Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.9%

Time: 10.7s

Alternatives: 14

Speedup: 2.0×

• 49.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (fma (* (* (sinh l) (cos (/ K -2.0))) 2.0) J U))

C

double code(double J, double l, double K, double U) {
	return fma(((sinh(l) * cos((K / -2.0))) * 2.0), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(Float64(sinh(l) * cos(Float64(K / -2.0))) * 2.0), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[Cos[N[(K / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]
5. Add Preprocessing

Alternative 2: 87.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.45:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;t\_0 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.45)
     (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
     (if (<= t_0 -0.02)
       (fma (* (+ J J) l) (cos (* -0.5 K)) U)
       (fma (* (* (sinh l) 1.0) 2.0) J U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.45) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else if (t_0 <= -0.02) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.45)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	elseif (t_0 <= -0.02)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.45], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[t$95$0, -0.02], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.450000000000000011

   1. Initial program 88.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}\right) \cdot 2, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{8}} + 1\right)\right) \cdot 2, J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right)}\right) \cdot 2, J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8}, 1\right)\right) \cdot 2, J, U\right) \]

      5. lower-*.f64 72.0

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right)\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 72.0%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}\right) \cdot 2, J, U\right) \]

   if -0.450000000000000011 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 78.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 84.6

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 84.6%

      \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 82.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.172:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.172)
   (fma (* (cos (* -0.5 K)) (* J (fma (* l l) 0.3333333333333333 2.0))) l U)
   (fma (* (* (sinh l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.172) {
		tmp = fma((cos((-0.5 * K)) * (J * fma((l * l), 0.3333333333333333, 2.0))), l, U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.172)
		tmp = fma(Float64(cos(Float64(-0.5 * K)) * Float64(J * fma(Float64(l * l), 0.3333333333333333, 2.0))), l, U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.172], N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.17199999999999999

   1. Initial program 86.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right), \ell, U\right)} \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(-0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right)\right), \ell, U\right)} \]

   if 0.17199999999999999 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 82.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 95.8%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (fma (* (+ J J) l) (cos (* -0.5 K)) U)
   (fma (* (* (sinh l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = fma(((J + J) * l), cos((-0.5 * K)), U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = fma(Float64(Float64(J + J) * l), cos(Float64(-0.5 * K)), U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(N[(N[(J + J), $MachinePrecision] * l), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 86.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 67.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.0%

      \[\leadsto \mathsf{fma}\left(\left(J + J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 82.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.45:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot 1\right) \cdot 2, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.45)
   (fma (* (* (fma (* K K) -0.125 1.0) l) 2.0) J U)
   (fma (* (* (sinh l) 1.0) 2.0) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.45) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * l) * 2.0), J, U);
	} else {
		tmp = fma(((sinh(l) * 1.0) * 2.0), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.45)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * l) * 2.0), J, U);
	else
		tmp = fma(Float64(Float64(sinh(l) * 1.0) * 2.0), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.45], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[Sinh[l], $MachinePrecision] * 1.0), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.450000000000000011

   1. Initial program 88.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      3. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot \ell\right) \cdot 2, J, U\right) \]

      7. lower-*.f64 60.9

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 60.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\ell + \color{blue}{\frac{-1}{8} \cdot \left({K}^{2} \cdot \ell\right)}\right) \cdot 2, J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 66.4%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \color{blue}{\ell}\right) \cdot 2, J, U\right) \]

   if -0.450000000000000011 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 81.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 92.5%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{1}\right) \cdot 2, J, U\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  (*
   (*
    J
    (*
     (fma
      (fma
       (fma 0.0003968253968253968 (* l l) 0.016666666666666666)
       (* l l)
       0.3333333333333333)
      (* l l)
      2.0)
     l))
   (cos (/ K 2.0)))
  U))

C

double code(double J, double l, double K, double U) {
	return ((J * (fma(fma(fma(0.0003968253968253968, (l * l), 0.016666666666666666), (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
}

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(fma(fma(fma(0.0003968253968253968, Float64(l * l), 0.016666666666666666), Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[(N[(N[(0.0003968253968253968 * N[(l * l), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. lower-*.f64 N/A

      \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. *-commutative N/A

      \[\leadsto \left(J \cdot \left(\left(\color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right) \cdot {\ell}^{2}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right), {\ell}^{2}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) + \frac{1}{3}}, {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. *-commutative N/A

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + \frac{1}{3}, {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}, {\ell}^{2}, \frac{1}{3}\right)}, {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. +-commutative N/A

      \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {\ell}^{2} + \frac{1}{60}}, {\ell}^{2}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   10. lower-fma.f64 N/A

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {\ell}^{2}, \frac{1}{60}\right)}, {\ell}^{2}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{\ell \cdot \ell}, \frac{1}{60}\right), {\ell}^{2}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   12. lower-*.f64 N/A

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{\ell \cdot \ell}, \frac{1}{60}\right), {\ell}^{2}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   13. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \ell \cdot \ell, \frac{1}{60}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   14. lower-*.f64 N/A

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \ell \cdot \ell, \frac{1}{60}\right), \color{blue}{\ell \cdot \ell}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   15. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \ell \cdot \ell, \frac{1}{60}\right), \ell \cdot \ell, \frac{1}{3}\right), \color{blue}{\ell \cdot \ell}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   16. lower-*.f64 95.8

       \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \color{blue}{\ell \cdot \ell}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Applied rewrites 95.8%

   \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, \ell \cdot \ell, 0.016666666666666666\right), \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
6. Add Preprocessing

Alternative 7: 82.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1e-6)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (+
    (*
     (*
      J
      (*
       (fma (fma 0.016666666666666666 (* l l) 0.3333333333333333) (* l l) 2.0)
       l))
     (cos (/ K 2.0)))
    U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1e-6) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = ((J * (fma(fma(0.016666666666666666, (l * l), 0.3333333333333333), (l * l), 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1e-6)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(fma(0.016666666666666666, Float64(l * l), 0.3333333333333333), Float64(l * l), 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1e-6], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(0.016666666666666666 * N[(l * l), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(l * l), $MachinePrecision] + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 9.99999999999999955e-7

   1. Initial program 81.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}\right) \cdot 2, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{8}} + 1\right)\right) \cdot 2, J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right)}\right) \cdot 2, J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8}, 1\right)\right) \cdot 2, J, U\right) \]

      5. lower-*.f64 77.6

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right)\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}\right) \cdot 2, J, U\right) \]

   if 9.99999999999999955e-7 < K

   1. Initial program 88.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right) \cdot {\ell}^{2}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}, {\ell}^{2}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{60} \cdot {\ell}^{2} + \frac{1}{3}}, {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60}, {\ell}^{2}, \frac{1}{3}\right)}, {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{\ell \cdot \ell}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \color{blue}{\ell \cdot \ell}, \frac{1}{3}\right), {\ell}^{2}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{60}, \ell \cdot \ell, \frac{1}{3}\right), \color{blue}{\ell \cdot \ell}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      11. lower-*.f64 95.4

          \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \color{blue}{\ell \cdot \ell}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 95.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, \ell \cdot \ell, 0.3333333333333333\right), \ell \cdot \ell, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 81.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 0.115:\\ \;\;\;\;\mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(K \cdot K, -0.125, 1\right)\right) \cdot 2, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 0.115)
   (fma (* (* (sinh l) (fma (* K K) -0.125 1.0)) 2.0) J U)
   (+ (* (* J (* (fma (* l l) 0.3333333333333333 2.0) l)) (cos (/ K 2.0))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 0.115) {
		tmp = fma(((sinh(l) * fma((K * K), -0.125, 1.0)) * 2.0), J, U);
	} else {
		tmp = ((J * (fma((l * l), 0.3333333333333333, 2.0) * l)) * cos((K / 2.0))) + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 0.115)
		tmp = fma(Float64(Float64(sinh(l) * fma(Float64(K * K), -0.125, 1.0)) * 2.0), J, U);
	else
		tmp = Float64(Float64(Float64(J * Float64(fma(Float64(l * l), 0.3333333333333333, 2.0) * l)) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 0.115], N[(N[(N[(N[Sinh[l], $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(J * N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333 + 2.0), $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 0.115000000000000005

   1. Initial program 81.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right) \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\left(\frac{-1}{8} \cdot {K}^{2} + 1\right)}\right) \cdot 2, J, U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \left(\color{blue}{{K}^{2} \cdot \frac{-1}{8}} + 1\right)\right) \cdot 2, J, U\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left({K}^{2}, \frac{-1}{8}, 1\right)}\right) \cdot 2, J, U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, \frac{-1}{8}, 1\right)\right) \cdot 2, J, U\right) \]

      5. lower-*.f64 77.6

         \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \mathsf{fma}\left(\color{blue}{K \cdot K}, -0.125, 1\right)\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\left(\sinh \ell \cdot \color{blue}{\mathsf{fma}\left(K \cdot K, -0.125, 1\right)}\right) \cdot 2, J, U\right) \]

   if 0.115000000000000005 < K

   1. Initial program 88.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. lower-*.f64 N/A

         \[\leadsto \left(J \cdot \color{blue}{\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. +-commutative N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\left(\color{blue}{{\ell}^{2} \cdot \frac{1}{3}} + 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(J \cdot \left(\color{blue}{\mathsf{fma}\left({\ell}^{2}, \frac{1}{3}, 2\right)} \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      6. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, \frac{1}{3}, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. lower-*.f64 93.0

         \[\leadsto \left(J \cdot \left(\mathsf{fma}\left(\color{blue}{\ell \cdot \ell}, 0.3333333333333333, 2\right) \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Applied rewrites 93.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\mathsf{fma}\left(\ell \cdot \ell, 0.3333333333333333, 2\right) \cdot \ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 61.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{J \cdot \ell}{U}, 2, 1\right) \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 5.5e-5)
   (fma (* (* 2.0 l) (* 2.0 l)) J U)
   (* (fma (/ (* J l) U) 2.0 1.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 5.5e-5) {
		tmp = fma(((2.0 * l) * (2.0 * l)), J, U);
	} else {
		tmp = fma(((J * l) / U), 2.0, 1.0) * U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 5.5e-5)
		tmp = fma(Float64(Float64(2.0 * l) * Float64(2.0 * l)), J, U);
	else
		tmp = Float64(fma(Float64(Float64(J * l) / U), 2.0, 1.0) * U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 5.5e-5], N[(N[(N[(2.0 * l), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(N[(J * l), $MachinePrecision] / U), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision] * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 5.5000000000000002e-5

   1. Initial program 86.4%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 66.1

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.3%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 37.3%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 57.8%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right) \]

   if 5.5000000000000002e-5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 82.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 71.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]

   9. Taylor expanded in U around inf

      \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \frac{J \cdot \ell}{U}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 71.1%

       \[\leadsto \mathsf{fma}\left(\frac{J \cdot \ell}{U}, 2, 1\right) \cdot U \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{+17} \lor \neg \left(\ell \leq 0.0033\right):\\ \;\;\;\;\mathsf{fma}\left({\left(2 \cdot \ell\right)}^{3}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -7.2e+254)
   (fma (* (* (fma (* K K) -0.125 1.0) l) 2.0) J U)
   (if (or (<= l -1.05e+17) (not (<= l 0.0033)))
     (fma (pow (* 2.0 l) 3.0) J U)
     (fma (+ l l) J U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -7.2e+254) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * l) * 2.0), J, U);
	} else if ((l <= -1.05e+17) || !(l <= 0.0033)) {
		tmp = fma(pow((2.0 * l), 3.0), J, U);
	} else {
		tmp = fma((l + l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -7.2e+254)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * l) * 2.0), J, U);
	elseif ((l <= -1.05e+17) || !(l <= 0.0033))
		tmp = fma((Float64(2.0 * l) ^ 3.0), J, U);
	else
		tmp = fma(Float64(l + l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -7.2e+254], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[Or[LessEqual[l, -1.05e+17], N[Not[LessEqual[l, 0.0033]], $MachinePrecision]], N[(N[Power[N[(2.0 * l), $MachinePrecision], 3.0], $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.19999999999999954e254

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      3. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot \ell\right) \cdot 2, J, U\right) \]

      7. lower-*.f64 51.7

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 51.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\ell + \color{blue}{\frac{-1}{8} \cdot \left({K}^{2} \cdot \ell\right)}\right) \cdot 2, J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 75.7%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \color{blue}{\ell}\right) \cdot 2, J, U\right) \]

   if -7.19999999999999954e254 < l < -1.05e17 or 0.0033 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 31.3

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 31.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.0%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 24.0%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 61.9%

       \[\leadsto \mathsf{fma}\left({\left(2 \cdot \ell\right)}^{3}, J, U\right) \]

   if -1.05e17 < l < 0.0033

   1. Initial program 69.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 99.2

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.3%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 86.3%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{+17} \lor \neg \left(\ell \leq 0.0033\right):\\ \;\;\;\;\mathsf{fma}\left({\left(2 \cdot \ell\right)}^{3}, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+185} \lor \neg \left(\ell \leq 0.0033\right):\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5e+185) (not (<= l 0.0033)))
   (fma (* (* 2.0 l) (* 2.0 l)) J U)
   (fma (+ l l) J U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5e+185) || !(l <= 0.0033)) {
		tmp = fma(((2.0 * l) * (2.0 * l)), J, U);
	} else {
		tmp = fma((l + l), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5e+185) || !(l <= 0.0033))
		tmp = fma(Float64(Float64(2.0 * l) * Float64(2.0 * l)), J, U);
	else
		tmp = fma(Float64(l + l), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5e+185], N[Not[LessEqual[l, 0.0033]], $MachinePrecision]], N[(N[(N[(2.0 * l), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.9999999999999999e185 or 0.0033 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 38.0

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 27.3%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 27.3%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 54.3%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right) \]

   if -4.9999999999999999e185 < l < 0.0033

   1. Initial program 74.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 86.2

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 86.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 74.7%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 74.7%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+185} \lor \neg \left(\ell \leq 0.0033\right):\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \ell\right) \cdot 2, J, U\right)\\ \mathbf{elif}\;\ell \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.4e+20)
   (fma (* (* (fma (* K K) -0.125 1.0) l) 2.0) J U)
   (if (<= l 0.0033) (fma (+ l l) J U) (fma (* (* 2.0 l) (* 2.0 l)) J U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.4e+20) {
		tmp = fma(((fma((K * K), -0.125, 1.0) * l) * 2.0), J, U);
	} else if (l <= 0.0033) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma(((2.0 * l) * (2.0 * l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.4e+20)
		tmp = fma(Float64(Float64(fma(Float64(K * K), -0.125, 1.0) * l) * 2.0), J, U);
	elseif (l <= 0.0033)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(Float64(2.0 * l) * Float64(2.0 * l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.4e+20], N[(N[(N[(N[(N[(K * K), $MachinePrecision] * -0.125 + 1.0), $MachinePrecision] * l), $MachinePrecision] * 2.0), $MachinePrecision] * J + U), $MachinePrecision], If[LessEqual[l, 0.0033], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.4e20

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)} + U \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), J, U\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{-2}\right)\right) \cdot 2, J, U\right)} \]

   5. Taylor expanded in l around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\ell \cdot \cos \left(\frac{-1}{2} \cdot K\right)\right)} \cdot 2, J, U\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

      3. cos-neg-rev N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\frac{-1}{2} \cdot K\right)\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right) \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\cos \left(\color{blue}{\frac{1}{2}} \cdot K\right) \cdot \ell\right) \cdot 2, J, U\right) \]

      7. lower-*.f64 31.2

         \[\leadsto \mathsf{fma}\left(\left(\cos \color{blue}{\left(0.5 \cdot K\right)} \cdot \ell\right) \cdot 2, J, U\right) \]

   7. Applied rewrites 31.2%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)} \cdot 2, J, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(\ell + \color{blue}{\frac{-1}{8} \cdot \left({K}^{2} \cdot \ell\right)}\right) \cdot 2, J, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 48.7%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{fma}\left(K \cdot K, -0.125, 1\right) \cdot \color{blue}{\ell}\right) \cdot 2, J, U\right) \]

   if -4.4e20 < l < 0.0033

   1. Initial program 69.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 98.5

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.7%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.7%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   if 0.0033 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 35.9

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.4%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.4%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 57.1%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 63.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(J \cdot 2\right), \ell, U\right)\\ \mathbf{elif}\;\ell \leq 0.0033:\\ \;\;\;\;\mathsf{fma}\left(\ell + \ell, J, U\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.4e+20)
   (fma (* (fma -0.125 (* K K) 1.0) (* J 2.0)) l U)
   (if (<= l 0.0033) (fma (+ l l) J U) (fma (* (* 2.0 l) (* 2.0 l)) J U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.4e+20) {
		tmp = fma((fma(-0.125, (K * K), 1.0) * (J * 2.0)), l, U);
	} else if (l <= 0.0033) {
		tmp = fma((l + l), J, U);
	} else {
		tmp = fma(((2.0 * l) * (2.0 * l)), J, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.4e+20)
		tmp = fma(Float64(fma(-0.125, Float64(K * K), 1.0) * Float64(J * 2.0)), l, U);
	elseif (l <= 0.0033)
		tmp = fma(Float64(l + l), J, U);
	else
		tmp = fma(Float64(Float64(2.0 * l) * Float64(2.0 * l)), J, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.4e+20], N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision] + 1.0), $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * l + U), $MachinePrecision], If[LessEqual[l, 0.0033], N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision], N[(N[(N[(2.0 * l), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * J + U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.4e20

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 31.2

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 31.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.2%

      \[\leadsto \mathsf{fma}\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right), \color{blue}{\ell}, U\right) \]

   8. Taylor expanded in K around 0

      \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(J \cdot 2\right), \ell, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 43.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, K \cdot K, 1\right) \cdot \left(J \cdot 2\right), \ell, U\right) \]

   if -4.4e20 < l < 0.0033

   1. Initial program 69.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 98.5

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 85.7%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.7%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]

   if 0.0033 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

      2. *-commutative N/A

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

      3. associate-*r* N/A

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

      12. cos-neg-rev N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      13. lower-cos.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

      14. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

      16. metadata-eval 35.9

          \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

   6. Taylor expanded in K around 0

      \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 30.4%

      \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 30.4%

       \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 57.1%

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot \ell\right) \cdot \left(2 \cdot \ell\right), J, U\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 54.0% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\ell + \ell, J, U\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (fma (+ l l) J U))

C

double code(double J, double l, double K, double U) {
	return fma((l + l), J, U);
}

Julia

function code(J, l, K, U)
	return fma(Float64(l + l), J, U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(l + l), $MachinePrecision] * J + U), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U} \]

   2. *-commutative N/A

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \ell\right)}\right) + U \]

   3. associate-*r* N/A

      \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \ell\right)} + U \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell} + U \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + U \]

   7. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)} + U \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell \cdot \left(2 \cdot J\right), \cos \left(\frac{1}{2} \cdot K\right), U\right)} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right) \cdot \ell}, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(2 \cdot J\right)} \cdot \ell, \cos \left(\frac{1}{2} \cdot K\right), U\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

   13. lower-cos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot K\right)\right)}, U\right) \]

   14. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot K\right)}, U\right) \]

   16. metadata-eval 69.6

       \[\leadsto \mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(\color{blue}{-0.5} \cdot K\right), U\right) \]

5. Applied rewrites 69.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(2 \cdot J\right) \cdot \ell, \cos \left(-0.5 \cdot K\right), U\right)} \]

6. Taylor expanded in K around 0

   \[\leadsto U + \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

8. Applied rewrites 58.4%

   \[\leadsto \mathsf{fma}\left(2 \cdot \ell, \color{blue}{J}, U\right) \]
9. Step-by-step derivation

10. Applied rewrites 58.4%

    \[\leadsto \mathsf{fma}\left(\ell + \ell, J, U\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024344 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))