math.cos on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.5s

Alternatives: 8

Speedup: 1.3×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

• Could not converge on a ground truth

  1. re = +nan.0
  2. im = 2.764749691375634e-223

Specification

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp(-im) + exp(im));
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp(-im) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp(-im) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp(-im) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp(-im) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\cosh im \cdot \cos re \]

FPCore

(FPCore (re im)
  :precision binary64
  (* (cosh im) (cos re)))

C

double code(double re, double im) {
	return cosh(im) * cos(re);
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = cosh(im) * cos(re)
end function

Java

public static double code(double re, double im) {
	return Math.cosh(im) * Math.cos(re);
}

Python

def code(re, im):
	return math.cosh(im) * math.cos(re)

Julia

function code(re, im)
	return Float64(cosh(im) * cos(re))
end

MATLAB

function tmp = code(re, im)
	tmp = cosh(im) * cos(re);
end

Wolfram

code[re_, im_] := N[(N[Cosh[im], $MachinePrecision] * N[Cos[re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

   5. metadata-eval N/A

      \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

   6. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

   7. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

   8. +-commutative N/A

      \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

   9. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

   10. lift-exp.f64 N/A

       \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

   12. cosh-def N/A

       \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   13. lower-*.f64 N/A

       \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   14. lower-cosh.f64 100.0%

       \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]
4. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := t\_0 \cdot \left(e^{-im} + e^{im}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right)\\ \mathbf{elif}\;t\_1 \leq 0.9999999801658838:\\ \;\;\;\;t\_0 \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (let* ((t_0 (* 0.5 (cos re))) (t_1 (* t_0 (+ (exp (- im)) (exp im)))))
  (if (<= t_1 (- INFINITY))
    (* (cosh im) (fma (* re re) -0.5 1.0))
    (if (<= t_1 0.9999999801658838) (* t_0 2.0) (* (cosh im) 1.0)))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = t_0 * (exp(-im) + exp(im));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = cosh(im) * fma((re * re), -0.5, 1.0);
	} else if (t_1 <= 0.9999999801658838) {
		tmp = t_0 * 2.0;
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(t_0 * Float64(exp(Float64(-im)) + exp(im)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(cosh(im) * fma(Float64(re * re), -0.5, 1.0));
	elseif (t_1 <= 0.9999999801658838)
		tmp = Float64(t_0 * 2.0);
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[Cosh[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.9999999801658838], N[(t$95$0 * 2.0), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -inf.0

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cosh im \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \cosh im \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right) \]

      3. lower-pow.f64 63.5%

         \[\leadsto \cosh im \cdot \left(1 + -0.5 \cdot {re}^{\color{blue}{2}}\right) \]

   6. Applied rewrites 63.5%

      \[\leadsto \cosh im \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \cosh im \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto \cosh im \cdot \left(\frac{-1}{2} \cdot {re}^{2} + \color{blue}{1}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \cosh im \cdot \left(\frac{-1}{2} \cdot {re}^{2} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \cosh im \cdot \left({re}^{2} \cdot \frac{-1}{2} + 1\right) \]

      5. lower-fma.f64 63.5%

         \[\leadsto \cosh im \cdot \mathsf{fma}\left({re}^{2}, \color{blue}{-0.5}, 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \cosh im \cdot \mathsf{fma}\left({re}^{2}, \frac{-1}{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{2}, 1\right) \]

      8. lower-*.f64 63.5%

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right) \]

   8. Applied rewrites 63.5%

      \[\leadsto \cosh im \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{-0.5}, 1\right) \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < 0.99999998016588376

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   if 0.99999998016588376 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.3%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;\cosh im \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.005)
  (* (cosh im) (fma (* re re) -0.5 1.0))
  (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.005) {
		tmp = cosh(im) * fma((re * re), -0.5, 1.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.005)
		tmp = Float64(cosh(im) * fma(Float64(re * re), -0.5, 1.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(N[Cosh[im], $MachinePrecision] * N[(N[(re * re), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {re}^{2}\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \cosh im \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \cosh im \cdot \left(1 + \frac{-1}{2} \cdot \color{blue}{{re}^{2}}\right) \]

      3. lower-pow.f64 63.5%

         \[\leadsto \cosh im \cdot \left(1 + -0.5 \cdot {re}^{\color{blue}{2}}\right) \]

   6. Applied rewrites 63.5%

      \[\leadsto \cosh im \cdot \color{blue}{\left(1 + -0.5 \cdot {re}^{2}\right)} \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \cosh im \cdot \left(1 + \color{blue}{\frac{-1}{2} \cdot {re}^{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto \cosh im \cdot \left(\frac{-1}{2} \cdot {re}^{2} + \color{blue}{1}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \cosh im \cdot \left(\frac{-1}{2} \cdot {re}^{2} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \cosh im \cdot \left({re}^{2} \cdot \frac{-1}{2} + 1\right) \]

      5. lower-fma.f64 63.5%

         \[\leadsto \cosh im \cdot \mathsf{fma}\left({re}^{2}, \color{blue}{-0.5}, 1\right) \]

      6. lift-pow.f64 N/A

         \[\leadsto \cosh im \cdot \mathsf{fma}\left({re}^{2}, \frac{-1}{2}, 1\right) \]

      7. unpow2 N/A

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(re \cdot re, \frac{-1}{2}, 1\right) \]

      8. lower-*.f64 63.5%

         \[\leadsto \cosh im \cdot \mathsf{fma}\left(re \cdot re, -0.5, 1\right) \]

   8. Applied rewrites 63.5%

      \[\leadsto \cosh im \cdot \mathsf{fma}\left(re \cdot re, \color{blue}{-0.5}, 1\right) \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.3%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-0.25, \sqrt{\left(\left(re \cdot re\right) \cdot re\right) \cdot re}, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.005)
  (* 2.0 (fma -0.25 (sqrt (* (* (* re re) re) re)) 0.5))
  (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.005) {
		tmp = 2.0 * fma(-0.25, sqrt((((re * re) * re) * re)), 0.5);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.005)
		tmp = Float64(2.0 * fma(-0.25, sqrt(Float64(Float64(Float64(re * re) * re) * re)), 0.5));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(2.0 * N[(-0.25 * N[Sqrt[N[(N[(N[(re * re), $MachinePrecision] * re), $MachinePrecision] * re), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.1%

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 33.1%

         \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right) \]

      7. lower-fma.f64 33.1%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \color{blue}{{re}^{2}}, 0.5\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, {re}^{\color{blue}{2}}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \]

      10. lower-*.f64 33.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot \color{blue}{re}, 0.5\right) \]

   9. Applied rewrites 33.1%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{re \cdot re} \cdot \color{blue}{\sqrt{re \cdot re}}, \frac{1}{2}\right) \]

       2. sqrt-unprod N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{1}{2}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{1}{2}\right) \]

       4. lift-sqrt.f64 35.8%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, 0.5\right) \]

       5. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{1}{2}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(re \cdot re\right) \cdot \left(re \cdot re\right)}, \frac{1}{2}\right) \]

       7. associate-*l* N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{re \cdot \left(re \cdot \left(re \cdot re\right)\right)}, \frac{1}{2}\right) \]

       8. *-commutative N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(re \cdot \left(re \cdot re\right)\right) \cdot re}, \frac{1}{2}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(re \cdot \left(re \cdot re\right)\right) \cdot re}, \frac{1}{2}\right) \]

       10. *-commutative N/A

           \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, \sqrt{\left(\left(re \cdot re\right) \cdot re\right) \cdot re}, \frac{1}{2}\right) \]

       11. lower-*.f64 35.8%

           \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \sqrt{\left(\left(re \cdot re\right) \cdot re\right) \cdot re}, 0.5\right) \]

   11. Applied rewrites 35.8%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \sqrt{\left(\left(re \cdot re\right) \cdot re\right) \cdot re}, 0.5\right) \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.3%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\cosh im \cdot 1\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.005)
  (* 2.0 (fma -0.25 (* re re) 0.0))
  (* (cosh im) 1.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.005) {
		tmp = 2.0 * fma(-0.25, (re * re), 0.0);
	} else {
		tmp = cosh(im) * 1.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.005)
		tmp = Float64(2.0 * fma(-0.25, Float64(re * re), 0.0));
	else
		tmp = Float64(cosh(im) * 1.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(2.0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[im], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.1%

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 33.1%

         \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right) \]

      7. lower-fma.f64 33.1%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \color{blue}{{re}^{2}}, 0.5\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, {re}^{\color{blue}{2}}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \]

      10. lower-*.f64 33.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot \color{blue}{re}, 0.5\right) \]

   9. Applied rewrites 33.1%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)} \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 8.3%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0\right) \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{-im} + e^{im}\right) \cdot \left(\frac{1}{2} \cdot \cos re\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(e^{-im} + e^{im}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos re\right)} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(e^{-im} + e^{im}\right) \cdot \frac{1}{2}\right) \cdot \cos re} \]

      5. metadata-eval N/A

         \[\leadsto \left(\left(e^{-im} + e^{im}\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos re \]

      6. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{e^{-im} + e^{im}}{2}} \cdot \cos re \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{-im} + e^{im}}}{2} \cdot \cos re \]

      8. +-commutative N/A

         \[\leadsto \frac{\color{blue}{e^{im} + e^{-im}}}{2} \cdot \cos re \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\color{blue}{e^{im}} + e^{-im}}{2} \cdot \cos re \]

      10. lift-exp.f64 N/A

          \[\leadsto \frac{e^{im} + \color{blue}{e^{-im}}}{2} \cdot \cos re \]

      11. lift-neg.f64 N/A

          \[\leadsto \frac{e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}}{2} \cdot \cos re \]

      12. cosh-def N/A

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

      14. lower-cosh.f64 100.0%

          \[\leadsto \color{blue}{\cosh im} \cdot \cos re \]

   3. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\cosh im \cdot \cos re} \]

   4. Taylor expanded in re around 0

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
   5. Step-by-step derivation

   6. Applied rewrites 65.3%

      \[\leadsto \cosh im \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 35.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 2\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.005)
  (* 2.0 (fma -0.25 (* re re) 0.0))
  (* 0.5 2.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.005) {
		tmp = 2.0 * fma(-0.25, (re * re), 0.0);
	} else {
		tmp = 0.5 * 2.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.005)
		tmp = Float64(2.0 * fma(-0.25, Float64(re * re), 0.0));
	else
		tmp = Float64(0.5 * 2.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(2.0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.1%

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 33.1%

         \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right) \]

      7. lower-fma.f64 33.1%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \color{blue}{{re}^{2}}, 0.5\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, {re}^{\color{blue}{2}}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \]

      10. lower-*.f64 33.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot \color{blue}{re}, 0.5\right) \]

   9. Applied rewrites 33.1%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)} \]

   10. Taylor expanded in undef-var around zero

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 8.3%

       \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0\right) \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
   6. Step-by-step derivation

   7. Applied rewrites 29.0%

      \[\leadsto \color{blue}{0.5} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 35.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \leq -0.005:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot 2\\ \end{array} \]

FPCore

(FPCore (re im)
  :precision binary64
  (if (<= (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))) -0.005)
  (* 2.0 (fma -0.25 (* re re) 0.5))
  (* 0.5 2.0)))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * cos(re)) * (exp(-im) + exp(im))) <= -0.005) {
		tmp = 2.0 * fma(-0.25, (re * re), 0.5);
	} else {
		tmp = 0.5 * 2.0;
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(-im)) + exp(im))) <= -0.005)
		tmp = Float64(2.0 * fma(-0.25, Float64(re * re), 0.5));
	else
		tmp = Float64(0.5 * 2.0);
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[(-im)], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.005], N[(2.0 * N[(-0.25 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(0.5 * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im))) < -0.0050000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \cdot 2 \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \cdot 2 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} + \frac{-1}{4} \cdot \color{blue}{{re}^{2}}\right) \cdot 2 \]

      3. lower-pow.f64 33.1%

         \[\leadsto \left(0.5 + -0.25 \cdot {re}^{\color{blue}{2}}\right) \cdot 2 \]

   7. Applied rewrites 33.1%

      \[\leadsto \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \cdot 2 \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right) \cdot 2} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \left(\frac{1}{2} + \frac{-1}{4} \cdot {re}^{2}\right)} \]

      3. lower-*.f64 33.1%

         \[\leadsto \color{blue}{2 \cdot \left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot {re}^{2}}\right) \]

      5. +-commutative N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \color{blue}{\frac{1}{2}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 2 \cdot \left(\frac{-1}{4} \cdot {re}^{2} + \frac{1}{2}\right) \]

      7. lower-fma.f64 33.1%

         \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, \color{blue}{{re}^{2}}, 0.5\right) \]

      8. lift-pow.f64 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, {re}^{\color{blue}{2}}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto 2 \cdot \mathsf{fma}\left(\frac{-1}{4}, re \cdot \color{blue}{re}, \frac{1}{2}\right) \]

      10. lower-*.f64 33.1%

          \[\leadsto 2 \cdot \mathsf{fma}\left(-0.25, re \cdot \color{blue}{re}, 0.5\right) \]

   9. Applied rewrites 33.1%

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(-0.25, re \cdot re, 0.5\right)} \]

   if -0.0050000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (cos.f64 re)) (+.f64 (exp.f64 (neg.f64 im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

   5. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
   6. Step-by-step derivation

   7. Applied rewrites 29.0%

      \[\leadsto \color{blue}{0.5} \cdot 2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 29.0% accurate, 16.6× speedup

Math

\[0.5 \cdot 2 \]

FPCore

(FPCore (re im)
  :precision binary64
  (* 0.5 2.0))

C

double code(double re, double im) {
	return 0.5 * 2.0;
}

Fortran

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * 2.0d0
end function

Java

public static double code(double re, double im) {
	return 0.5 * 2.0;
}

Python

def code(re, im):
	return 0.5 * 2.0

Julia

function code(re, im)
	return Float64(0.5 * 2.0)
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * 2.0;
end

Wolfram

code[re_, im_] := N[(0.5 * 2.0), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} + e^{im}\right) \]

2. Taylor expanded in im around 0

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]
3. Step-by-step derivation

4. Applied rewrites 51.3%

   \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{2} \]

5. Taylor expanded in re around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \cdot 2 \]
6. Step-by-step derivation

7. Applied rewrites 29.0%

   \[\leadsto \color{blue}{0.5} \cdot 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (re im)
  :name "math.cos on complex, real part"
  :precision binary64
  (* (* 0.5 (cos re)) (+ (exp (- im)) (exp im))))