Logarithmic Transform

Percentage Accurate: 41.5% → 93.1%

Time: 20.7s

Alternatives: 7

Speedup: 19.8×

• could not determine a ground truth

  1. c = -3.444090105351809e+21
  2. x = 2.989501629624694e+213
  3. y = 1.6500308391369249e-254

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 (* (- (pow E x) 1) 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 + N[(N[(N[Power[E, x], $MachinePrecision] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 41.5% 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 (* (- (pow E x) 1) 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 + N[(N[(N[Power[E, x], $MachinePrecision] - 1), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{e}^{x} - 1 \leq \frac{-5764607523034235}{1152921504606846976}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(1 - e^{x}\right) \cdot y\right)\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(\left(\left(\frac{1}{6} \cdot x - \frac{-1}{2}\right) \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= (- (pow E x) 1) -5764607523034235/1152921504606846976)
  (* (30-log1z0 (* (- 1 (exp x)) y)) c)
  (* (30-log1z0 (* (* (- (* (- (* 1/6 x) -1/2) x) -1) x) (- y))) c)))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) < -0.0050000000000000001

   1. Initial program 41.5%

      \[c \cdot \log \left(1 + \left({e}^{x} - 1\right) \cdot y\right) \]
   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 41.5%

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

   3. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(1 - e^{x}\right) \cdot y\right)\right) \cdot c} \]

   if -0.0050000000000000001 < (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64))

   1. Initial program 41.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

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

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

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

         \[\leadsto c \cdot \log \left(1 + \frac{\color{blue}{{\left({e}^{x}\right)}^{3} - {1}^{3}}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      5. lower-unsound-pow.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{\color{blue}{{\left({e}^{x}\right)}^{3}} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left({e}^{x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      7. lift-E.f64 N/A

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

      8. e-exp-1 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\left({\color{blue}{\left(e^{1}\right)}}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      9. pow-exp N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left(e^{1 \cdot x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{\color{blue}{x}}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      11. lower-exp.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left(e^{x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      12. lower-unsound-pow.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{x}\right)}^{3} - \color{blue}{{1}^{3}}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      13. lower-unsound-+.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

   3. Applied rewrites 41.4%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}} \cdot y\right) \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 37.3%

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

   6. Applied rewrites 37.3%

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

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\left(\frac{1}{6} \cdot x - \frac{-1}{2}\right) \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{e}^{x} - 1 \leq \frac{-5764607523034235}{1152921504606846976}:\\ \;\;\;\;\left(e^{x} - 1\right) \cdot \left(y \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(\left(\left(\frac{1}{6} \cdot x - \frac{-1}{2}\right) \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<= (- (pow E x) 1) -5764607523034235/1152921504606846976)
  (* (- (exp x) 1) (* y c))
  (* (30-log1z0 (* (* (- (* (- (* 1/6 x) -1/2) x) -1) x) (- y))) c)))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64)) < -0.0050000000000000001

   1. Initial program 41.5%

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

   2. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-E.f64 46.1%

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

   4. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-E.f64 N/A

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

      8. e-exp-1 N/A

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

      9. pow-exp N/A

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

      10. *-lft-identity N/A

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

      11. lift-exp.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 45.9%

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

   6. Applied rewrites 45.9%

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

   if -0.0050000000000000001 < (-.f64 (pow.f64 (E.f64) x) #s(literal 1 binary64))

   1. Initial program 41.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

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

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

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

         \[\leadsto c \cdot \log \left(1 + \frac{\color{blue}{{\left({e}^{x}\right)}^{3} - {1}^{3}}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      5. lower-unsound-pow.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{\color{blue}{{\left({e}^{x}\right)}^{3}} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left({e}^{x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      7. lift-E.f64 N/A

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

      8. e-exp-1 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\left({\color{blue}{\left(e^{1}\right)}}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      9. pow-exp N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left(e^{1 \cdot x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{\color{blue}{x}}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      11. lower-exp.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left(e^{x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      12. lower-unsound-pow.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{x}\right)}^{3} - \color{blue}{{1}^{3}}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      13. lower-unsound-+.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

   3. Applied rewrites 41.4%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}} \cdot y\right) \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 37.3%

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

   6. Applied rewrites 37.3%

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

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\left(\frac{1}{6} \cdot x - \frac{-1}{2}\right) \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq \frac{-5767152963771295}{22181357552966518876627313473144669627491496603006532601363836644916970462445004984319795248833116624779129687691228574631793262592}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \mathbf{elif}\;y \leq \frac{3916911482881289}{100433627766186892221372630771322662657637687111424552206336}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(\left(\left(\frac{1}{6} \cdot x - \frac{-1}{2}\right) \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<=
     y
     -5767152963771295/22181357552966518876627313473144669627491496603006532601363836644916970462445004984319795248833116624779129687691228574631793262592)
  (* (30-log1z0 (* (* 1 x) (- y))) c)
  (if (<=
       y
       3916911482881289/100433627766186892221372630771322662657637687111424552206336)
    (* (* y c) x)
    (*
     (30-log1z0 (* (* (- (* (- (* 1/6 x) -1/2) x) -1) x) (- y)))
     c))))

Derivation

1. Split input into 3 regimes

2. if y < -2.6e-115

   1. Initial program 41.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-E.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-E.f64 36.8%

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

   4. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.8%

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

   6. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\frac{1}{2} \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c \]

   if -2.6e-115 < y < 3.9000000000000002e-44

   1. Initial program 41.5%

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

   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 56.4%

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

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log e\right)\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.9%

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

   6. Applied rewrites 61.9%

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

   if 3.9000000000000002e-44 < y

   1. Initial program 41.5%

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

      1. lift--.f64 N/A

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

      2. flip3-- N/A

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

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

         \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left({e}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

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

         \[\leadsto c \cdot \log \left(1 + \frac{\color{blue}{{\left({e}^{x}\right)}^{3} - {1}^{3}}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      5. lower-unsound-pow.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{\color{blue}{{\left({e}^{x}\right)}^{3}} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left({e}^{x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      7. lift-E.f64 N/A

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

      8. e-exp-1 N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\left({\color{blue}{\left(e^{1}\right)}}^{x}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      9. pow-exp N/A

         \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left(e^{1 \cdot x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      10. *-lft-identity N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{\color{blue}{x}}\right)}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      11. lower-exp.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\color{blue}{\left(e^{x}\right)}}^{3} - {1}^{3}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      12. lower-unsound-pow.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{x}\right)}^{3} - \color{blue}{{1}^{3}}}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)} \cdot y\right) \]

      13. lower-unsound-+.f64 N/A

          \[\leadsto c \cdot \log \left(1 + \frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{{e}^{x} \cdot {e}^{x} + \left(1 \cdot 1 + {e}^{x} \cdot 1\right)}} \cdot y\right) \]

   3. Applied rewrites 41.4%

      \[\leadsto c \cdot \log \left(1 + \color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}} \cdot y\right) \]

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 37.3%

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

   6. Applied rewrites 37.3%

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

   8. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\left(\frac{1}{6} \cdot x - \frac{-1}{2}\right) \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq \frac{-5767152963771295}{22181357552966518876627313473144669627491496603006532601363836644916970462445004984319795248833116624779129687691228574631793262592}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \mathbf{elif}\;y \leq \frac{4097692012860425}{803469022129495137770981046170581301261101496891396417650688}:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{30\_log1z0}\left(\left(\left(\left(\frac{1}{2} \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (if (<=
     y
     -5767152963771295/22181357552966518876627313473144669627491496603006532601363836644916970462445004984319795248833116624779129687691228574631793262592)
  (* (30-log1z0 (* (* 1 x) (- y))) c)
  (if (<=
       y
       4097692012860425/803469022129495137770981046170581301261101496891396417650688)
    (* (* y c) x)
    (* (30-log1z0 (* (* (- (* 1/2 x) -1) x) (- y))) c))))

Derivation

1. Split input into 3 regimes

2. if y < -2.6e-115

   1. Initial program 41.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-E.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-E.f64 36.8%

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

   4. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.8%

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

   6. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\frac{1}{2} \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c \]

   if -2.6e-115 < y < 5.0999999999999997e-45

   1. Initial program 41.5%

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

   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 56.4%

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

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log e\right)\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.9%

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

   6. Applied rewrites 61.9%

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

   if 5.0999999999999997e-45 < y

   1. Initial program 41.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-E.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-E.f64 36.8%

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

   4. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.8%

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

   6. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\frac{1}{2} \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c\\ \mathbf{if}\;y \leq \frac{-5767152963771295}{22181357552966518876627313473144669627491496603006532601363836644916970462445004984319795248833116624779129687691228574631793262592}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 210000000000000000:\\ \;\;\;\;\left(y \cdot c\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (let* ((t_0 (* (30-log1z0 (* (* 1 x) (- y))) c)))
  (if (<=
       y
       -5767152963771295/22181357552966518876627313473144669627491496603006532601363836644916970462445004984319795248833116624779129687691228574631793262592)
    t_0
    (if (<= y 210000000000000000) (* (* y c) x) t_0))))

Derivation

1. Split input into 2 regimes

2. if y < -2.6e-115 or 2.1e17 < y

   1. Initial program 41.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-E.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-E.f64 36.8%

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

   4. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.8%

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

   6. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{30\_log1z0}\left(\left(\left(\left(\frac{1}{2} \cdot x - -1\right) \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c \]
   8. Step-by-step derivation

   9. Applied rewrites 66.1%

      \[\leadsto \mathsf{30\_log1z0}\left(\left(\left(1 \cdot x\right) \cdot \left(-y\right)\right)\right) \cdot c \]

   if -2.6e-115 < y < 2.1e17

   1. Initial program 41.5%

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

   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 56.4%

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

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log e\right)\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.9%

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

   6. Applied rewrites 61.9%

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

Alternative 6: 64.5% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, c\right) \cdot \begin{array}{l} \mathbf{if}\;\left|c\right| \leq \frac{6257774519299541}{521481209941628438084722096232800809229175908778479680162851955034721612739414196782949728256}:\\ \;\;\;\;\left(y \cdot \left|c\right|\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left|c\right|\right) \cdot y\\ \end{array} \]

FPCore

(FPCore (c x y)
  :precision binary64
  (*
 (copysign 1 c)
 (if (<=
      (fabs c)
      6257774519299541/521481209941628438084722096232800809229175908778479680162851955034721612739414196782949728256)
   (* (* y (fabs c)) x)
   (* (* x (fabs c)) y))))

C

double code(double c, double x, double y) {
	double tmp;
	if (fabs(c) <= 1.2e-77) {
		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) <= 1.2e-77) {
		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) <= 1.2e-77:
		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) <= 1.2e-77)
		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) <= 1.2e-77)
		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], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[c], $MachinePrecision], 6257774519299541/521481209941628438084722096232800809229175908778479680162851955034721612739414196782949728256], 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 < 1.2e-77

   1. Initial program 41.5%

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

   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 56.4%

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

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log e\right)\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-E.f64 N/A

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

      7. log-E N/A

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

      8. *-rgt-identity N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 61.9%

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

   6. Applied rewrites 61.9%

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

   if 1.2e-77 < c

   1. Initial program 41.5%

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

   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 56.4%

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

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log e\right)\right)} \]
   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 59.6%

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

   6. Applied rewrites 59.6%

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

Alternative 7: 59.6% accurate, 19.8× 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 41.5%

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

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 56.4%

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

4. Applied rewrites 56.4%

   \[\leadsto \color{blue}{c \cdot \left(x \cdot \left(y \cdot \log e\right)\right)} \]
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 59.6%

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

6. Applied rewrites 59.6%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (c x y)
  :name "Logarithmic Transform"
  :precision binary64
  (* c (log (+ 1 (* (- (pow E x) 1) y)))))