Logarithmic Transform

Percentage Accurate: 41.8% → 99.3%

Time: 5.3s

Alternatives: 9

Speedup: 5.0×

• could not determine a ground truth

  1. c = -6.498978509714293e+143
  2. x = 8.793715475836643e+297
  3. y = 7.085334732443707e+71

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-28}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* c (log1p (* y (expm1 x))))))
  (if (<= y -4e-43) t_0 (if (<= y 5e-28) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((y * expm1(x)));
	double tmp;
	if (y <= -4e-43) {
		tmp = t_0;
	} else if (y <= 5e-28) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((y * Math.expm1(x)));
	double tmp;
	if (y <= -4e-43) {
		tmp = t_0;
	} else if (y <= 5e-28) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((y * math.expm1(x)))
	tmp = 0
	if y <= -4e-43:
		tmp = t_0
	elif y <= 5e-28:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(y * expm1(x))))
	tmp = 0.0
	if (y <= -4e-43)
		tmp = t_0;
	elseif (y <= 5e-28)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e-43], t$95$0, If[LessEqual[y, 5e-28], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000003e-43 or 5.0000000000000002e-28 < y

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   if -4.0000000000000003e-43 < y < 5.0000000000000002e-28

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

      1. add-flip 73.0%

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

      2. lift-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-expm1.f64 73.0%

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

      6. *-lft-identity 73.0%

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

      7. pow-exp 73.0%

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

      8. e-exp-1 73.0%

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

      9. distribute-rgt-neg-in 73.0%

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

      10. add-flip 73.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 76.2%

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

   8. Applied rewrites 76.2%

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

Alternative 2: 92.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left({e}^{x} - 1\right) \cdot y\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-10}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(y, \mathsf{expm1}\left(x\right), 1\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* (- (pow E x) 1.0) y)))
  (if (<= t_0 -5e-304)
    (* (* c (expm1 x)) y)
    (if (<= t_0 0.0)
      (* c (log1p (* x y)))
      (if (<= t_0 1e-10)
        (* c (* y (expm1 x)))
        (* (log (fma y (expm1 x) 1.0)) c))))))

C

double code(double c, double x, double y) {
	double t_0 = (pow(((double) M_E), x) - 1.0) * y;
	double tmp;
	if (t_0 <= -5e-304) {
		tmp = (c * expm1(x)) * y;
	} else if (t_0 <= 0.0) {
		tmp = c * log1p((x * y));
	} else if (t_0 <= 1e-10) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = log(fma(y, expm1(x), 1.0)) * c;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(Float64((exp(1) ^ x) - 1.0) * y)
	tmp = 0.0
	if (t_0 <= -5e-304)
		tmp = Float64(Float64(c * expm1(x)) * y);
	elseif (t_0 <= 0.0)
		tmp = Float64(c * log1p(Float64(x * y)));
	elseif (t_0 <= 1e-10)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(log(fma(y, expm1(x), 1.0)) * c);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[(N[Power[E, x], $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-304], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-10], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(y * N[(Exp[x] - 1), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -4.9999999999999997e-304

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

      1. add-flip 73.0%

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

      2. lift-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-expm1.f64 73.0%

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

      6. *-lft-identity 73.0%

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

      7. pow-exp 73.0%

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

      8. e-exp-1 73.0%

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

      9. distribute-rgt-neg-in 73.0%

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

      10. add-flip 73.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 76.2%

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

   8. Applied rewrites 76.2%

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

   if -4.9999999999999997e-304 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < -0.0

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

         \[\leadsto c \cdot \log \left(\left({e}^{x} - 1\right) \cdot y + \color{blue}{\frac{2}{2}}\right) \]

      5. add-to-fraction N/A

         \[\leadsto c \cdot \log \color{blue}{\left(\frac{\left(\left({e}^{x} - 1\right) \cdot y\right) \cdot 2 + 2}{2}\right)} \]

      6. log-div N/A

         \[\leadsto c \cdot \color{blue}{\left(\log \left(\left(\left({e}^{x} - 1\right) \cdot y\right) \cdot 2 + 2\right) - \log 2\right)} \]

      7. lower-unsound--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(\log \left(\left(\left({e}^{x} - 1\right) \cdot y\right) \cdot 2 + 2\right) - \log 2\right)} \]

   3. Applied rewrites 51.7%

      \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(y \cdot \mathsf{expm1}\left(x\right), 2, 2\right)\right) - \log 2\right)} \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 39.8%

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

   6. Applied rewrites 39.8%

      \[\leadsto c \cdot \left(\log \left(\mathsf{fma}\left(\color{blue}{x \cdot y}, 2, 2\right)\right) - \log 2\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-fma.f64 N/A

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

      6. add-to-fraction-rev N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 65.7%

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

   8. Applied rewrites 65.7%

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

   if -0.0 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y) < 1e-10

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

   if 1e-10 < (*.f64 (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) y)

   1. Initial program 41.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.8%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.8%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.7%

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

   3. Applied rewrites 51.7%

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

Alternative 3: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := c \cdot \mathsf{log1p}\left(x \cdot y\right)\\ \mathbf{if}\;y \leq -8 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+21}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* c (log1p (* x y)))))
  (if (<= y -8e-11)
    t_0
    (if (<= y 2.7e+21) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = c * log1p((x * y));
	double tmp;
	if (y <= -8e-11) {
		tmp = t_0;
	} else if (y <= 2.7e+21) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double t_0 = c * Math.log1p((x * y));
	double tmp;
	if (y <= -8e-11) {
		tmp = t_0;
	} else if (y <= 2.7e+21) {
		tmp = (c * Math.expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(c, x, y):
	t_0 = c * math.log1p((x * y))
	tmp = 0
	if y <= -8e-11:
		tmp = t_0
	elif y <= 2.7e+21:
		tmp = (c * math.expm1(x)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(c, x, y)
	t_0 = Float64(c * log1p(Float64(x * y)))
	tmp = 0.0
	if (y <= -8e-11)
		tmp = t_0;
	elseif (y <= 2.7e+21)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(c * N[Log[1 + N[(x * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8e-11], t$95$0, If[LessEqual[y, 2.7e+21], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.9999999999999995e-11 or 2.7e21 < y

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

         \[\leadsto c \cdot \log \left(\left({e}^{x} - 1\right) \cdot y + \color{blue}{\frac{2}{2}}\right) \]

      5. add-to-fraction N/A

         \[\leadsto c \cdot \log \color{blue}{\left(\frac{\left(\left({e}^{x} - 1\right) \cdot y\right) \cdot 2 + 2}{2}\right)} \]

      6. log-div N/A

         \[\leadsto c \cdot \color{blue}{\left(\log \left(\left(\left({e}^{x} - 1\right) \cdot y\right) \cdot 2 + 2\right) - \log 2\right)} \]

      7. lower-unsound--.f64 N/A

         \[\leadsto c \cdot \color{blue}{\left(\log \left(\left(\left({e}^{x} - 1\right) \cdot y\right) \cdot 2 + 2\right) - \log 2\right)} \]

   3. Applied rewrites 51.7%

      \[\leadsto c \cdot \color{blue}{\left(\log \left(\mathsf{fma}\left(y \cdot \mathsf{expm1}\left(x\right), 2, 2\right)\right) - \log 2\right)} \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 39.8%

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

   6. Applied rewrites 39.8%

      \[\leadsto c \cdot \left(\log \left(\mathsf{fma}\left(\color{blue}{x \cdot y}, 2, 2\right)\right) - \log 2\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-log.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. diff-log N/A

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

      5. lift-fma.f64 N/A

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

      6. add-to-fraction-rev N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lower-log1p.f64 65.7%

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

   8. Applied rewrites 65.7%

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

   if -7.9999999999999995e-11 < y < 2.7e21

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

      1. add-flip 73.0%

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

      2. lift-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-expm1.f64 73.0%

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

      6. *-lft-identity 73.0%

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

      7. pow-exp 73.0%

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

      8. e-exp-1 73.0%

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

      9. distribute-rgt-neg-in 73.0%

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

      10. add-flip 73.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 76.2%

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

   8. Applied rewrites 76.2%

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

Alternative 4: 81.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(y, x, 1\right)\right) \cdot c\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+189}:\\ \;\;\;\;\left(c \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* (log (fma y x 1.0)) c)))
  (if (<= y -1.1e+101)
    t_0
    (if (<= y 1.25e+189) (* (* c (expm1 x)) y) t_0))))

C

double code(double c, double x, double y) {
	double t_0 = log(fma(y, x, 1.0)) * c;
	double tmp;
	if (y <= -1.1e+101) {
		tmp = t_0;
	} else if (y <= 1.25e+189) {
		tmp = (c * expm1(x)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(c, x, y)
	t_0 = Float64(log(fma(y, x, 1.0)) * c)
	tmp = 0.0
	if (y <= -1.1e+101)
		tmp = t_0;
	elseif (y <= 1.25e+189)
		tmp = Float64(Float64(c * expm1(x)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := Block[{t$95$0 = N[(N[Log[N[(y * x + 1.0), $MachinePrecision]], $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[y, -1.1e+101], t$95$0, If[LessEqual[y, 1.25e+189], N[(N[(c * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e101 or 1.2500000000000001e189 < y

   1. Initial program 41.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 41.8%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 41.8%

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

      9. lift--.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. lift-E.f64 N/A

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

      12. e-exp-1 N/A

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

      13. pow-exp N/A

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

      14. *-lft-identity N/A

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

      15. lower-expm1.f64 51.7%

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

   3. Applied rewrites 51.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 39.8%

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

   6. Applied rewrites 39.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 39.8%

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

   8. Applied rewrites 39.8%

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

   if -1.1e101 < y < 1.2500000000000001e189

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

      1. add-flip 73.0%

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

      2. lift-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-expm1.f64 73.0%

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

      6. *-lft-identity 73.0%

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

      7. pow-exp 73.0%

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

      8. e-exp-1 73.0%

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

      9. distribute-rgt-neg-in 73.0%

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

      10. add-flip 73.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 76.2%

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

   8. Applied rewrites 76.2%

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

Alternative 5: 78.7% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\left|c\right| \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left|c\right| \cdot \mathsf{expm1}\left(x\right)\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1.0 c)
 (if (<= (fabs c) 5e-71)
   (* (fabs c) (* y (expm1 x)))
   (* (* (fabs c) (expm1 x)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 5e-71) {
		tmp = fabs(c) * (y * expm1(x));
	} else {
		tmp = (fabs(c) * expm1(x)) * y;
	}
	return copysign(1.0, c) * tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (Math.abs(c) <= 5e-71) {
		tmp = Math.abs(c) * (y * Math.expm1(x));
	} else {
		tmp = (Math.abs(c) * Math.expm1(x)) * y;
	}
	return Math.copySign(1.0, c) * tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if math.fabs(c) <= 5e-71:
		tmp = math.fabs(c) * (y * math.expm1(x))
	else:
		tmp = (math.fabs(c) * math.expm1(x)) * y
	return math.copysign(1.0, c) * tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (abs(c) <= 5e-71)
		tmp = Float64(abs(c) * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(abs(c) * expm1(x)) * y);
	end
	return Float64(copysign(1.0, c) * tmp)
end

Wolfram

code[c_, x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 5e-71], N[(N[Abs[c], $MachinePrecision] * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[c], $MachinePrecision] * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 5e-71

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

   if 5e-71 < c

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

      1. add-flip 73.0%

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

      2. lift-expm1.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lift-expm1.f64 73.0%

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

      6. *-lft-identity 73.0%

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

      7. pow-exp 73.0%

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

      8. e-exp-1 73.0%

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

      9. distribute-rgt-neg-in 73.0%

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

      10. add-flip 73.0%

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 76.2%

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

   8. Applied rewrites 76.2%

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

Alternative 6: 75.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-112}:\\ \;\;\;\;c \cdot \left(y \cdot \mathsf{expm1}\left(x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot c\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= x -4.1e-112) (* c (* y (expm1 x))) (* (* x c) y)))

C

double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.1e-112) {
		tmp = c * (y * expm1(x));
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (x <= -4.1e-112) {
		tmp = c * (y * Math.expm1(x));
	} else {
		tmp = (x * c) * y;
	}
	return tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if x <= -4.1e-112:
		tmp = c * (y * math.expm1(x))
	else:
		tmp = (x * c) * y
	return tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (x <= -4.1e-112)
		tmp = Float64(c * Float64(y * expm1(x)));
	else
		tmp = Float64(Float64(x * c) * y);
	end
	return tmp
end

Wolfram

code[c_, x_, y_] := If[LessEqual[x, -4.1e-112], N[(c * N[(y * N[(Exp[x] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * c), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1e-112

   1. Initial program 41.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-log1p.f64 56.7%

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 56.7%

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

      7. lift--.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-E.f64 N/A

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

      10. e-exp-1 N/A

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

      11. pow-exp N/A

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

      12. *-lft-identity N/A

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

      13. lower-expm1.f64 93.4%

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

   3. Applied rewrites 93.4%

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

   4. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 73.0%

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

   6. Applied rewrites 73.0%

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

   if -4.1e-112 < x

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 55.4%

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.5%

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

   6. Applied rewrites 58.5%

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

Alternative 7: 64.0% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;\left(y \cdot \left|c\right|\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left|c\right|\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1.0 c)
 (if (<= (fabs c) 1.5e+14) (* (* y (fabs c)) x) (* (* x (fabs c)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 1.5e+14) {
		tmp = (y * fabs(c)) * x;
	} else {
		tmp = (x * fabs(c)) * y;
	}
	return copysign(1.0, c) * tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (Math.abs(c) <= 1.5e+14) {
		tmp = (y * Math.abs(c)) * x;
	} else {
		tmp = (x * Math.abs(c)) * y;
	}
	return Math.copySign(1.0, c) * tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if math.fabs(c) <= 1.5e+14:
		tmp = (y * math.fabs(c)) * x
	else:
		tmp = (x * math.fabs(c)) * y
	return math.copysign(1.0, c) * tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (abs(c) <= 1.5e+14)
		tmp = Float64(Float64(y * abs(c)) * x);
	else
		tmp = Float64(Float64(x * abs(c)) * y);
	end
	return Float64(copysign(1.0, c) * tmp)
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (abs(c) <= 1.5e+14)
		tmp = (y * abs(c)) * x;
	else
		tmp = (x * abs(c)) * y;
	end
	tmp_2 = (sign(c) * abs(1.0)) * tmp;
end

Wolfram

code[c_, x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 1.5e+14], N[(N[(y * N[Abs[c], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x * N[Abs[c], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 1.5e14

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 55.4%

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.5%

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

   6. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 55.4%

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

   8. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.0%

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

   10. Applied rewrites 61.0%

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

   if 1.5e14 < c

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 55.4%

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.5%

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

   6. Applied rewrites 58.5%

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

Alternative 8: 61.1% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq 5 \cdot 10^{-71}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \left|c\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left|c\right|\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1.0 c)
 (if (<= (fabs c) 5e-71) (* (* x y) (fabs c)) (* (* x (fabs c)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 5e-71) {
		tmp = (x * y) * fabs(c);
	} else {
		tmp = (x * fabs(c)) * y;
	}
	return copysign(1.0, c) * tmp;
}

Java

public static double code(double c, double x, double y) {
	double tmp;
	if (Math.abs(c) <= 5e-71) {
		tmp = (x * y) * Math.abs(c);
	} else {
		tmp = (x * Math.abs(c)) * y;
	}
	return Math.copySign(1.0, c) * tmp;
}

Python

def code(c, x, y):
	tmp = 0
	if math.fabs(c) <= 5e-71:
		tmp = (x * y) * math.fabs(c)
	else:
		tmp = (x * math.fabs(c)) * y
	return math.copysign(1.0, c) * tmp

Julia

function code(c, x, y)
	tmp = 0.0
	if (abs(c) <= 5e-71)
		tmp = Float64(Float64(x * y) * abs(c));
	else
		tmp = Float64(Float64(x * abs(c)) * y);
	end
	return Float64(copysign(1.0, c) * tmp)
end

MATLAB

function tmp_2 = code(c, x, y)
	tmp = 0.0;
	if (abs(c) <= 5e-71)
		tmp = (x * y) * abs(c);
	else
		tmp = (x * abs(c)) * y;
	end
	tmp_2 = (sign(c) * abs(1.0)) * tmp;
end

Wolfram

code[c_, x_, y_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 5e-71], N[(N[(x * y), $MachinePrecision] * N[Abs[c], $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Abs[c], $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if c < 5e-71

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 55.4%

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.5%

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

   6. Applied rewrites 58.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lower-*.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if 5e-71 < c

   1. Initial program 41.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-E.f64 55.4%

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 58.5%

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

   6. Applied rewrites 58.5%

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

Alternative 9: 58.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double c, double x, double y) {
	return (x * c) * 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 = (x * c) * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.8%

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

2. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-log.f64 N/A

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

   5. lower-E.f64 55.4%

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

4. Applied rewrites 55.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. lift-E.f64 N/A

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

   7. log-E N/A

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

   8. *-rgt-identity N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 58.5%

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

6. Applied rewrites 58.5%

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

Developer Target 1: 93.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2025217 
(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)))))