math.sin on complex, real part

Percentage Accurate: 100.0% → 100.0%

Time: 2.4s

Alternatives: 8

Speedup: 1.0×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * sin(re)) * (exp((0.0d0 - im)) + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.sin(re)) * (Math.exp((0.0 - im)) + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.sin(re)) * (math.exp((0.0 - im)) + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]
2. Add Preprocessing

Alternative 2: 77.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(e^{im} + 1\right) \cdot \left(\sin re \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (+ (exp im) 1.0) (* (sin re) 0.5)))

C

double code(double re, double im) {
	return (exp(im) + 1.0) * (sin(re) * 0.5);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (exp(im) + 1.0d0) * (sin(re) * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return (Math.exp(im) + 1.0) * (Math.sin(re) * 0.5);
}

Python

def code(re, im):
	return (math.exp(im) + 1.0) * (math.sin(re) * 0.5)

Julia

function code(re, im)
	return Float64(Float64(exp(im) + 1.0) * Float64(sin(re) * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = (exp(im) + 1.0) * (sin(re) * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(N[Exp[im], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

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

4. Applied rewrites 74.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 74.4

      \[\leadsto \color{blue}{\left(1 + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 74.4

      \[\leadsto \color{blue}{\left(e^{im} + 1\right)} \cdot \left(0.5 \cdot \sin re\right) \]

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 74.4

      \[\leadsto \left(e^{im} + 1\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

6. Applied rewrites 74.4%

   \[\leadsto \color{blue}{\left(e^{im} + 1\right) \cdot \left(\sin re \cdot 0.5\right)} \]
7. Add Preprocessing

Alternative 3: 74.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right)\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+23}:\\ \;\;\;\;\left(\sin re \cdot 0.5\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im)))))
   (if (<= t_0 (- INFINITY))
     (* 2.0 (* (cosh im) (* (fma -0.08333333333333333 (* re re) 0.5) re)))
     (if (<= t_0 1e+23)
       (* (* (sin re) 0.5) 2.0)
       (* (* 0.5 re) (+ 1.0 (exp im)))))))

C

double code(double re, double im) {
	double t_0 = (0.5 * sin(re)) * (exp((0.0 - im)) + exp(im));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 2.0 * (cosh(im) * (fma(-0.08333333333333333, (re * re), 0.5) * re));
	} else if (t_0 <= 1e+23) {
		tmp = (sin(re) * 0.5) * 2.0;
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	t_0 = Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(cosh(im) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re)));
	elseif (t_0 <= 1e+23)
		tmp = Float64(Float64(sin(re) * 0.5) * 2.0);
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+23], N[(N[(N[Sin[re], $MachinePrecision] * 0.5), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. lower-pow.f64 63.4

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      3. lower-*.f64 63.4

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      9. sub0-neg N/A

         \[\leadsto \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      10. cosh-undef N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      12. lower-cosh.f64 63.4

          \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      15. lower-*.f64 63.4

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

      17. +-commutative N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      19. *-commutative N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \]

      20. lower-fma.f64 63.4

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left({re}^{2}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

      21. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      22. unpow2 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      23. lower-*.f64 63.4

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   6. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)\right)} \]

      5. lower-*.f64 63.4

         \[\leadsto 2 \cdot \color{blue}{\left(\cosh im \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\cosh im \cdot \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\cosh im \cdot \left(\left(\frac{-1}{12} \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right)\right) \]

      8. lower-fma.f64 63.4

         \[\leadsto 2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \]

   8. Applied rewrites 63.4%

      \[\leadsto \color{blue}{2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right)} \]

   if -inf.0 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 9.9999999999999992e22

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

      \[\leadsto \left(\frac{1}{2} \cdot \sin re\right) \cdot \color{blue}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \sin re\right)} \cdot 2 \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin re \cdot \frac{1}{2}\right)} \cdot 2 \]

      3. lower-*.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if 9.9999999999999992e22 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 74.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} + e^{im}\right) \]

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 44.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \leq 0.1:\\ \;\;\;\;2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))) 0.1)
   (* 2.0 (* (cosh im) (* (fma -0.08333333333333333 (* re re) 0.5) re)))
   (* (* 0.5 re) (+ 1.0 (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (((0.5 * sin(re)) * (exp((0.0 - im)) + exp(im))) <= 0.1) {
		tmp = 2.0 * (cosh(im) * (fma(-0.08333333333333333, (re * re), 0.5) * re));
	} else {
		tmp = (0.5 * re) * (1.0 + exp(im));
	}
	return tmp;
}

Julia

function code(re, im)
	tmp = 0.0
	if (Float64(Float64(0.5 * sin(re)) * Float64(exp(Float64(0.0 - im)) + exp(im))) <= 0.1)
		tmp = Float64(2.0 * Float64(cosh(im) * Float64(fma(-0.08333333333333333, Float64(re * re), 0.5) * re)));
	else
		tmp = Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)));
	end
	return tmp
end

Wolfram

code[re_, im_] := If[LessEqual[N[(N[(0.5 * N[Sin[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.1], N[(2.0 * N[(N[Cosh[im], $MachinePrecision] * N[(N[(-0.08333333333333333 * N[(re * re), $MachinePrecision] + 0.5), $MachinePrecision] * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im))) < 0.10000000000000001

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in re around 0

      \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{12} \cdot {re}^{2}}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot \color{blue}{{re}^{2}}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

      4. lower-pow.f64 63.4

         \[\leadsto \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{\color{blue}{2}}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   4. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \cdot \left(e^{0 - im} + e^{im}\right) \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \cdot \left(e^{0 - im} + e^{im}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right)} \]

      3. lower-*.f64 63.4

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(e^{0 - im} + e^{im}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(e^{im} + e^{0 - im}\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      6. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{e^{im}} + e^{0 - im}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      7. lift-exp.f64 N/A

         \[\leadsto \left(e^{im} + \color{blue}{e^{0 - im}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      8. lift--.f64 N/A

         \[\leadsto \left(e^{im} + e^{\color{blue}{0 - im}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      9. sub0-neg N/A

         \[\leadsto \left(e^{im} + e^{\color{blue}{\mathsf{neg}\left(im\right)}}\right) \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      10. cosh-undef N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(re \cdot \left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)\right) \]

      12. lower-cosh.f64 63.4

          \[\leadsto \left(2 \cdot \color{blue}{\cosh im}\right) \cdot \left(re \cdot \left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right)\right) \]

      13. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(re \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right)}\right) \]

      14. *-commutative N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      15. lower-*.f64 63.4

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(0.5 + -0.08333333333333333 \cdot {re}^{2}\right) \cdot \color{blue}{re}\right) \]

      16. lift-+.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{1}{2} + \frac{-1}{12} \cdot {re}^{2}\right) \cdot re\right) \]

      17. +-commutative N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left(\frac{-1}{12} \cdot {re}^{2} + \frac{1}{2}\right) \cdot re\right) \]

      19. *-commutative N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\left({re}^{2} \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right) \]

      20. lower-fma.f64 63.4

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left({re}^{2}, -0.08333333333333333, 0.5\right) \cdot re\right) \]

      21. lift-pow.f64 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left({re}^{2}, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      22. unpow2 N/A

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      23. lower-*.f64 63.4

          \[\leadsto \left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right) \]

   6. Applied rewrites 63.4%

      \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right) \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(2 \cdot \cosh im\right)} \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(re \cdot re, \frac{-1}{12}, \frac{1}{2}\right) \cdot re\right)\right)} \]

      5. lower-*.f64 63.4

         \[\leadsto 2 \cdot \color{blue}{\left(\cosh im \cdot \left(\mathsf{fma}\left(re \cdot re, -0.08333333333333333, 0.5\right) \cdot re\right)\right)} \]

      6. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \left(\cosh im \cdot \left(\left(\left(re \cdot re\right) \cdot \frac{-1}{12} + \frac{1}{2}\right) \cdot re\right)\right) \]

      7. *-commutative N/A

         \[\leadsto 2 \cdot \left(\cosh im \cdot \left(\left(\frac{-1}{12} \cdot \left(re \cdot re\right) + \frac{1}{2}\right) \cdot re\right)\right) \]

      8. lower-fma.f64 63.4

         \[\leadsto 2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right) \]

   8. Applied rewrites 63.4%

      \[\leadsto \color{blue}{2 \cdot \left(\cosh im \cdot \left(\mathsf{fma}\left(-0.08333333333333333, re \cdot re, 0.5\right) \cdot re\right)\right)} \]

   if 0.10000000000000001 < (*.f64 (*.f64 #s(literal 1/2 binary64) (sin.f64 re)) (+.f64 (exp.f64 (-.f64 #s(literal 0 binary64) im)) (exp.f64 im)))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

   2. Taylor expanded in im around 0

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

   4. Applied rewrites 74.4%

      \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} + e^{im}\right) \]

   5. Taylor expanded in re around 0

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

   7. Applied rewrites 44.5%

      \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 54.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \]

   6. lower-neg.f64 63.2

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

4. Applied rewrites 63.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 63.2

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-neg.f64 N/A

      \[\leadsto \left(\left(e^{\mathsf{neg}\left(im\right)} + e^{im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   9. sub0-neg N/A

      \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   10. lift--.f64 N/A

       \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   11. lift-+.f64 N/A

       \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   12. lower-*.f64 63.2

       \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot re\right) \cdot 0.5 \]

   13. lift-+.f64 N/A

       \[\leadsto \left(\left(e^{0 - im} + e^{im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   14. +-commutative N/A

       \[\leadsto \left(\left(e^{im} + e^{0 - im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   15. lift-exp.f64 N/A

       \[\leadsto \left(\left(e^{im} + e^{0 - im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   16. lift-exp.f64 N/A

       \[\leadsto \left(\left(e^{im} + e^{0 - im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   17. lift--.f64 N/A

       \[\leadsto \left(\left(e^{im} + e^{0 - im}\right) \cdot re\right) \cdot \frac{1}{2} \]

   18. sub0-neg N/A

       \[\leadsto \left(\left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right) \cdot re\right) \cdot \frac{1}{2} \]

   19. cosh-undef N/A

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

   20. lower-*.f64 N/A

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

   21. lower-cosh.f64 63.2

       \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot 0.5 \]

6. Applied rewrites 63.2%

   \[\leadsto \left(\left(2 \cdot \cosh im\right) \cdot re\right) \cdot \color{blue}{0.5} \]
7. Add Preprocessing

Alternative 6: 44.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot re\right) \cdot \left(1 + e^{im}\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (* 0.5 re) (+ 1.0 (exp im))))

C

double code(double re, double im) {
	return (0.5 * re) * (1.0 + exp(im));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * re) * (1.0d0 + exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * re) * (1.0 + Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * re) * (1.0 + math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * re) * Float64(1.0 + exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * re) * (1.0 + exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * re), $MachinePrecision] * N[(1.0 + N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

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

4. Applied rewrites 74.4%

   \[\leadsto \left(0.5 \cdot \sin re\right) \cdot \left(\color{blue}{1} + e^{im}\right) \]

5. Taylor expanded in re around 0

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

7. Applied rewrites 44.5%

   \[\leadsto \left(0.5 \cdot \color{blue}{re}\right) \cdot \left(1 + e^{im}\right) \]
8. Add Preprocessing

Alternative 7: 32.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(1 + im\right) + 1\right) \cdot \left(re \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* (+ (+ 1.0 im) 1.0) (* re 0.5)))

C

double code(double re, double im) {
	return ((1.0 + im) + 1.0) * (re * 0.5);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(re, im)
use fmin_fmax_functions
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = ((1.0d0 + im) + 1.0d0) * (re * 0.5d0)
end function

Java

public static double code(double re, double im) {
	return ((1.0 + im) + 1.0) * (re * 0.5);
}

Python

def code(re, im):
	return ((1.0 + im) + 1.0) * (re * 0.5)

Julia

function code(re, im)
	return Float64(Float64(Float64(1.0 + im) + 1.0) * Float64(re * 0.5))
end

MATLAB

function tmp = code(re, im)
	tmp = ((1.0 + im) + 1.0) * (re * 0.5);
end

Wolfram

code[re_, im_] := N[(N[(N[(1.0 + im), $MachinePrecision] + 1.0), $MachinePrecision] * N[(re * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in im around 0

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

4. Applied rewrites 74.4%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 74.4

      \[\leadsto \color{blue}{\left(1 + e^{im}\right) \cdot \left(0.5 \cdot \sin re\right)} \]

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 74.4

      \[\leadsto \color{blue}{\left(e^{im} + 1\right)} \cdot \left(0.5 \cdot \sin re\right) \]

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 74.4

      \[\leadsto \left(e^{im} + 1\right) \cdot \color{blue}{\left(\sin re \cdot 0.5\right)} \]

6. Applied rewrites 74.4%

   \[\leadsto \color{blue}{\left(e^{im} + 1\right) \cdot \left(\sin re \cdot 0.5\right)} \]

7. Taylor expanded in re around 0

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

9. Applied rewrites 44.5%

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

10. Taylor expanded in im around 0

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

    1. lower-+.f64 32.1

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

12. Applied rewrites 32.1%

    \[\leadsto \left(\color{blue}{\left(1 + im\right)} + 1\right) \cdot \left(re \cdot 0.5\right) \]
13. Add Preprocessing

Alternative 8: 26.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \left(re + re\right) \cdot 0.5 \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(0.5 \cdot \sin re\right) \cdot \left(e^{0 - im} + e^{im}\right) \]

2. Taylor expanded in re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(re \cdot \left(e^{im} + e^{\mathsf{neg}\left(im\right)}\right)\right) \]

   6. lower-neg.f64 63.2

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

4. Applied rewrites 63.2%

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

5. Taylor expanded in im around 0

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

   1. lower-*.f64 26.5

      \[\leadsto 0.5 \cdot \left(2 \cdot re\right) \]

7. Applied rewrites 26.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 26.5

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

   4. lift-*.f64 N/A

      \[\leadsto \left(2 \cdot re\right) \cdot \frac{1}{2} \]

   5. count-2-rev N/A

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

   6. lower-+.f64 26.5

      \[\leadsto \left(re + re\right) \cdot 0.5 \]

9. Applied rewrites 26.5%

   \[\leadsto \color{blue}{\left(re + re\right) \cdot 0.5} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (re im)
  :name "math.sin on complex, real part"
  :precision binary64
  (* (* 0.5 (sin re)) (+ (exp (- 0.0 im)) (exp im))))