666666666666666 * x + 0.5), $MachinePrecision]), $MachinePrecision] * x + y), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -17

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.9

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 57.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 57.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 52.8%

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

   if -17 < y < 1

   1. Initial program 47.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 72.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.9%

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

   if 1 < y

   1. Initial program 14.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 14.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 14.8

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

Alternative 3: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(y \cdot \left(\mathsf{fma}\left(0.5, x, 1\right) \cdot x\right)\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= y -17.0)
   (* (log1p (* y x)) c)
   (if (<= y 1.0)
     (* (* (expm1 x) c) y)
     (* (log1p (* y (* (fma 0.5 x 1.0) x))) c))))

C

double code(double c, double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = log1p((y * x)) * c;
	} else if (y <= 1.0) {
		tmp = (expm1(x) * c) * y;
	} else {
		tmp = log1p((y * (fma(0.5, x, 1.0) * x))) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(log1p(Float64(y * x)) * c);
	elseif (y <= 1.0)
		tmp = Float64(Float64(expm1(x) * c) * y);
	else
		tmp = Float64(log1p(Float64(y * Float64(fma(0.5, x, 1.0) * x))) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[y, -17.0], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision], N[(N[Log[1 + N[(y * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -17

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.9

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 57.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 57.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 52.8%

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

   if -17 < y < 1

   1. Initial program 47.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 72.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.9%

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

   if 1 < y

   1. Initial program 14.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 14.8

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 14.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 14.8

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 99.8

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

   7. Applied rewrites 99.8%

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

Alternative 4: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17 \lor \neg \left(y \leq 1.05\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (or (<= y -17.0) (not (<= y 1.05)))
   (* (log1p (* y x)) c)
   (* (* (expm1 x) c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if ((y <= -17.0) || !(y <= 1.05)) {
		tmp = log1p((y * x)) * c;
	} else {
		tmp = (expm1(x) * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if ((y <= -17.0) || !(y <= 1.05)) {
		tmp = Math.log1p((y * x)) * c;
	} else {
		tmp = (Math.expm1(x) * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if (y <= -17.0) or not (y <= 1.05):
		tmp = math.log1p((y * x)) * c
	else:
		tmp = (math.expm1(x) * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if ((y <= -17.0) || !(y <= 1.05))
		tmp = Float64(log1p(Float64(y * x)) * c);
	else
		tmp = Float64(Float64(expm1(x) * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[Or[LessEqual[y, -17.0], N[Not[LessEqual[y, 1.05]], $MachinePrecision]], N[(N[Log[1 + N[(y * x), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -17 or 1.05000000000000004 < y

   1. Initial program 36.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.9

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 36.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 36.9

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 75.4%

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

   if -17 < y < 1.05000000000000004

   1. Initial program 47.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.6

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

      4. lift-log.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lower-log1p.f64 72.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 72.2

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

      10. lift--.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow-to-exp N/A

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

      13. lower-expm1.f64 N/A

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

      14. lift-E.f64 N/A

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

      15. log-E N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 91.1

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

   4. Applied rewrites 91.1%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 98.9%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17 \lor \neg \left(y \leq 1.05\right):\\ \;\;\;\;\mathsf{log1p}\left(y \cdot x\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{expm1}\left(x\right) \cdot c\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 44.1

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

   4. lift-log.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lower-log1p.f64 60.6

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 60.6

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

   10. lift--.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow-to-exp N/A

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

   13. lower-expm1.f64 N/A

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

   14. lift-E.f64 N/A

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

   15. log-E N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 93.9

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

4. Applied rewrites 93.9%

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

5. Taylor expanded in y around 0

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

7. Applied rewrites 81.2%

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

Alternative 6: 63.8% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;\left(c \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (c x y)
 :precision binary64
 (if (<= c 2.15e-32) (* (* c y) x) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.15e-32) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(c, x, y)
use fmin_fmax_functions
    real(8), intent (in) :: c
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (c <= 2.15d-32) then
        tmp = (c * y) * x
    else
        tmp = (x * c) * y
    end if
    code = tmp
end function

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (c <= 2.15e-32) {
		tmp = (c * y) * x;
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if c <= 2.15e-32:
		tmp = (c * y) * x
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (c <= 2.15e-32)
		tmp = Float64(Float64(c * y) * x);
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (c <= 2.15e-32)
		tmp = (c * y) * x;
	else
		tmp = (x * c) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[c_, x_, y_] := If[LessEqual[c, 2.15e-32], N[(N[(c * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.14999999999999995e-32

   1. Initial program 53.2%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. log-E N/A

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

      9. lower-*.f64 N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. log-E N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if 2.14999999999999995e-32 < c

   1. Initial program 11.6%

      \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
   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. *-rgt-identity N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. log-E N/A

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

      9. lower-*.f64 N/A

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

      10. log-E N/A

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

      11. metadata-eval N/A

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

      12. log-E N/A

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

      13. log-E N/A

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

      14. metadata-eval N/A

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

      15. *-rgt-identity N/A

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

      16. lower-*.f64 66.3

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

   5. Applied rewrites 66.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 74.9

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

   7. Applied rewrites 74.9%

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

Alternative 7: 62.4% 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 44.1%

   \[c \cdot \log \left(1 + \left({\mathsf{E}\left(\right)}^{x} - 1\right) \cdot y\right) \]
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. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. *-rgt-identity N/A

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

   8. log-E N/A

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

   9. lower-*.f64 N/A

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

   10. log-E N/A

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

   11. metadata-eval N/A

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

   12. log-E N/A

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

   13. log-E N/A

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

   14. metadata-eval N/A

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

   15. *-rgt-identity N/A

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

   16. lower-*.f64 62.9

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

5. Applied rewrites 62.9%

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

Developer Target 1: 93.3% 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 2025026 
(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)))))